Squared_Difference.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Measuring the divergence in empirical and theoretical probabilities of permutations using squared difference"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\Delta(a, b) = \\sum_{i}(Prob(i, a) - Prob(j, a))^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "importing Jupyter notebook from Mallows_Notebook.ipynb\n",
      "importing Jupyter notebook from PL_Notebook.ipynb\n",
      "0.125\n",
      "[0.30201484 0.2058898  0.27630837 0.21578699]\n",
      "defaultdict(<class 'int'>, {4: 308, 3: 126, 1: 19, 2: 22})\n",
      "initial weights [0.3279398  0.10113504 0.37556066 0.1953645 ]\n"
     ]
    },
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       "HBox(children=(IntProgress(value=0, description='plackettCost'), HTML(value='')))"
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     "metadata": {},
     "output_type": "display_data"
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "deque([2518.2061604003343, 2567.299698085064, 2554.6195404226, 2495.811155212152, 2559.2797001553017, 2557.7745434433127, 2571.4460034928684, 2588.236059718956, 2595.9627316862643, 2594.8959694439354, 2649.940994678969, 2654.1612692255385, 2678.391640577149, 2686.2030898864427, 2681.832976776231], maxlen=15)\n",
      "PL loaded\n"
     ]
    }
   ],
   "source": [
    "import import_ipynb\n",
    "from Mallows_Notebook import *\n",
    "from PL_Notebook import *\n",
    "import metropolis\n",
    "import math\n",
    "from tqdm import tqdm_notebook\n",
    "import itertools"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import os\n",
    "netflix = os.listdir('netflix_soc')\n",
    "files = ['netflix_soc/' + x for x in netflix]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "irish = [#'ED-00001-00000001.soi',\\\n",
    "         'ED-00001-00000002.soi',\\\n",
    "         #'ED-00001-00000003.soi',]\n",
    "        ]\n",
    "\n",
    "files = ['irish_soi/' + x for x in irish]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['preflib_soi/ED-00002-00000001.soi',\n",
       " 'preflib_soi/ED-00002-00000002.soi',\n",
       " 'preflib_soi/ED-00002-00000003.soi',\n",
       " 'preflib_soi/ED-00002-00000004.soi',\n",
       " 'preflib_soi/ED-00002-00000005.soi',\n",
       " 'preflib_soi/ED-00002-00000006.soi',\n",
       " 'preflib_soi/ED-00002-00000007.soi',\n",
       " 'preflib_soi/ED-02-Logo.soi']"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "files = ['ED-00002-00000001.soi',\\\n",
    "         'ED-00002-00000002.soi',\\\n",
    "         'ED-00002-00000003.soi',\\\n",
    "         'ED-00002-00000004.soi',\\\n",
    "         'ED-00002-00000005.soi',\\\n",
    "         'ED-00002-00000006.soi',\\\n",
    "         'ED-00002-00000007.soi',\\\n",
    "         'ED-02-Logo.soi']\n",
    "\n",
    "files = ['preflib_soi/' + x for x in files]\n",
    "files"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "files = ['preflib_soi/ED-02-Logo.soi']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['all_soi/ED-00001-00000002.soi',\n",
       " 'all_soi/ED-00002-00000001.soi',\n",
       " 'all_soi/ED-00002-00000002.soi',\n",
       " 'all_soi/ED-00002-00000003.soi',\n",
       " 'all_soi/ED-00002-00000004.soi',\n",
       " 'all_soi/ED-00002-00000005.soi',\n",
       " 'all_soi/ED-00002-00000006.soi',\n",
       " 'all_soi/ED-00002-00000007.soi',\n",
       " 'all_soi/ED-00002-00000008.soi',\n",
       " 'all_soi/ED-00008-00000001.soi',\n",
       " 'all_soi/ED-00008-00000010.soi',\n",
       " 'all_soi/ED-00008-00000012.soi',\n",
       " 'all_soi/ED-00008-00000014.soi',\n",
       " 'all_soi/ED-00008-00000015.soi',\n",
       " 'all_soi/ED-00008-00000017.soi',\n",
       " 'all_soi/ED-00008-00000018.soi',\n",
       " 'all_soi/ED-00008-00000020.soi',\n",
       " 'all_soi/ED-00028-00000001.soi',\n",
       " 'all_soi/ED-00028-00000002.soi',\n",
       " 'all_soi/ED-00028-00000003.soi',\n",
       " 'all_soi/ED-00028-00000004.soi',\n",
       " 'all_soi/ED-00028-00000005.soi',\n",
       " 'all_soi/ED-00028-00000006.soi',\n",
       " 'all_soi/ED-00028-00000007.soi',\n",
       " 'all_soi/ED-00028-00000008.soi',\n",
       " 'all_soi/ED-00028-00000009.soi',\n",
       " 'all_soi/ED-00028-00000010.soi',\n",
       " 'all_soi/ED-00028-00000011.soi',\n",
       " 'all_soi/ED-00028-00000012.soi',\n",
       " 'all_soi/ED-00030-00000001.soi']"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The BIG run\n",
    "import os\n",
    "bigrun = os.listdir('all_soi')\n",
    "soi_files = ['all_soi/' + x for x in bigrun]\n",
    "\n",
    "u10 = []\n",
    "u11 = []\n",
    "for file in soi_files:\n",
    "    cands, lengths, num_votes, votes = readPreflib.soiInputwithNumVotes(file)\n",
    "    complete = True\n",
    "    if len(cands) != len(lengths.keys()):\n",
    "        complete = False\n",
    "    if len(cands) < 10 and complete:\n",
    "        u10.append(file)\n",
    "#     if len(cands) < 11:\n",
    "#         u11 += 1\n",
    "\n",
    "# u10 = u10[9:]\n",
    "# u10 = u10[:8]\n",
    "u10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Projected Time = 14300.0 seconds, which is ~ 150.0 minutes\n"
     ]
    },
    {
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       "HBox(children=(IntProgress(value=0, description='All Files', max=8), HTML(value='')))"
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     "text": [
      "stopped after  94\n",
      "deque([1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918, 1461.392631137918], maxlen=15)\n",
      "defaultdict(<class 'int'>, {1: 0.25130434782608696, 2: 0.2108695652173913, 3: 0.21753623188405796, 4: 0.16144927536231884, 5: 0.0463768115942029, 9: 0.07942028985507246, 7: 0.005797101449275362, 6: 0.02072463768115942, 8: 0.006521739130434782, 0: 0})\n",
      "initial weights [0.04478802 0.03921405 0.00544811 0.04808484 0.06353734 0.06066137\n",
      " 0.20786799 0.22852714 0.30187115]\n"
     ]
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     "text": [
      "deque([242.71849779622502, 242.70839704012403, 242.61691344738446, 242.5424418048822, 242.4947478684619, 242.58864670730452, 242.8795215776017, 243.51307659766178, 243.49804679205425, 243.3601216813386, 243.8301416603849, 243.58671116840122, 243.4899935071771, 243.51493411696705, 243.60284756831618], maxlen=15)\n"
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     "output_type": "stream",
     "text": [
      "stopped after  1118\n",
      "deque([2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303, 2083.3822270037303], maxlen=15)\n",
      "defaultdict(<class 'int'>, {1: 0.23600552868002764, 3: 0.3214697074406819, 2: 0.1422483298779083, 4: 0.1529601474314674, 5: 0.035245335176226675, 9: 0.08880442294402212, 8: 0.005413499193734163, 6: 0.012554710896106889, 7: 0.005298318359824925, 0: 0})\n",
      "initial weights [0.06952365 0.06106911 0.03866189 0.12841019 0.01429481 0.0219702\n",
      " 0.21934648 0.18990682 0.25681685]\n"
     ]
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     "output_type": "stream",
     "text": [
      "deque([239.36074515762138, 238.3464022947685, 238.09198550023862, 237.91062073050412, 237.44655362505796, 237.11090639274863, 237.37046345587544, 237.36365969794338, 237.303958254794, 237.38679399479093, 237.2626473342217, 237.4449520470864, 237.43324712421625, 237.44951218570245, 237.43867697998607], maxlen=15)\n"
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stopped after  1037\n",
      "deque([1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898, 1321.3747865348898], maxlen=15)\n",
      "defaultdict(<class 'int'>, {1: 0.20762802656953075, 2: 0.23184058281551317, 3: 0.233983286908078, 4: 0.15856010284979644, 8: 0.10531390614956075, 5: 0.0432826226698093, 6: 0.013284765373901865, 7: 0.006106706663809728, 0: 0})\n",
      "initial weights [0.02554838 0.02459545 0.04155341 0.0827931  0.06910992 0.2213617\n",
      " 0.21737423 0.31766382]\n"
     ]
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     "text": [
      "deque([319.45915578723384, 319.4088871093089, 319.32907131317563, 320.2073969375261, 321.09650783865993, 321.00923122663505, 321.61800328050487, 321.4439603947462, 320.4292340857812, 320.25254762891615, 319.1617883384472, 319.4036696802128, 320.40268557901055, 320.3346409055351, 320.3372538795689], maxlen=15)\n"
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     "text": [
      "stopped after  376\n",
      "deque([2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745, 2385.961185284745], maxlen=15)\n",
      "defaultdict(<class 'int'>, {1: 0.2435107564892435, 2: 0.23886476113523886, 3: 0.22361377638622362, 4: 0.14897485102514899, 8: 0.10130289869710131, 5: 0.02918897081102919, 6: 0.007675992324007676, 7: 0.006867993132006868, 0: 0})\n",
      "initial weights [0.02100581 0.01899788 0.02492011 0.05110541 0.05672758 0.11864418\n",
      " 0.25857674 0.45002229]\n"
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     "text": [
      "deque([408.77395737696094, 408.6679208957001, 409.3534341174679, 409.02244352195106, 408.69346906802224, 408.37270530518464, 408.66125094242955, 408.582213962935, 408.54337591976815, 408.5329505142582, 408.5677571979717, 407.1475016966415, 406.77355243374654, 406.0290847822092, 405.1229959671183], maxlen=15)\n"
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     "output_type": "stream",
     "text": [
      "stopped after  792\n",
      "deque([1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746, 1976.644853743746], maxlen=15)\n",
      "defaultdict(<class 'int'>, {2: 0.25583545181419, 1: 0.24000462214005083, 3: 0.223249364455743, 4: 0.1509128726600416, 5: 0.03547492489022418, 9: 0.06655881673214699, 6: 0.014097527155072799, 8: 0.008319852091518374, 7: 0.005546568061012249, 0: 0})\n",
      "initial weights [0.07162657 0.03935966 0.04996177 0.01322019 0.08762523 0.13583694\n",
      " 0.04386445 0.26029428 0.2982109 ]\n"
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     "text": [
      "deque([378.35031651819816, 377.99815217452624, 377.5243950930522, 377.50732121067784, 377.4573241641672, 377.4351163118371, 376.59393489334343, 376.51275999319, 377.53553575677256, 378.5703478233859, 377.4220859925677, 377.1886025285528, 377.5035603338977, 376.7316089335835, 376.6754560537024], maxlen=15)\n"
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     "text": [
      "stopped after  204\n",
      "deque([1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594, 1615.1296612224594], maxlen=15)\n",
      "defaultdict(<class 'int'>, {1: 0.19695543000627747, 2: 0.2405053358443189, 3: 0.24144695543000627, 4: 0.1746704331450094, 5: 0.04770872567482737, 9: 0.0709353421217828, 6: 0.016242937853107344, 8: 0.0045511613308223476, 7: 0.006983678593848085, 0: 0})\n",
      "initial weights [0.0406574  0.03708796 0.00276465 0.00943863 0.08457463 0.15335306\n",
      " 0.12947953 0.09340015 0.44924399]\n"
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     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "deque([451.2274007513951, 451.84775222226193, 451.54102885287836, 451.5638199071742, 452.4964748625616, 451.5331588234789, 451.09710133157824, 450.31493592414006, 449.1216361873097, 448.75298517357623, 449.45625549145024, 450.18884092032283, 450.1474159514422, 451.01776387259554, 451.0520182220203], maxlen=15)\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "8a1e81c4e51c4b0d890e102f2cbd3082",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "HBox(children=(IntProgress(value=0, description='mallowsCost', max=5000), HTML(value='')))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stopped after  55\n",
      "deque([1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028, 1896.460509353028], maxlen=15)\n",
      "defaultdict(<class 'int'>, {1: 0.30615657991010764, 2: 0.18386118950559213, 3: 0.23047977422389465, 4: 0.14205079962370648, 5: 0.038883662590153655, 9: 0.07410891606564231, 8: 0.006062506532873419, 6: 0.013379324762203408, 7: 0.005017246785826278, 0: 0})\n",
      "initial weights [0.03025419 0.01706623 0.03226513 0.07341649 0.04253632 0.02128432\n",
      " 0.21663392 0.2651145  0.3014289 ]\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "24df9dd899df4ddea03b14164a95d70b",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "HBox(children=(IntProgress(value=0, description='plackettCost', max=5000), HTML(value='')))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "deque([358.0343911221313, 357.9993483690067, 358.08013069603646, 357.95873776342705, 357.95013619170254, 357.09149064679974, 356.9077206535719, 357.75578990671943, 358.421615035414, 358.56668870192397, 357.6972957741706, 356.8183286877191, 356.8704426987623, 356.3182856457828, 356.2698364798358], maxlen=15)\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "bed7df69cd54468196ecbf222f8f9439",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "HBox(children=(IntProgress(value=0, description='mallowsCost', max=5000), HTML(value='')))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stopped after  93\n",
      "deque([2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427, 2291.883251201427], maxlen=15)\n",
      "defaultdict(<class 'int'>, {1: 0.25028610666056306, 3: 0.28633554589150834, 2: 0.17360952162966353, 4: 0.14602883955138476, 9: 0.07862211032272831, 5: 0.04005493247882811, 6: 0.015106431677729458, 8: 0.004463263904783703, 7: 0.005493247882810712, 0: 0})\n",
      "initial weights [0.04629577 0.04588879 0.03373506 0.03054067 0.1020307  0.07202173\n",
      " 0.23093086 0.2430361  0.19552033]\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "87436bfe24e144758e1594ad5a553f36",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "HBox(children=(IntProgress(value=0, description='plackettCost', max=5000), HTML(value='')))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "deque([282.39467778111623, 282.4551679592222, 282.6297142991159, 282.2092752346396, 282.21933167265496, 282.2106132999239, 282.46581772665144, 281.12440908501014, 281.15985744620195, 280.8709526884568, 280.4413104104735, 281.0810558828179, 281.25635931682837, 280.7376030547887, 280.9055160091895], maxlen=15)\n"
     ]
    }
   ],
   "source": [
    "list_of_votes = []\n",
    "mallows_params = []\n",
    "pl_params = []\n",
    "\n",
    "nruns = 5_000\n",
    "\n",
    "# print(files[0])\n",
    "\n",
    "print('Projected Time =',143/50.0 * nruns,'seconds, which is ~', 30/1000.0 * nruns, 'minutes')\n",
    "for file in tqdm_notebook(u10, desc = 'All Files'):\n",
    "    # print(file)\n",
    "    cands, lengths, num_votes, votes = readPreflib.soiInputwithNumVotes(file)\n",
    "    num_alternatives = len(cands)\n",
    "    #print(num_alternatives)\n",
    "    #print(num_votes)\n",
    "    #print(list(lengths.keys()))\n",
    "    list_of_votes.append((num_votes, lengths, votes))\n",
    "    # print(list_of_votes[0])\n",
    "    p_mal = runMallows(votes, num_alternatives, nruns, lengths)\n",
    "    mallows_params.append(p_mal)\n",
    "    p_pl = runPL(votes, nruns, lengths)\n",
    "    pl_params.append(p_pl)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "([[array([3, 6, 8, 5, 1, 7, 4, 2, 9]), 0.002517998703411961],\n",
       "  [array([9, 4, 5, 8, 3, 7, 2, 1, 6]), 0.0006146129814220413],\n",
       "  [array([1, 4, 2, 8, 7, 5, 6, 3]), 0.00017515828370942566],\n",
       "  [array([7, 8, 2, 6, 3, 1, 5, 4]), 0.0010341632271258783],\n",
       "  [array([4, 3, 9, 6, 1, 5, 8, 2, 7]), 0.0014362817086728263],\n",
       "  [array([2, 9, 7, 8, 6, 1, 4, 5, 3]), 5.6681369158761985e-05],\n",
       "  [array([6, 3, 7, 4, 1, 8, 9, 2, 5]), 0.008013644303874662],\n",
       "  [array([9, 4, 6, 8, 5, 1, 7, 2, 3]), 0.0009473147827070705]],\n",
       " [array([0.06658622, 0.12265479, 0.09405724, 0.07447783, 0.09679563,\n",
       "         0.10078141, 0.16521486, 0.172877  , 0.10655501]),\n",
       "  array([0.10292062, 0.10321315, 0.15874575, 0.1089393 , 0.10436468,\n",
       "         0.1006994 , 0.11121339, 0.08793239, 0.12197131]),\n",
       "  array([0.14559781, 0.14542865, 0.12626349, 0.14756259, 0.08033008,\n",
       "         0.0998198 , 0.13258328, 0.12241429]),\n",
       "  array([0.13157881, 0.10885867, 0.13273439, 0.13601211, 0.08735751,\n",
       "         0.05516434, 0.15572035, 0.19257382]),\n",
       "  array([0.07326227, 0.12730612, 0.12595902, 0.12781756, 0.14348806,\n",
       "         0.12193746, 0.10433057, 0.08772786, 0.08817107]),\n",
       "  array([0.05701956, 0.03453678, 0.02437645, 0.01480665, 0.08920137,\n",
       "         0.15391254, 0.127072  , 0.09729135, 0.40178331]),\n",
       "  array([0.09756553, 0.08500583, 0.11548008, 0.12522904, 0.08410051,\n",
       "         0.1278272 , 0.12820449, 0.13050132, 0.106086  ]),\n",
       "  array([0.05008485, 0.06984053, 0.03496663, 0.05917418, 0.08673044,\n",
       "         0.07497622, 0.20775585, 0.22325743, 0.19321388])])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mallows_params, pl_params\n",
    "pickle.dump([mallows_params, pl_params], open('pickle/u10_28_5k.p','wb'))\n",
    "# prob pickle things here\n",
    "mallows_params, pl_params\n",
    "# print(pl_params[2])\n",
    "# mallows_params.pop(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "109600"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def all_possible_votes(n):\n",
    "    all_votes = []\n",
    "    for i in range(1,n+1):\n",
    "        # print(i)\n",
    "        sub = list(itertools.permutations(range(1,n+1),i))\n",
    "        #sub = itertools.permutations(range(1,i+1))\n",
    "        for p in sub:\n",
    "            all_votes.append(p)\n",
    "    return all_votes\n",
    "# import sys\n",
    "len(all_possible_votes(8))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(6900,\n",
       "  defaultdict(int,\n",
       "              {1: 0.25130434782608696,\n",
       "               2: 0.2108695652173913,\n",
       "               3: 0.21753623188405796,\n",
       "               4: 0.16144927536231884,\n",
       "               5: 0.0463768115942029,\n",
       "               9: 0.07942028985507246,\n",
       "               7: 0.005797101449275362,\n",
       "               6: 0.02072463768115942,\n",
       "               8: 0.006521739130434782,\n",
       "               0: 0}),\n",
       "  [(476, [7]),\n",
       "   (389, [8]),\n",
       "   (304, [3]),\n",
       "   (299, [3, 8]),\n",
       "   (263, [8, 3]),\n",
       "   (174, [2]),\n",
       "   (149, [1]),\n",
       "   (119, [6]),\n",
       "   (82, [7, 1]),\n",
       "   (73, [1, 6]),\n",
       "   (67, [6, 1]),\n",
       "   (53, [5]),\n",
       "   (49, [9]),\n",
       "   (42, [7, 6]),\n",
       "   (39, [7, 1, 6]),\n",
       "   (39, [7, 5]),\n",
       "   (33, [3, 8, 1]),\n",
       "   (31, [7, 5, 1]),\n",
       "   (31, [8, 7]),\n",
       "   (28, [1, 6, 7]),\n",
       "   (27, [8, 3, 6]),\n",
       "   (26, [7, 5, 9]),\n",
       "   (26, [8, 3, 1]),\n",
       "   (25, [1, 7]),\n",
       "   (25, [6, 7]),\n",
       "   (24, [3, 7]),\n",
       "   (23, [2, 1]),\n",
       "   (23, [3, 8, 6]),\n",
       "   (22, [7, 3]),\n",
       "   (21, [6, 7, 1]),\n",
       "   (21, [2, 4]),\n",
       "   (21, [4]),\n",
       "   (21, [6, 3, 8]),\n",
       "   (20, [3, 6, 8]),\n",
       "   (20, [3, 6]),\n",
       "   (20, [3, 8, 5]),\n",
       "   (20, [7, 3, 8]),\n",
       "   (20, [7, 8]),\n",
       "   (19, [2, 6]),\n",
       "   (19, [7, 6, 1]),\n",
       "   (19, [3, 1]),\n",
       "   (19, [3, 8, 6, 1]),\n",
       "   (18, [5, 7]),\n",
       "   (18, [1, 6, 3]),\n",
       "   (17, [6, 1, 4]),\n",
       "   (17, [7, 9]),\n",
       "   (16, [7, 1, 5]),\n",
       "   (16, [1, 3, 8]),\n",
       "   (16, [1, 9]),\n",
       "   (15, [8, 3, 6, 1]),\n",
       "   (15, [7, 8, 3]),\n",
       "   (15, [3, 8, 7]),\n",
       "   (15, [6, 3, 1]),\n",
       "   (15, [3, 7, 8]),\n",
       "   (15, [1, 7, 6]),\n",
       "   (14, [6, 1, 3, 8]),\n",
       "   (14, [6, 8, 3]),\n",
       "   (14, [9, 1]),\n",
       "   (14, [8, 3, 5]),\n",
       "   (14, [1, 6, 9]),\n",
       "   (14, [7, 2]),\n",
       "   (13, [7, 1, 9]),\n",
       "   (13, [2, 7]),\n",
       "   (13, [1, 5, 9]),\n",
       "   (13, [6, 2]),\n",
       "   (13, [6, 3]),\n",
       "   (13, [8, 3, 7]),\n",
       "   (13, [7, 1, 5, 9]),\n",
       "   (13, [8, 6]),\n",
       "   (12, [1, 6, 3, 8]),\n",
       "   (12, [6, 8]),\n",
       "   (12, [1, 3]),\n",
       "   (12, [3, 5, 8]),\n",
       "   (11, [1, 9, 5]),\n",
       "   (11, [7, 1, 4]),\n",
       "   (11, [7, 6, 2]),\n",
       "   (11, [2, 6, 1]),\n",
       "   (11, [1, 3, 6]),\n",
       "   (11, [1, 6, 8, 3]),\n",
       "   (11, [1, 2]),\n",
       "   (11, [6, 8, 3, 1]),\n",
       "   (10, [6, 1, 7]),\n",
       "   (10, [8, 6, 3]),\n",
       "   (10, [2, 3]),\n",
       "   (10, [3, 1, 8]),\n",
       "   (10, [8, 6, 1]),\n",
       "   (10, [4, 1]),\n",
       "   (10, [3, 8, 2]),\n",
       "   (10, [9, 5, 1]),\n",
       "   (10, [8, 1]),\n",
       "   (10, [7, 9, 5]),\n",
       "   (10, [7, 9, 5, 1]),\n",
       "   (10, [1, 4]),\n",
       "   (10, [8, 7, 3]),\n",
       "   (10, [8, 2]),\n",
       "   (10, [1, 6, 8]),\n",
       "   (10, [7, 4]),\n",
       "   (9, [6, 1, 2]),\n",
       "   (9, [6, 1, 9]),\n",
       "   (9, [1, 3, 8, 6]),\n",
       "   (9, [3, 5]),\n",
       "   (9, [3, 2]),\n",
       "   (9, [5, 1]),\n",
       "   (9, [7, 9, 1]),\n",
       "   (9, [2, 1, 6]),\n",
       "   (8, [7, 5, 3]),\n",
       "   (8, [6, 7, 8]),\n",
       "   (8, [6, 1, 8]),\n",
       "   (8, [7, 6, 5]),\n",
       "   (8, [7, 5, 9, 1]),\n",
       "   (8, [3, 8, 9]),\n",
       "   (8, [1, 5]),\n",
       "   (8, [7, 1, 6, 9]),\n",
       "   (8, [1, 6, 2]),\n",
       "   (8, [7, 2, 1]),\n",
       "   (8, [7, 5, 1, 9]),\n",
       "   (8, [3, 1, 6, 8]),\n",
       "   (7, [2, 6, 4]),\n",
       "   (7, [4, 6, 1]),\n",
       "   (7, [8, 3, 2]),\n",
       "   (7, [7, 3, 8, 1]),\n",
       "   (7, [6, 1, 5]),\n",
       "   (7, [2, 3, 8]),\n",
       "   (7, [6, 3, 8, 1]),\n",
       "   (7, [8, 3, 2, 6]),\n",
       "   (7, [7, 1, 9, 5]),\n",
       "   (7, [6, 2, 1]),\n",
       "   (7, [5, 9]),\n",
       "   (7, [3, 8, 4]),\n",
       "   (7, [6, 4]),\n",
       "   (7, [1, 6, 4]),\n",
       "   (7, [7, 6, 1, 4]),\n",
       "   (7, [8, 3, 1, 6]),\n",
       "   (7, [1, 7, 9]),\n",
       "   (7, [7, 6, 3]),\n",
       "   (6, [3, 6, 1]),\n",
       "   (6, [1, 9, 6]),\n",
       "   (6, [7, 1, 3]),\n",
       "   (6, [2, 6, 4, 1]),\n",
       "   (6, [6, 7, 2]),\n",
       "   (6, [5, 1, 7]),\n",
       "   (6, [3, 8, 5, 9]),\n",
       "   (6, [3, 8, 9, 1]),\n",
       "   (6, [3, 1, 6]),\n",
       "   (6, [9, 1, 7]),\n",
       "   (6, [4, 1, 6]),\n",
       "   (6, [8, 3, 6, 7]),\n",
       "   (6, [7, 3, 6]),\n",
       "   (6, [4, 2]),\n",
       "   (6, [3, 8, 1, 6]),\n",
       "   (6, [8, 1, 6]),\n",
       "   (6, [8, 3, 9]),\n",
       "   (6, [8, 3, 4]),\n",
       "   (6, [2, 7, 6]),\n",
       "   (6, [2, 1, 4]),\n",
       "   (6, [6, 2, 7]),\n",
       "   (6, [1, 7, 8]),\n",
       "   (5, [1, 9, 7]),\n",
       "   (5, [7, 1, 8]),\n",
       "   (5, [2, 4, 1]),\n",
       "   (5, [2, 6, 3]),\n",
       "   (5, [6, 1, 3]),\n",
       "   (5, [6, 4, 1]),\n",
       "   (5, [6, 8, 1]),\n",
       "   (5, [1, 4, 6]),\n",
       "   (5, [7, 5, 9, 6]),\n",
       "   (5, [7, 3, 1]),\n",
       "   (5, [1, 7, 6, 9]),\n",
       "   (5, [5, 8]),\n",
       "   (5, [7, 8, 5]),\n",
       "   (5, [2, 8, 3]),\n",
       "   (5, [5, 7, 9]),\n",
       "   (5, [6, 9, 1]),\n",
       "   (5, [7, 1, 6, 8]),\n",
       "   (5, [1, 7, 4]),\n",
       "   (5, [6, 2, 1, 4]),\n",
       "   (5, [6, 3, 8, 2]),\n",
       "   (5, [5, 7, 9, 1]),\n",
       "   (5, [3, 6, 8, 1]),\n",
       "   (5, [9, 7]),\n",
       "   (5, [7, 5, 6]),\n",
       "   (5, [1, 3, 6, 8]),\n",
       "   (5, [1, 5, 6]),\n",
       "   (5, [3, 8, 6, 2]),\n",
       "   (4, [3, 8, 1, 7]),\n",
       "   (4, [3, 6, 2]),\n",
       "   (4, [8, 1, 5]),\n",
       "   (4, [2, 4, 6]),\n",
       "   (4, [5, 9, 1, 7]),\n",
       "   (4, [6, 2, 8, 3]),\n",
       "   (4, [6, 7, 3]),\n",
       "   (4, [5, 9, 7, 1]),\n",
       "   (4, [2, 3, 8, 6]),\n",
       "   (4, [7, 6, 2, 1]),\n",
       "   (4, [5, 1, 9]),\n",
       "   (4, [2, 8]),\n",
       "   (4, [1, 9, 5, 6]),\n",
       "   (4, [7, 6, 1, 8]),\n",
       "   (4, [6, 1, 7, 9]),\n",
       "   (4, [3, 1, 8, 6]),\n",
       "   (4, [6, 8, 7]),\n",
       "   (4, [1, 3, 5]),\n",
       "   (4, [7, 1, 4, 9]),\n",
       "   (4, [4, 6]),\n",
       "   (4, [3, 7, 8, 9]),\n",
       "   (4, [8, 3, 7, 9]),\n",
       "   (4, [5, 3]),\n",
       "   (4, [1, 6, 9, 5]),\n",
       "   (4, [1, 5, 9, 6]),\n",
       "   (4, [6, 5, 1]),\n",
       "   (4, [3, 5, 8, 1]),\n",
       "   (4, [7, 8, 6, 3]),\n",
       "   (4, [3, 5, 8, 9]),\n",
       "   (4, [7, 1, 6, 4]),\n",
       "   (4, [8, 7, 9]),\n",
       "   (4, [8, 7, 1]),\n",
       "   (4, [6, 8, 3, 2]),\n",
       "   (4, [1, 2, 6]),\n",
       "   (4, [7, 8, 3, 1]),\n",
       "   (4, [9, 5, 7]),\n",
       "   (4, [7, 2, 6]),\n",
       "   (4, [7, 2, 4]),\n",
       "   (4, [3, 7, 2]),\n",
       "   (4, [3, 7, 6]),\n",
       "   (4, [6, 2, 3]),\n",
       "   (4, [6, 3, 7]),\n",
       "   (4, [7, 8, 1]),\n",
       "   (4, [7, 8, 6]),\n",
       "   (4, [7, 6, 8]),\n",
       "   (4, [3, 2, 6]),\n",
       "   (4, [3, 2, 1]),\n",
       "   (4, [8, 7, 3, 6]),\n",
       "   (4, [3, 2, 8, 6]),\n",
       "   (4, [3, 8, 6, 9]),\n",
       "   (4, [7, 6, 1, 3]),\n",
       "   (3, [7, 5, 4]),\n",
       "   (3, [6, 1, 8, 3]),\n",
       "   (3, [1, 8, 3, 6]),\n",
       "   (3, [7, 1, 2]),\n",
       "   (3, [2, 6, 8]),\n",
       "   (3, [2, 6, 7]),\n",
       "   (3, [5, 3, 8, 9]),\n",
       "   (3, [8, 1, 3, 7]),\n",
       "   (3, [7, 1, 8, 6]),\n",
       "   (3, [6, 7, 5]),\n",
       "   (3, [6, 1, 8, 7]),\n",
       "   (3, [7, 6, 4, 1]),\n",
       "   (3, [7, 5, 8]),\n",
       "   (3, [8, 3, 6, 4]),\n",
       "   (3, [3, 6, 7, 8]),\n",
       "   (3, [8, 3, 6, 2]),\n",
       "   (3, [5, 1, 6]),\n",
       "   (3, [6, 1, 9, 7]),\n",
       "   (3, [3, 8, 5, 6]),\n",
       "   (3, [3, 8, 5, 1]),\n",
       "   (3, [1, 3, 7, 6]),\n",
       "   (3, [8, 5]),\n",
       "   (3, [7, 6, 2, 4]),\n",
       "   (3, [6, 3, 1, 8]),\n",
       "   (3, [7, 6, 8, 1]),\n",
       "   (3, [7, 6, 8, 3]),\n",
       "   (3, [2, 3, 6]),\n",
       "   (3, [1, 6, 9, 7]),\n",
       "   (3, [6, 7, 9, 1]),\n",
       "   (3, [6, 4, 3, 8]),\n",
       "   (3, [9, 1, 6]),\n",
       "   (3, [7, 6, 1, 2]),\n",
       "   (3, [7, 5, 1, 8]),\n",
       "   (3, [6, 8, 2]),\n",
       "   (3, [5, 1, 7, 9]),\n",
       "   (3, [7, 4, 1]),\n",
       "   (3, [7, 1, 9, 6]),\n",
       "   (3, [3, 8, 7, 6]),\n",
       "   (3, [3, 9]),\n",
       "   (3, [3, 4]),\n",
       "   (3, [4, 7]),\n",
       "   (3, [1, 7, 6, 3]),\n",
       "   (3, [3, 8, 2, 1]),\n",
       "   (3, [3, 8, 2, 6]),\n",
       "   (3, [7, 1, 6, 8, 3]),\n",
       "   (3, [7, 6, 4]),\n",
       "   (3, [5, 8, 3]),\n",
       "   (3, [5, 8, 7]),\n",
       "   (3, [1, 6, 7, 9]),\n",
       "   (3, [8, 1, 7, 3]),\n",
       "   (3, [3, 4, 8]),\n",
       "   (3, [7, 3, 2]),\n",
       "   (3, [7, 2, 1, 4]),\n",
       "   (3, [5, 7, 1]),\n",
       "   (3, [8, 1, 6, 3]),\n",
       "   (3, [3, 8, 1, 9]),\n",
       "   (3, [2, 4, 6, 1]),\n",
       "   (3, [1, 7, 5, 9]),\n",
       "   (3, [6, 3, 2]),\n",
       "   (3, [3, 8, 1, 4]),\n",
       "   (3, [3, 8, 1, 5]),\n",
       "   (3, [8, 1, 3]),\n",
       "   (3, [7, 9, 5, 6]),\n",
       "   (3, [7, 9, 5, 4]),\n",
       "   (3, [1, 8]),\n",
       "   (3, [7, 1, 6, 2]),\n",
       "   (3, [8, 7, 5]),\n",
       "   (3, [8, 7, 6]),\n",
       "   (3, [3, 5, 6]),\n",
       "   (3, [3, 5, 7]),\n",
       "   (3, [9, 1, 5]),\n",
       "   (3, [1, 4, 7]),\n",
       "   (3, [1, 8, 3]),\n",
       "   (3, [3, 8, 7, 1]),\n",
       "   (3, [9, 1, 4]),\n",
       "   (3, [1, 7, 9, 5]),\n",
       "   (3, [1, 5, 7]),\n",
       "   (3, [5, 9, 1]),\n",
       "   (3, [5, 9, 7]),\n",
       "   (3, [1, 6, 5]),\n",
       "   (3, [7, 2, 8]),\n",
       "   (3, [2, 7, 4]),\n",
       "   (3, [2, 6, 7, 1]),\n",
       "   (3, [7, 3, 1, 6]),\n",
       "   (3, [2, 1, 3]),\n",
       "   (3, [3, 8, 6, 2, 1]),\n",
       "   (3, [6, 2, 4]),\n",
       "   (3, [7, 9, 6, 1]),\n",
       "   (3, [6, 3, 8, 4]),\n",
       "   (3, [7, 1, 3, 8]),\n",
       "   (3, [7, 8, 4]),\n",
       "   (3, [7, 8, 9]),\n",
       "   (3, [8, 3, 1, 5]),\n",
       "   (3, [8, 3, 7, 6]),\n",
       "   (3, [8, 3, 7, 1]),\n",
       "   (3, [7, 1, 5, 6]),\n",
       "   (3, [9, 3]),\n",
       "   (3, [3, 7, 1]),\n",
       "   (3, [3, 2, 8]),\n",
       "   (3, [8, 7, 3, 1]),\n",
       "   (3, [1, 5, 3]),\n",
       "   (3, [1, 7, 3]),\n",
       "   (3, [3, 5, 7, 8]),\n",
       "   (3, [7, 8, 3, 2]),\n",
       "   (3, [7, 9, 1, 5]),\n",
       "   (3, [5, 6, 9]),\n",
       "   (3, [9, 5, 1, 7]),\n",
       "   (2, [6, 7, 1, 2]),\n",
       "   (2, [3, 6, 7]),\n",
       "   (2, [3, 6, 4]),\n",
       "   (2, [5, 3, 8, 7]),\n",
       "   (2, [1, 6, 3, 8, 7]),\n",
       "   (2, [1, 6, 3, 8, 2]),\n",
       "   (2, [7, 5, 2]),\n",
       "   (2, [1, 6, 3, 8, 9]),\n",
       "   (2, [3, 8, 4, 9]),\n",
       "   (2, [9, 8, 3]),\n",
       "   (2, [9, 8, 1]),\n",
       "   (2, [1, 6, 4, 9]),\n",
       "   (2, [3, 1, 8, 6, 4]),\n",
       "   (2, [7, 8, 6, 1]),\n",
       "   (2, [8, 1, 3, 4]),\n",
       "   (2, [7, 8, 3, 6, 5, 2, 1, 9, 4]),\n",
       "   (2, [6, 8, 3, 2, 1]),\n",
       "   (2, [4, 6, 2]),\n",
       "   (2, [8, 6, 2]),\n",
       "   (2, [3, 8, 6, 1, 4]),\n",
       "   (2, [3, 5, 6, 8]),\n",
       "   (2, [8, 5, 3]),\n",
       "   (2, [3, 1, 7, 8]),\n",
       "   (2, [6, 2, 4, 1]),\n",
       "   (2, [8, 2, 3]),\n",
       "   (2, [5, 3, 8]),\n",
       "   (2, [7, 6, 8, 5]),\n",
       "   (2, [3, 8, 1, 2, 6]),\n",
       "   (2, [7, 3, 8, 9]),\n",
       "   (2, [8, 9, 1]),\n",
       "   (2, [4, 1, 3, 6]),\n",
       "   (2, [6, 3, 1, 2]),\n",
       "   (2, [6, 5, 1, 9]),\n",
       "   (2, [9, 8]),\n",
       "   (2, [6, 7, 1, 8, 3]),\n",
       "   (2, [9, 6]),\n",
       "   (2, [3, 8, 9, 5]),\n",
       "   (2, [2, 5, 9]),\n",
       "   (2, [8, 3, 4, 1]),\n",
       "   (2, [1, 7, 3, 6]),\n",
       "   (2, [1, 7, 3, 8]),\n",
       "   (2, [9, 8, 3, 1]),\n",
       "   (2, [1, 9, 5, 3]),\n",
       "   (2, [2, 3, 1]),\n",
       "   (2, [8, 6, 7]),\n",
       "   (2, [3, 1, 7]),\n",
       "   (2, [3, 1, 2]),\n",
       "   (2, [6, 1, 3, 4]),\n",
       "   (2, [3, 1, 9]),\n",
       "   (2, [1, 6, 9, 2]),\n",
       "   (2, [2, 1, 6, 4]),\n",
       "   (2, [1, 5, 9, 7, 6]),\n",
       "   (2, [7, 6, 2, 3]),\n",
       "   (2, [5, 7, 1, 6]),\n",
       "   (2, [3, 7, 5, 6, 8]),\n",
       "   (2, [6, 8, 1, 3, 4]),\n",
       "   (2, [8, 3, 6, 9]),\n",
       "   (2, [3, 5, 9]),\n",
       "   (2, [1, 6, 7, 8, 9, 4, 2, 3, 5]),\n",
       "   (2, [7, 6, 9]),\n",
       "   (2, [5, 1, 3, 8]),\n",
       "   (2, [7, 8, 1, 6]),\n",
       "   (2, [9, 1, 6, 3]),\n",
       "   (2, [4, 1, 7]),\n",
       "   (2, [8, 3, 9, 5]),\n",
       "   (2, [1, 7, 9, 5, 6]),\n",
       "   (2, [7, 8, 9, 6]),\n",
       "   (2, [7, 4, 9]),\n",
       "   (2, [7, 4, 6]),\n",
       "   (2, [7, 5, 9, 3]),\n",
       "   (2, [7, 5, 9, 4]),\n",
       "   (2, [1, 3, 7]),\n",
       "   (2, [9, 1, 4, 7]),\n",
       "   (2, [3, 8, 7, 2]),\n",
       "   (2, [1, 7, 6, 2]),\n",
       "   (2, [6, 7, 8, 1, 3]),\n",
       "   (2, [5, 9, 8, 7]),\n",
       "   (2, [6, 7, 1, 9, 5]),\n",
       "   (2, [4, 3]),\n",
       "   (2, [4, 9]),\n",
       "   (2, [3, 8, 5, 1, 2, 4, 6, 7, 9]),\n",
       "   (2, [3, 7, 8, 1]),\n",
       "   (2, [3, 7, 8, 2]),\n",
       "   (2, [7, 6, 4, 3, 1]),\n",
       "   (2, [8, 5, 9]),\n",
       "   (2, [2, 6, 3, 8, 4]),\n",
       "   (2, [1, 2, 8, 6]),\n",
       "   (2, [2, 4, 1, 6]),\n",
       "   (2, [6, 7, 5, 1]),\n",
       "   (2, [6, 7, 5, 2]),\n",
       "   (2, [8, 4]),\n",
       "   (2, [9, 5, 7, 1]),\n",
       "   (2, [7, 5, 6, 4]),\n",
       "   (2, [8, 7, 1, 6]),\n",
       "   (2, [7, 9, 2]),\n",
       "   (2, [7, 9, 4]),\n",
       "   (2, [8, 2, 6, 3]),\n",
       "   (2, [4, 7, 1]),\n",
       "   (2, [8, 3, 7, 5, 2]),\n",
       "   (2, [9, 6, 1]),\n",
       "   (2, [2, 8, 4]),\n",
       "   (2, [2, 8, 1]),\n",
       "   (2, [8, 2, 3, 6]),\n",
       "   (2, [9, 5]),\n",
       "   (2, [2, 4, 6, 1, 7]),\n",
       "   (2, [4, 8, 3]),\n",
       "   (2, [5, 7, 8]),\n",
       "   (2, [5, 7, 4]),\n",
       "   (2, [5, 9, 7, 3, 8, 1, 6]),\n",
       "   (2, [7, 5, 9, 6, 1, 2, 4, 3, 8]),\n",
       "   (2, [1, 6, 8, 7]),\n",
       "   (2, [7, 5, 9, 1, 4]),\n",
       "   (2, [7, 4, 1, 2]),\n",
       "   (2, [8, 6, 1, 7]),\n",
       "   (2, [8, 6, 1, 3]),\n",
       "   (2, [6, 5, 3]),\n",
       "   (2, [7, 9, 1, 2]),\n",
       "   (2, [3, 9, 8]),\n",
       "   (2, [3, 8, 1, 2]),\n",
       "   (2, [8, 1, 7]),\n",
       "   (2, [3, 5, 8, 6]),\n",
       "   (2, [6, 7, 1, 3]),\n",
       "   (2, [7, 1, 6, 5]),\n",
       "   (2, [3, 1, 6, 8, 9]),\n",
       "   (2, [7, 8, 6, 5, 4, 3, 2, 1, 9]),\n",
       "   (2, [7, 1, 6, 9, 5]),\n",
       "   (2, [8, 6, 5, 1]),\n",
       "   (2, [3, 1, 5, 6]),\n",
       "   (2, [6, 8, 3, 4]),\n",
       "   (2, [6, 1, 5, 7]),\n",
       "   (2, [1, 6, 7, 9, 5]),\n",
       "   (2, [6, 1, 7, 8]),\n",
       "   (2, [6, 1, 7, 2]),\n",
       "   (2, [6, 1, 7, 5]),\n",
       "   (2, [1, 2, 3]),\n",
       "   (2, [1, 8, 7]),\n",
       "   (2, [1, 8, 5]),\n",
       "   (2, [3, 8, 7, 9]),\n",
       "   (2, [9, 7, 1]),\n",
       "   (2, [9, 7, 6]),\n",
       "   (2, [1, 7, 9, 6]),\n",
       "   (2, [8, 9]),\n",
       "   (2, [6, 7, 8, 3]),\n",
       "   (2, [2, 6, 1, 4]),\n",
       "   (2, [2, 6, 1, 8]),\n",
       "   (2, [1, 5, 3, 7]),\n",
       "   (2, [5, 9, 3]),\n",
       "   (2, [5, 9, 4]),\n",
       "   (2, [1, 5, 7, 9]),\n",
       "   (2, [7, 8, 3, 5]),\n",
       "   (2, [8, 3, 5, 4]),\n",
       "   (2, [3, 8, 6, 7]),\n",
       "   (2, [6, 7, 2, 4]),\n",
       "   (2, [7, 6, 1, 3, 8]),\n",
       "   (2, [7, 2, 9]),\n",
       "   (2, [1, 9, 7, 5]),\n",
       "   (2, [2, 6, 7, 8]),\n",
       "   (2, [8, 3, 5, 1]),\n",
       "   (2, [1, 6, 7, 8]),\n",
       "   (2, [4, 9, 7]),\n",
       "   (2, [1, 9, 6, 7]),\n",
       "   (2, [1, 6, 7, 2]),\n",
       "   (2, [1, 6, 7, 5]),\n",
       "   (2, [2, 1, 4, 6]),\n",
       "   (2, [4, 1, 6, 8, 3]),\n",
       "   (2, [2, 7, 9]),\n",
       "   (2, [7, 9, 3, 1]),\n",
       "   (2, [2, 1, 3, 6, 8]),\n",
       "   (2, [7, 6, 3, 8]),\n",
       "   (2, [6, 4, 1, 2]),\n",
       "   (2, [6, 2, 8]),\n",
       "   (2, [6, 3, 8, 9]),\n",
       "   (2, [6, 3, 8, 7]),\n",
       "   (2, [7, 6, 1, 9]),\n",
       "   (2, [7, 5, 1, 6]),\n",
       "   (2, [7, 6, 1, 5]),\n",
       "   (2, [7, 8, 2]),\n",
       "   (2, [7, 1, 3, 6]),\n",
       "   (2, [7, 8, 5, 6]),\n",
       "   (2, [1, 6, 8, 9, 3]),\n",
       "   (2, [1, 9, 7, 3]),\n",
       "   (2, [8, 9, 5, 3]),\n",
       "   (2, [3, 6, 8, 5]),\n",
       "   (2, [3, 6, 8, 7]),\n",
       "   (2, [3, 6, 8, 2]),\n",
       "   (2, [3, 6, 8, 9]),\n",
       "   (2, [7, 4, 6, 1]),\n",
       "   (2, [6, 2, 3, 8]),\n",
       "   (2, [3, 1, 6, 4]),\n",
       "   (2, [6, 2, 3, 1]),\n",
       "   (2, [8, 7, 1, 4]),\n",
       "   (2, [6, 1, 3, 8, 2]),\n",
       "   (2, [6, 2, 7, 1]),\n",
       "   (2, [4, 6, 7]),\n",
       "   (2, [6, 1, 2, 4]),\n",
       "   (2, [6, 7, 3, 8]),\n",
       "   (2, [7, 9, 6, 3]),\n",
       "   (2, [8, 7, 3, 2]),\n",
       "   (2, [4, 6, 1, 7]),\n",
       "   (2, [8, 7, 3, 9]),\n",
       "   (2, [5, 7, 9, 1, 6]),\n",
       "   (2, [7, 6, 9, 1]),\n",
       "   (2, [7, 8, 3, 6]),\n",
       "   (2, [6, 5, 7]),\n",
       "   (2, [3, 8, 6, 4]),\n",
       "   (2, [3, 7, 5]),\n",
       "   (2, [1, 4, 6, 7, 3, 8, 2, 9, 5]),\n",
       "   (2, [3, 6, 2, 7]),\n",
       "   (2, [5, 1, 6, 7, 8, 4, 2, 3, 9]),\n",
       "   (2, [3, 1, 8, 2]),\n",
       "   (2, [1, 6, 2, 3, 8]),\n",
       "   (2, [8, 3, 5, 9]),\n",
       "   (1, [2, 4, 8, 3, 6, 7, 1, 9, 5]),\n",
       "   (1, [6, 7, 1, 4]),\n",
       "   (1, [6, 7, 1, 5]),\n",
       "   (1, [6, 7, 1, 9]),\n",
       "   (1, [2, 7, 4, 6, 9, 5, 1, 3, 8]),\n",
       "   (1, [6, 7, 3, 1, 4]),\n",
       "   (1, [8, 7, 5, 1]),\n",
       "   (1, [1, 2, 4, 6, 7, 8, 9, 3, 5]),\n",
       "   (1, [7, 8, 1, 3, 5, 2, 6, 4, 9]),\n",
       "   (1, [3, 9, 5, 6, 1]),\n",
       "   (1, [4, 7, 1, 6, 2]),\n",
       "   (1, [5, 9, 1, 3]),\n",
       "   (1, [8, 3, 7, 1, 6, 5]),\n",
       "   (1, [3, 1, 8, 9]),\n",
       "   (1, [2, 6, 8, 3, 1]),\n",
       "   (1, [7, 1, 5, 6, 9, 4, 2, 3, 8]),\n",
       "   (1, [1, 6, 4, 9, 8, 3, 2, 7, 5]),\n",
       "   (1, [8, 3, 2, 1, 9, 7, 6, 4, 5]),\n",
       "   (1, [7, 5, 3, 2]),\n",
       "   (1, [6, 2, 7, 8, 3]),\n",
       "   (1, [4, 2, 6, 7, 8, 1, 3, 9, 5]),\n",
       "   (1, [1, 6, 3, 8, 4]),\n",
       "   (1, [1, 6, 3, 8, 5]),\n",
       "   (1, [2, 7, 4, 3, 8]),\n",
       "   (1, [6, 8, 3, 4, 1]),\n",
       "   (1, [6, 1, 8, 2]),\n",
       "   (1, [1, 9, 4]),\n",
       "   (1, [1, 9, 2]),\n",
       "   (1, [1, 8, 3, 2]),\n",
       "   (1, [7, 6, 1, 3, 8, 4, 5, 9, 2]),\n",
       "   (1, [1, 9, 8]),\n",
       "   (1, [2, 4, 8]),\n",
       "   (1, [7, 8, 1, 2, 3, 6, 4, 9, 5]),\n",
       "   (1, [3, 8, 2, 4, 1, 6]),\n",
       "   (1, [3, 5, 6, 8, 9, 1, 4, 7, 2]),\n",
       "   (1, [2, 4, 5]),\n",
       "   (1, [1, 5, 8, 3]),\n",
       "   (1, [3, 8, 4, 5]),\n",
       "   (1, [3, 1, 7, 6, 2]),\n",
       "   (1, [8, 3, 6, 2, 1, 4]),\n",
       "   (1, [9, 5, 7, 8, 3, 1, 6, 4, 2]),\n",
       "   (1, [5, 7, 8, 1, 2, 6]),\n",
       "   (1, [5, 7, 6, 4]),\n",
       "   (1, [8, 3, 2, 1]),\n",
       "   (1, [2, 9, 4, 3, 1, 6, 7, 8, 5]),\n",
       "   (1, [1, 6, 2, 9]),\n",
       "   (1, [7, 8, 3, 2, 6, 1]),\n",
       "   (1, [6, 3, 5, 1]),\n",
       "   (1, [4, 1, 2, 6]),\n",
       "   (1, [1, 6, 2, 7]),\n",
       "   (1, [4, 1, 3, 2, 6, 9, 5, 8, 7]),\n",
       "   (1, [8, 7, 3, 1, 2, 6]),\n",
       "   (1, [2, 6, 1, 5, 4]),\n",
       "   (1, [3, 8, 7, 6, 1]),\n",
       "   (1, [2, 1, 6, 4, 3]),\n",
       "   (1, [1, 5, 8, 6, 3, 9, 7, 4, 2]),\n",
       "   (1, [2, 1, 6, 4, 7]),\n",
       "   (1, [3, 2, 7, 8, 1, 6]),\n",
       "   (1, [6, 4, 1, 3, 5, 8, 7, 9, 2]),\n",
       "   (1, [8, 3, 2, 7]),\n",
       "   (1, [7, 2, 3, 8]),\n",
       "   (1, [1, 5, 4, 2, 3, 6, 8]),\n",
       "   (1, [2, 3, 1, 8]),\n",
       "   (1, [5, 7, 1, 9, 6, 8, 3, 2, 4]),\n",
       "   (1, [9, 1, 5, 7, 4]),\n",
       "   (1, [7, 2, 3, 1]),\n",
       "   (1, [5, 9, 4, 1, 6, 2, 7, 8, 3]),\n",
       "   (1, [6, 5, 4, 1, 2, 9, 8, 3]),\n",
       "   (1, [7, 2, 9, 8, 3, 6, 1]),\n",
       "   (1, [9, 1, 5, 7, 6, 8, 3, 4, 2]),\n",
       "   (1, [3, 7, 5, 6, 8, 9, 1, 4, 2]),\n",
       "   (1, [3, 7, 8, 6, 4, 1, 2, 5, 9]),\n",
       "   (1, [2, 6, 1, 7, 3, 8, 4, 9]),\n",
       "   (1, [3, 6, 2, 4]),\n",
       "   (1, [7, 8, 2, 6]),\n",
       "   (1, [1, 6, 7, 8, 3]),\n",
       "   (1, [3, 8, 1, 4, 6, 7]),\n",
       "   (1, [9, 1, 5, 3, 8]),\n",
       "   (1, [1, 9, 8, 5]),\n",
       "   (1, [9, 8, 7, 6, 5, 4, 3, 2, 1]),\n",
       "   (1, [2, 1, 7, 5]),\n",
       "   (1, [1, 6, 7, 8, 9]),\n",
       "   (1, [9, 4, 1]),\n",
       "   (1, [2, 1, 7, 3]),\n",
       "   (1, [1, 9, 5, 4, 7, 6, 8, 3, 2]),\n",
       "   (1, [1, 9, 7, 5, 6, 2]),\n",
       "   (1, [7, 3, 2, 5]),\n",
       "   (1, [1, 8, 9, 3, 5]),\n",
       "   (1, [9, 7, 8, 3, 1, 6, 5, 4, 2]),\n",
       "   (1, [7, 5, 9, 4, 1, 6, 2, 3, 8]),\n",
       "   (1, [7, 5, 9, 1, 2, 4, 8, 6, 3]),\n",
       "   (1, [2, 7, 8, 9, 3]),\n",
       "   (1, [3, 4, 8, 1]),\n",
       "   (1, [6, 1, 7, 8, 3, 2, 4, 5, 9]),\n",
       "   (1, [7, 5, 1, 4, 6, 9, 2, 3, 8]),\n",
       "   (1, [1, 8, 3, 7, 4, 9, 5, 6, 2]),\n",
       "   (1, [7, 6, 2, 3, 5, 4, 8, 1, 9]),\n",
       "   (1, [7, 9, 1, 2, 4, 5, 3]),\n",
       "   (1, [8, 9, 3, 7, 2]),\n",
       "   (1, [3, 8, 1, 9, 5, 4, 6, 7, 2]),\n",
       "   (1, [6, 9, 3, 8]),\n",
       "   (1, [8, 3, 1, 2, 4, 6, 9, 5, 7]),\n",
       "   (1, [1, 6, 4, 2]),\n",
       "   (1, [7, 6, 5, 8, 3, 1, 9, 4, 2]),\n",
       "   (1, [1, 6, 4, 7]),\n",
       "   (1, [7, 3, 8, 2]),\n",
       "   (1, [7, 9, 1, 3, 8]),\n",
       "   (1, [3, 8, 6, 2, 4]),\n",
       "   (1, [8, 3, 6, 7, 1, 2, 5, 9, 4]),\n",
       "   (1, [3, 7, 6, 8, 5, 9, 4, 1, 2]),\n",
       "   (1, [8, 9, 6]),\n",
       "   (1, [2, 6, 8, 4]),\n",
       "   (1, [3, 1, 8, 6, 2]),\n",
       "   (1, [4, 1, 2, 6, 7, 8]),\n",
       "   (1, [1, 3, 8, 6, 7, 9, 4, 5, 2]),\n",
       "   (1, [7, 8, 3, 1, 6, 2, 4, 9, 5]),\n",
       "   (1, [8, 9, 5]),\n",
       "   (1, [2, 6, 8, 3]),\n",
       "   (1, [8, 1, 3, 9]),\n",
       "   (1, [3, 7, 8, 9, 6, 1, 5, 4, 2]),\n",
       "   (1, [1, 3, 5, 8, 7]),\n",
       "   (1, [8, 1, 3, 6, 7]),\n",
       "   (1, [8, 1, 3, 6]),\n",
       "   (1, [1, 6, 7, 9, 8, 3, 5, 4, 2]),\n",
       "   (1, [1, 8, 7, 9, 3, 4, 5, 6, 2]),\n",
       "   (1, [7, 9, 8, 3, 6, 5, 1, 2, 4]),\n",
       "   (1, [1, 3, 4]),\n",
       "   (1, [8, 3, 6, 9, 1, 2, 4, 5, 7]),\n",
       "   (1, [1, 9, 5, 7, 4]),\n",
       "   (1, [4, 1, 7, 3, 6, 8, 9, 2, 5]),\n",
       "   (1, [6, 4, 2, 1]),\n",
       "   (1, [2, 6, 7, 3, 8, 9, 5, 1, 4]),\n",
       "   (1, [6, 4, 2, 7]),\n",
       "   (1, [7, 2, 1, 6, 3, 8]),\n",
       "   (1, [7, 9, 6, 8, 3, 4, 5, 1, 2]),\n",
       "   (1, [1, 4, 5, 6, 7, 9, 3, 2, 8]),\n",
       "   (1, [3, 8, 1, 2, 6, 7]),\n",
       "   (1, [7, 9, 8, 1, 3]),\n",
       "   (1, [1, 9, 4, 5]),\n",
       "   (1, [1, 9, 4, 6]),\n",
       "   (1, [7, 9, 1, 6, 3, 8, 5, 4, 2]),\n",
       "   (1, [9, 1, 2, 7]),\n",
       "   (1, [8, 6, 9, 3, 1]),\n",
       "   (1, [3, 5, 1, 6, 8, 9, 4, 7, 2]),\n",
       "   (1, [5, 1, 6, 9, 7]),\n",
       "   (1, [2, 6, 4, 7]),\n",
       "   (1, [2, 6, 4, 8]),\n",
       "   (1, [3, 8, 1, 5, 6, 9, 2, 4, 7]),\n",
       "   (1, [7, 6, 3, 8, 1, 2, 5, 4, 9]),\n",
       "   (1, [4, 1, 6, 7, 3]),\n",
       "   (1, [4, 1, 3, 6, 8, 5, 9, 2]),\n",
       "   (1, [7, 1, 2, 4, 5, 6, 9, 8, 3]),\n",
       "   (1, [8, 6, 3, 9]),\n",
       "   (1, [3, 8, 7, 5, 9, 6, 1, 2, 4]),\n",
       "   (1, [7, 6, 8, 9, 2]),\n",
       "   (1, [4, 3, 8, 6]),\n",
       "   (1, [4, 3, 8, 5]),\n",
       "   (1, [5, 6, 1, 9, 7, 2, 3, 4, 8]),\n",
       "   (1, [8, 6, 3, 1]),\n",
       "   (1, [4, 3, 8, 2]),\n",
       "   (1, [6, 9, 4]),\n",
       "   (1, [8, 3, 5, 7, 1, 2, 4, 6, 9]),\n",
       "   (1, [3, 8, 5, 2, 1, 6, 7, 4, 9]),\n",
       "   (1, [6, 7, 1, 2, 4]),\n",
       "   (1, [6, 7, 4]),\n",
       "   (1, [1, 9, 3, 6, 8, 5, 2, 7, 4]),\n",
       "   (1, [6, 5, 7, 4]),\n",
       "   (1, [8, 3, 9, 6]),\n",
       "   (1, [3, 1, 8, 7, 9, 2, 4, 5, 6]),\n",
       "   (1, [8, 9, 5, 3, 1]),\n",
       "   (1, [1, 6, 7, 4, 3]),\n",
       "   (1, [7, 3, 1, 2, 8]),\n",
       "   (1, [8, 3, 2, 1, 5, 6, 9, 7, 4]),\n",
       "   (1, [8, 3, 1, 7, 6, 4, 2, 5, 9]),\n",
       "   (1, [6, 8, 3, 2, 1, 4, 7, 5, 9]),\n",
       "   (1, [7, 6, 4, 2]),\n",
       "   (1, [1, 6, 7, 3, 8, 9, 4, 5, 2]),\n",
       "   (1, [7, 2, 3, 6, 8, 1, 9, 5, 4]),\n",
       "   (1, [2, 6, 1, 3, 5, 8]),\n",
       "   (1, [7, 6, 3, 8, 9, 1, 2, 4, 5]),\n",
       "   (1, [7, 3, 5, 4, 6, 8, 1, 9, 2]),\n",
       "   (1, [6, 1, 8, 7, 5]),\n",
       "   (1, [6, 3, 7, 2]),\n",
       "   (1, [2, 8, 1, 3, 4, 6, 9, 7, 5]),\n",
       "   (1, [8, 3, 1, 2]),\n",
       "   (1, [9, 7, 3, 8, 6, 1, 2, 4, 5]),\n",
       "   (1, [7, 1, 6, 2, 4, 9, 5, 8, 3]),\n",
       "   (1, [3, 8, 6, 1, 2]),\n",
       "   (1, [8, 2, 6]),\n",
       "   (1, [3, 5, 9, 8, 7, 1, 2, 4]),\n",
       "   (1, [2, 6, 8, 3, 7, 9, 4, 1, 5]),\n",
       "   (1, [4, 9, 2, 6, 1, 8, 3, 7, 5]),\n",
       "   (1, [5, 1, 9, 4, 6]),\n",
       "   (1, [8, 3, 6, 5]),\n",
       "   (1, [3, 1, 5, 4, 9, 8, 7, 6, 2]),\n",
       "   (1, [3, 6, 7, 2]),\n",
       "   (1, [4, 5, 9, 1]),\n",
       "   (1, [3, 2, 8, 4, 9, 5, 6, 7, 1]),\n",
       "   (1, [8, 2, 7]),\n",
       "   (1, [3, 6, 7, 8, 2, 9, 1, 4, 5]),\n",
       "   (1, [7, 8, 2, 9]),\n",
       "   (1, [3, 8, 1, 6, 2, 9, 5, 7]),\n",
       "   (1, [2, 7, 1, 3, 4, 9, 6, 5, 8]),\n",
       "   (1, [9, 5, 7, 6, 4]),\n",
       "   (1, [7, 5, 1, 6, 4, 9, 8, 3, 2]),\n",
       "   (1, [1, 6, 9, 5, 4, 7, 2, 3, 8]),\n",
       "   (1, [7, 5, 1, 4, 9, 8, 6, 3, 2]),\n",
       "   (1, [3, 2, 5, 6]),\n",
       "   (1, [6, 2, 1, 3, 8, 7, 5, 9, 4]),\n",
       "   (1, [7, 6, 2, 1, 4, 8, 3, 9, 5]),\n",
       "   (1, [3, 8, 5, 9, 1, 2, 6, 7, 4]),\n",
       "   (1, [6, 4, 3, 8, 5, 1]),\n",
       "   (1, [8, 5, 1]),\n",
       "   (1, [5, 7, 8, 3, 1, 9, 2, 6, 4]),\n",
       "   (1, [8, 5, 7]),\n",
       "   (1, [8, 5, 6]),\n",
       "   (1, [7, 8, 9, 6, 5]),\n",
       "   (1, [8, 5, 6, 9]),\n",
       "   (1, [8, 7, 1, 5, 6, 2, 4, 9, 3]),\n",
       "   (1, [6, 7, 8, 4, 3]),\n",
       "   (1, [7, 6, 4, 1, 9, 5, 8, 3, 2]),\n",
       "   (1, [8, 1, 5, 4]),\n",
       "   (1, [8, 9, 1, 7, 6, 5, 4, 3, 2]),\n",
       "   (1, [6, 3, 8, 2, 1, 5, 9, 4, 7]),\n",
       "   (1, [3, 1, 5, 8, 6, 9, 4, 2, 7]),\n",
       "   (1, [5, 9, 7, 6, 1]),\n",
       "   (1, [2, 7, 1, 4]),\n",
       "   (1, [2, 7, 1, 5]),\n",
       "   (1, [7, 2, 9, 6]),\n",
       "   (1, [3, 8, 1, 9, 6, 5]),\n",
       "   (1, [1, 7, 3, 9, 5]),\n",
       "   (1, [7, 4, 3, 2]),\n",
       "   (1, [2, 4, 6, 3, 8]),\n",
       "   (1, [8, 3, 2, 1, 4, 6, 7, 5, 9]),\n",
       "   (1, [3, 6, 1, 8, 9, 5, 7]),\n",
       "   (1, [5, 8, 7, 3, 1, 2, 9, 6, 4]),\n",
       "   (1, [6, 3, 1, 2, 7, 8, 9, 5, 4]),\n",
       "   (1, [4, 2, 1, 6, 7]),\n",
       "   (1, [7, 1, 9, 5, 6, 4, 8, 3, 2]),\n",
       "   (1, [2, 7, 1, 4, 3, 6, 8, 9, 5]),\n",
       "   (1, [7, 1, 6, 2, 4]),\n",
       "   (1, [5, 3, 9]),\n",
       "   (1, [5, 1, 8]),\n",
       "   (1, [4, 6, 3, 8, 1, 2, 9, 5, 7]),\n",
       "   (1, [8, 2, 3, 4]),\n",
       "   (1, [4, 6, 2, 8]),\n",
       "   (1, [2, 3, 6, 8, 7, 1, 9, 5, 4]),\n",
       "   (1, [2, 1, 6, 7, 8, 3, 9]),\n",
       "   (1, [5, 3, 7]),\n",
       "   (1, [5, 3, 8, 9, 7, 6, 1, 2, 4]),\n",
       "   (1, [6, 9, 1, 3, 4]),\n",
       "   (1, [7, 1, 5, 9, 6]),\n",
       "   (1, [3, 8, 4, 1, 6, 7, 2, 9, 5]),\n",
       "   (1, [4, 9, 1, 3, 8]),\n",
       "   (1, [6, 1, 9, 3]),\n",
       "   (1, [6, 8, 7, 3]),\n",
       "   (1, [3, 1, 6, 7, 2, 4, 8, 9, 5]),\n",
       "   (1, [6, 1, 9, 8]),\n",
       "   (1, [7, 3, 8, 4]),\n",
       "   (1, [8, 3, 6, 9, 5, 4]),\n",
       "   (1, [6, 4, 2, 7, 1, 8, 3, 5, 9]),\n",
       "   (1, [1, 7, 9, 6, 5]),\n",
       "   (1, [8, 9, 7]),\n",
       "   (1, [7, 6, 2, 1, 4]),\n",
       "   (1, [1, 7, 6, 8, 3, 5]),\n",
       "   (1, [2, 6, 1, 3, 4, 8, 7, 5, 9]),\n",
       "   (1, [3, 8, 9, 5, 6, 7, 4, 2, 1]),\n",
       "   (1, [3, 8, 5, 7]),\n",
       "   (1, [2, 6, 4, 8, 1]),\n",
       "   (1, [7, 6, 9, 1, 4, 8, 3, 5, 2]),\n",
       "   (1, [6, 1, 7, 8, 3]),\n",
       "   (1, [7, 9, 8, 3, 1]),\n",
       "   (1, [7, 5, 4, 1, 9, 6, 2, 3, 8]),\n",
       "   (1, [6, 7, 3, 8, 5]),\n",
       "   (1, [7, 3, 8, 9, 1]),\n",
       "   (1, [2, 7, 4, 6, 8, 3]),\n",
       "   (1, [5, 9, 3, 1, 6]),\n",
       "   (1, [3, 1, 8, 9, 5, 4, 6, 7, 2]),\n",
       "   (1, [5, 6, 7, 8, 1, 3, 2, 9]),\n",
       "   (1, [1, 9, 5, 6, 7]),\n",
       "   (1, [1, 3, 8, 7, 6]),\n",
       "   (1, [5, 9, 7, 6, 3, 4, 8, 1, 2]),\n",
       "   (1, [3, 4, 7, 9, 1, 6, 8, 2]),\n",
       "   (1, [4, 1, 2, 6, 7, 3, 8, 5, 9]),\n",
       "   (1, [5, 6, 7, 1, 3, 9]),\n",
       "   (1, [3, 6, 8, 2, 1, 4, 5, 9, 7]),\n",
       "   (1, [8, 3, 9, 7, 6, 5, 4, 1]),\n",
       "   (1, [5, 7, 8, 1, 3, 9, 2, 6, 4]),\n",
       "   (1, [7, 1, 3, 8, 6]),\n",
       "   (1, [2, 3, 6, 1, 8, 4, 9, 7, 5]),\n",
       "   (1, [7, 4, 1, 9, 5]),\n",
       "   (1, [1, 5, 6, 7, 8, 9, 3, 4, 2]),\n",
       "   (1, [3, 6, 1, 4]),\n",
       "   (1, [3, 6, 1, 5]),\n",
       "   (1, [4, 1, 7, 9]),\n",
       "   (1, [7, 1, 6, 4, 2, 3, 8, 9, 5]),\n",
       "   (1, [8, 6, 1, 3, 4, 2, 9, 7, 5]),\n",
       "   (1, [2, 3, 8, 1]),\n",
       "   (1, [7, 1, 5, 9, 6, 4, 3, 8, 2]),\n",
       "   (1, [5, 1, 3, 8, 9, 6]),\n",
       "   (1, [1, 2, 5, 9, 3]),\n",
       "   (1, [6, 3, 1, 7]),\n",
       "   (1, [7, 6, 1, 4, 9, 5, 2, 3, 8]),\n",
       "   (1, [3, 7, 1, 8]),\n",
       "   (1, [1, 3, 5, 8]),\n",
       "   (1, [6, 1, 5, 9, 2, 4, 7]),\n",
       "   (1, [8, 3, 5, 6, 9, 7, 4, 1, 2]),\n",
       "   (1, [1, 5, 6, 9, 8, 7, 3, 2, 4]),\n",
       "   (1, [8, 4, 7, 3]),\n",
       "   (1, [1, 7, 9, 8, 3, 2, 6, 4, 5]),\n",
       "   (1, [3, 8, 1, 6, 4, 7, 5, 9, 2]),\n",
       "   (1, [1, 2, 6, 3, 4, 8]),\n",
       "   (1, [5, 9, 3, 8]),\n",
       "   (1, [7, 6, 3, 8, 1]),\n",
       "   (1, [7, 6, 9, 5, 4, 1, 3, 8, 2]),\n",
       "   (1, [7, 9, 1, 5, 6, 4]),\n",
       "   (1, [1, 3, 6, 8, 9, 7]),\n",
       "   (1, [7, 6, 3, 8, 9]),\n",
       "   (1, [7, 6, 2, 9]),\n",
       "   (1, [7, 6, 9, 2, 1, 5]),\n",
       "   (1, [2, 8, 3, 9, 7, 1, 6, 4, 5]),\n",
       "   (1, [2, 6, 8, 3, 4, 1]),\n",
       "   (1, [6, 3, 8, 2, 1, 4, 9, 5, 7]),\n",
       "   (1, [1, 7, 8, 9, 5, 2, 3]),\n",
       "   (1, [8, 3, 9, 6, 1, 2, 4, 5, 7]),\n",
       "   (1, [8, 7, 4, 3]),\n",
       "   (1, [6, 1, 3, 9, 2, 8]),\n",
       "   (1, [7, 4, 5, 6, 1, 9, 2, 8, 3]),\n",
       "   (1, [6, 3, 8, 2, 7, 1, 9, 4, 5]),\n",
       "   (1, [7, 9, 8, 3, 1, 6, 5, 4, 2]),\n",
       "   (1, [9, 8, 7, 5]),\n",
       "   (1, [7, 9, 5, 6, 8, 4, 3, 2, 1]),\n",
       "   (1, [3, 8, 2, 7, 1, 6]),\n",
       "   (1, [3, 8, 4, 1, 6, 5, 7, 2, 9]),\n",
       "   (1, [5, 4, 7, 6]),\n",
       "   (1, [3, 6, 7, 8, 2, 4, 1, 9, 5]),\n",
       "   (1, [8, 5, 1, 3]),\n",
       "   (1, [5, 4, 7, 1]),\n",
       "   (1, [5, 7, 3, 9]),\n",
       "   (1, [5, 4, 7, 9]),\n",
       "   (1, [3, 8, 9, 2]),\n",
       "   (1, [7, 8, 5, 1]),\n",
       "   (1, [3, 8, 9, 6]),\n",
       "   (1, [9, 7, 1, 5]),\n",
       "   (1, [1, 9, 7, 6, 8, 3, 4]),\n",
       "   (1, [8, 5, 3, 9, 6]),\n",
       "   (1, [9, 5, 4, 7, 8, 3, 6, 1, 2]),\n",
       "   (1, [9, 3, 6, 1]),\n",
       "   (1, [7, 8, 4, 1]),\n",
       "   (1, [3, 8, 7, 1, 4, 9, 5, 6, 2]),\n",
       "   (1, [8, 3, 7, 9, 5, 1, 2]),\n",
       "   (1, [7, 6, 9, 8, 3]),\n",
       "   (1, [7, 1, 8, 2, 3, 6, 4, 9, 5]),\n",
       "   (1, [4, 7, 9, 5, 2, 1, 3, 8, 6]),\n",
       "   (1, [1, 3, 5, 9, 8]),\n",
       "   (1, [8, 2, 7, 6]),\n",
       "   (1, [1, 5, 4, 6, 7, 9, 8, 3, 2]),\n",
       "   (1, [7, 3, 8, 9, 6, 1, 2]),\n",
       "   (1, [7, 5, 4, 9, 6, 1, 2, 3, 8]),\n",
       "   (1, [1, 9, 6, 5]),\n",
       "   (1, [3, 8, 1, 6, 4, 2, 7, 9, 5]),\n",
       "   (1, [2, 9]),\n",
       "   (1, [2, 7, 1, 9, 6, 8]),\n",
       "   (1, [4, 1, 5, 9]),\n",
       "   (1, [3, 6, 7, 1, 2, 8, 9, 4, 5]),\n",
       "   (1, [7, 5, 3, 2, 1, 6, 8, 9]),\n",
       "   (1, [5, 8, 3, 6]),\n",
       "   (1, [3, 8, 1, 5, 9, 7, 6, 4, 2]),\n",
       "   (1, [7, 4, 6, 2, 3]),\n",
       "   (1, [1, 9, 5, 4]),\n",
       "   (1, [1, 9, 5, 7]),\n",
       "   (1, [3, 8, 5, 1, 9, 6, 2]),\n",
       "   (1, [7, 1, 6, 5, 2]),\n",
       "   (1, [9, 8, 7, 1]),\n",
       "   (1, [4, 7, 3, 5, 6, 8, 9, 2, 1]),\n",
       "   (1, [8, 9, 7, 5, 6, 3, 1]),\n",
       "   (1, [1, 6, 2, 4, 3, 8, 7, 9, 5]),\n",
       "   (1, [6, 7, 3, 1, 4, 8, 5, 9, 2]),\n",
       "   (1, [5, 3, 2, 1, 4, 6, 7, 8, 9]),\n",
       "   (1, [2, 8, 1, 3, 6]),\n",
       "   (1, [2, 3, 7]),\n",
       "   (1, [1, 6, 3, 8, 7, 9, 5, 4, 2]),\n",
       "   (1, [9, 7, 8, 5, 3, 1]),\n",
       "   (1, [2, 8, 3, 6, 7, 1, 4, 9, 5]),\n",
       "   (1, [2, 8, 9, 7, 6]),\n",
       "   (1, [7, 9, 5, 6, 3, 8, 1, 4, 2]),\n",
       "   (1, [8, 6, 7, 3, 1, 2, 9, 4, 5]),\n",
       "   (1, [1, 8, 3, 7, 4, 5, 9, 6, 2]),\n",
       "   (1, [1, 8, 6, 5, 3, 9]),\n",
       "   (1, [7, 9, 8, 6]),\n",
       "   (1, [3, 5, 8, 1, 6]),\n",
       "   (1, [7, 9, 8, 2]),\n",
       "   (1, [7, 9, 8, 3]),\n",
       "   (1, [7, 9, 8, 1]),\n",
       "   (1, [3, 4, 2, 6]),\n",
       "   (1, [6, 1, 8, 7, 9, 3, 4, 2, 5]),\n",
       "   (1, [3, 4, 8, 1, 6, 9]),\n",
       "   (1, [6, 1, 3, 5]),\n",
       "   (1, [6, 1, 3, 7]),\n",
       "   (1, [1, 9, 7, 8, 6, 3]),\n",
       "   (1, [5, 3, 8, 1, 6, 2, 4, 7, 9]),\n",
       "   (1, [1, 6, 9, 3]),\n",
       "   (1, [6, 3, 8, 7, 9, 1]),\n",
       "   (1, [7, 1, 3, 8, 2, 4]),\n",
       "   (1, [8, 7, 9, 5, 3, 6]),\n",
       "   (1, [3, 8, 1, 9, 6, 4, 7, 5, 2]),\n",
       "   (1, [2, 1, 6, 3]),\n",
       "   (1, [6, 2, 4, 1, 7, 3, 8]),\n",
       "   (1, [7, 6, 8, 3, 1]),\n",
       "   (1, [2, 4, 3, 8, 5, 6, 7, 9, 1]),\n",
       "   (1, [8, 4, 2, 3, 6, 7, 1]),\n",
       "   (1, [9, 7, 1, 6]),\n",
       "   (1, [8, 3, 6, 1, 4]),\n",
       "   (1, [6, 1, 8, 2, 3, 4, 5, 9, 7]),\n",
       "   (1, [7, 9, 6, 5, 1]),\n",
       "   (1, [6, 3, 8, 7, 2, 1, 9, 5, 4]),\n",
       "   (1, [2, 1, 7, 9]),\n",
       "   (1, [1, 5, 6, 9, 7, 8]),\n",
       "   (1, [5, 7, 1, 9]),\n",
       "   (1, [1, 3, 6, 8, 4, 9, 7, 5, 2]),\n",
       "   (1, [5, 7, 1, 4]),\n",
       "   (1, [6, 1, 4, 9, 8, 3, 5, 2, 7]),\n",
       "   (1, [7, 6, 1, 4, 5, 3, 8, 2, 9]),\n",
       "   (1, [8, 2, 6, 1, 3, 4, 9, 5, 7]),\n",
       "   (1, [7, 5, 8, 1, 4, 3, 6, 9]),\n",
       "   (1, [8, 5, 3, 1, 6, 9]),\n",
       "   (1, [6, 3, 8, 4, 1]),\n",
       "   (1, [7, 8, 3, 1, 9, 6]),\n",
       "   (1, [4, 1, 5, 6, 8, 3, 9, 7, 2]),\n",
       "   (1, [1, 3, 9, 8, 6]),\n",
       "   (1, [6, 5, 3, 8, 9, 7, 2, 1, 4]),\n",
       "   (1, [6, 3, 7, 1, 2]),\n",
       "   (1, [6, 1, 3, 2, 8, 7]),\n",
       "   (1, [7, 6, 1, 3, 5, 4, 8, 9, 2]),\n",
       "   (1, [5, 9, 1, 7, 4, 6, 3, 2, 8]),\n",
       "   (1, [9, 8, 3, 6, 1, 5, 2]),\n",
       "   (1, [1, 4, 3, 8, 9, 6]),\n",
       "   (1, [3, 1, 2, 6, 8, 9, 5, 4, 7]),\n",
       "   (1, [1, 6, 7, 2, 9, 8, 3, 4, 5]),\n",
       "   (1, [7, 5, 1, 9, 4, 3, 8, 2]),\n",
       "   (1, [5, 7, 9, 8, 3, 6, 4, 1, 2]),\n",
       "   (1, [7, 5, 2, 9]),\n",
       "   (1, [8, 5, 3, 7]),\n",
       "   (1, [3, 1, 6, 8, 4, 9, 7, 5, 2]),\n",
       "   (1, [6, 2, 9, 1]),\n",
       "   (1, [7, 1, 8, 2, 9]),\n",
       "   ...]),\n",
       " (8682,\n",
       "  defaultdict(int,\n",
       "              {1: 0.23600552868002764,\n",
       "               3: 0.3214697074406819,\n",
       "               2: 0.1422483298779083,\n",
       "               4: 0.1529601474314674,\n",
       "               5: 0.035245335176226675,\n",
       "               9: 0.08880442294402212,\n",
       "               8: 0.005413499193734163,\n",
       "               6: 0.012554710896106889,\n",
       "               7: 0.005298318359824925,\n",
       "               0: 0}),\n",
       "  [(693, [3]),\n",
       "   (385, [4, 5, 7]),\n",
       "   (306, [4]),\n",
       "   (271, [7, 5, 4]),\n",
       "   (243, [7]),\n",
       "   (230, [7, 4, 5]),\n",
       "   (221, [5, 4, 7]),\n",
       "   (204, [6]),\n",
       "   (194, [5]),\n",
       "   (148, [9]),\n",
       "   (134, [1]),\n",
       "   (130, [4, 7, 5]),\n",
       "   (130, [5, 7, 4]),\n",
       "   (113, [3, 9]),\n",
       "   (79, [2]),\n",
       "   (76, [4, 7]),\n",
       "   (68, [4, 5]),\n",
       "   (62, [7, 4]),\n",
       "   (58, [3, 8]),\n",
       "   (51, [3, 1]),\n",
       "   (50, [5, 4]),\n",
       "   (48, [8]),\n",
       "   (41, [3, 2]),\n",
       "   (36, [3, 4]),\n",
       "   (36, [5, 7]),\n",
       "   (35, [3, 9, 1]),\n",
       "   (35, [6, 2]),\n",
       "   (35, [3, 4, 5, 7]),\n",
       "   (34, [3, 6]),\n",
       "   (33, [4, 5, 7, 2]),\n",
       "   (31, [7, 5]),\n",
       "   (29, [3, 1, 9]),\n",
       "   (29, [4, 5, 7, 9]),\n",
       "   (28, [3, 9, 8]),\n",
       "   (28, [1, 3]),\n",
       "   (28, [1, 9]),\n",
       "   (27, [3, 8, 2]),\n",
       "   (25, [3, 7]),\n",
       "   (25, [3, 2, 8]),\n",
       "   (24, [3, 4, 5]),\n",
       "   (23, [5, 4, 7, 2]),\n",
       "   (23, [2, 6]),\n",
       "   (23, [6, 3]),\n",
       "   (22, [2, 8]),\n",
       "   (22, [7, 5, 4, 9]),\n",
       "   (22, [9, 3, 1]),\n",
       "   (21, [5, 4, 7, 9]),\n",
       "   (21, [9, 3]),\n",
       "   (20, [3, 5]),\n",
       "   (20, [9, 1]),\n",
       "   (19, [4, 5, 7, 3]),\n",
       "   (19, [4, 5, 7, 8]),\n",
       "   (18, [9, 1, 3]),\n",
       "   (17, [3, 8, 9]),\n",
       "   (17, [6, 8]),\n",
       "   (16, [3, 6, 8]),\n",
       "   (16, [8, 3]),\n",
       "   (16, [5, 4, 7, 3]),\n",
       "   (16, [3, 1, 2]),\n",
       "   (16, [1, 3, 8]),\n",
       "   (16, [8, 2]),\n",
       "   (16, [9, 7]),\n",
       "   (16, [3, 2, 6]),\n",
       "   (15, [6, 2, 8]),\n",
       "   (14, [3, 1, 8]),\n",
       "   (14, [3, 4, 7]),\n",
       "   (14, [1, 4, 5, 7]),\n",
       "   (14, [7, 9]),\n",
       "   (13, [1, 9, 3]),\n",
       "   (13, [4, 5, 7, 6]),\n",
       "   (13, [7, 5, 4, 1]),\n",
       "   (13, [4, 5, 9]),\n",
       "   (12, [7, 4, 3]),\n",
       "   (12, [6, 8, 2]),\n",
       "   (12, [1, 3, 9]),\n",
       "   (12, [7, 5, 4, 2]),\n",
       "   (12, [7, 5, 4, 3]),\n",
       "   (12, [7, 1]),\n",
       "   (11, [5, 9]),\n",
       "   (11, [2, 3, 8]),\n",
       "   (11, [3, 5, 7]),\n",
       "   (11, [5, 7, 4, 9]),\n",
       "   (11, [4, 9]),\n",
       "   (11, [3, 8, 1]),\n",
       "   (11, [4, 7, 5, 8]),\n",
       "   (11, [4, 3]),\n",
       "   (11, [9, 7, 5, 4]),\n",
       "   (10, [3, 6, 2]),\n",
       "   (10, [3, 8, 9, 1]),\n",
       "   (10, [2, 7]),\n",
       "   (10, [2, 3]),\n",
       "   (10, [2, 4, 5, 7]),\n",
       "   (10, [5, 7, 4, 3]),\n",
       "   (10, [5, 3]),\n",
       "   (10, [4, 5, 7, 1]),\n",
       "   (10, [5, 7, 9]),\n",
       "   (10, [7, 4, 5, 9]),\n",
       "   (10, [3, 9, 7]),\n",
       "   (10, [9, 4]),\n",
       "   (10, [3, 2, 9]),\n",
       "   (9, [3, 5, 4, 7]),\n",
       "   (9, [5, 4, 7, 1]),\n",
       "   (9, [2, 5]),\n",
       "   (9, [9, 1, 7]),\n",
       "   (9, [1, 3, 4]),\n",
       "   (9, [4, 7, 9]),\n",
       "   (9, [4, 1]),\n",
       "   (9, [4, 2]),\n",
       "   (9, [4, 6]),\n",
       "   (9, [7, 4, 5, 1]),\n",
       "   (9, [3, 9, 5]),\n",
       "   (9, [1, 7, 5]),\n",
       "   (9, [6, 2, 3]),\n",
       "   (9, [4, 5, 6, 7]),\n",
       "   (9, [3, 2, 1]),\n",
       "   (8, [5, 3, 7]),\n",
       "   (8, [5, 4, 7, 8]),\n",
       "   (8, [2, 4]),\n",
       "   (8, [9, 5, 4, 7]),\n",
       "   (8, [4, 3, 5, 7]),\n",
       "   (8, [5, 1]),\n",
       "   (8, [3, 2, 6, 8]),\n",
       "   (8, [9, 8]),\n",
       "   (8, [2, 8, 9]),\n",
       "   (8, [4, 7, 5, 3]),\n",
       "   (8, [7, 4, 5, 3]),\n",
       "   (8, [7, 4, 5, 8]),\n",
       "   (8, [6, 3, 2]),\n",
       "   (8, [3, 9, 6]),\n",
       "   (8, [1, 7]),\n",
       "   (8, [1, 8]),\n",
       "   (8, [9, 7, 4]),\n",
       "   (8, [9, 3, 8]),\n",
       "   (8, [9, 7, 1]),\n",
       "   (8, [4, 5, 3]),\n",
       "   (8, [4, 3, 7]),\n",
       "   (8, [3, 9, 4]),\n",
       "   (7, [3, 8, 6]),\n",
       "   (7, [2, 8, 3]),\n",
       "   (7, [7, 4, 5, 2]),\n",
       "   (7, [5, 7, 4, 2]),\n",
       "   (7, [5, 7, 4, 1]),\n",
       "   (7, [3, 1, 8, 9]),\n",
       "   (7, [1, 4]),\n",
       "   (7, [8, 3, 2]),\n",
       "   (7, [3, 5, 4]),\n",
       "   (7, [1, 4, 7]),\n",
       "   (7, [1, 4, 5]),\n",
       "   (7, [1, 4, 5, 9]),\n",
       "   (7, [6, 3, 8]),\n",
       "   (7, [1, 5, 7]),\n",
       "   (6, [1, 9, 8]),\n",
       "   (6, [7, 1, 4]),\n",
       "   (6, [2, 4, 5]),\n",
       "   (6, [2, 4, 7]),\n",
       "   (6, [8, 2, 6]),\n",
       "   (6, [7, 5, 1]),\n",
       "   (6, [5, 9, 7, 4]),\n",
       "   (6, [5, 4, 7, 6]),\n",
       "   (6, [2, 9]),\n",
       "   (6, [2, 3, 4]),\n",
       "   (6, [5, 4, 9, 7]),\n",
       "   (6, [7, 4, 9]),\n",
       "   (6, [3, 2, 4, 5]),\n",
       "   (6, [1, 9, 3, 8]),\n",
       "   (6, [4, 7, 1]),\n",
       "   (6, [4, 7, 3]),\n",
       "   (6, [4, 7, 2]),\n",
       "   (6, [1, 5, 4, 7]),\n",
       "   (6, [7, 5, 4, 8]),\n",
       "   (6, [5, 4, 7, 9, 1]),\n",
       "   (6, [7, 9, 4]),\n",
       "   (6, [3, 4, 9]),\n",
       "   (6, [2, 8, 6]),\n",
       "   (6, [5, 7, 1]),\n",
       "   (6, [3, 9, 8, 1]),\n",
       "   (6, [6, 4]),\n",
       "   (6, [3, 8, 1, 9]),\n",
       "   (6, [7, 9, 5, 4]),\n",
       "   (6, [4, 5, 7, 2, 8]),\n",
       "   (6, [9, 7, 5]),\n",
       "   (6, [1, 4, 9]),\n",
       "   (6, [1, 8, 3]),\n",
       "   (6, [6, 5]),\n",
       "   (6, [7, 3]),\n",
       "   (6, [3, 7, 5]),\n",
       "   (6, [4, 5, 8]),\n",
       "   (6, [4, 5, 6]),\n",
       "   (6, [3, 1, 9, 8]),\n",
       "   (6, [4, 3, 5]),\n",
       "   (6, [3, 7, 9]),\n",
       "   (5, [1, 9, 7]),\n",
       "   (5, [2, 4, 8]),\n",
       "   (5, [2, 6, 8]),\n",
       "   (5, [4, 5, 9, 7]),\n",
       "   (5, [7, 1, 9]),\n",
       "   (5, [5, 1, 7]),\n",
       "   (5, [6, 5, 4]),\n",
       "   (5, [3, 1, 5, 9]),\n",
       "   (5, [4, 1, 5, 7]),\n",
       "   (5, [9, 4, 5, 7]),\n",
       "   (5, [7, 9, 5]),\n",
       "   (5, [9, 1, 8]),\n",
       "   (5, [3, 9, 1, 8]),\n",
       "   (5, [4, 1, 7]),\n",
       "   (5, [5, 4, 3]),\n",
       "   (5, [5, 4, 2]),\n",
       "   (5, [9, 1, 3, 8]),\n",
       "   (5, [6, 8, 2, 3]),\n",
       "   (5, [5, 8]),\n",
       "   (5, [8, 9]),\n",
       "   (5, [3, 4, 8]),\n",
       "   (5, [3, 4, 1]),\n",
       "   (5, [3, 4, 2]),\n",
       "   (5, [4, 7, 5, 9]),\n",
       "   (5, [4, 7, 5, 1]),\n",
       "   (5, [7, 4, 5, 6]),\n",
       "   (5, [7, 3, 4, 5]),\n",
       "   (5, [9, 5]),\n",
       "   (5, [3, 8, 2, 1]),\n",
       "   (5, [3, 5, 1]),\n",
       "   (5, [3, 5, 8]),\n",
       "   (5, [3, 5, 9]),\n",
       "   (5, [6, 1]),\n",
       "   (5, [9, 3, 4]),\n",
       "   (5, [9, 3, 2]),\n",
       "   (5, [1, 3, 9, 8]),\n",
       "   (5, [5, 9, 7]),\n",
       "   (5, [5, 9, 4]),\n",
       "   (5, [1, 7, 5, 4]),\n",
       "   (5, [3, 7, 2]),\n",
       "   (5, [3, 7, 4]),\n",
       "   (5, [9, 1, 4]),\n",
       "   (5, [4, 5, 1]),\n",
       "   (5, [6, 4, 5]),\n",
       "   (5, [3, 1, 2, 8]),\n",
       "   (5, [3, 2, 7]),\n",
       "   (5, [3, 2, 4]),\n",
       "   (5, [3, 5, 7, 4]),\n",
       "   (5, [3, 6, 2, 8]),\n",
       "   (4, [3, 6, 1]),\n",
       "   (4, [7, 5, 6]),\n",
       "   (4, [7, 5, 9]),\n",
       "   (4, [9, 3, 8, 1]),\n",
       "   (4, [7, 5, 4, 1, 9]),\n",
       "   (4, [4, 6, 5]),\n",
       "   (4, [3, 2, 9, 1]),\n",
       "   (4, [4, 2, 5, 7]),\n",
       "   (4, [7, 4, 3, 5]),\n",
       "   (4, [8, 2, 3]),\n",
       "   (4, [3, 8, 9, 2]),\n",
       "   (4, [2, 3, 6]),\n",
       "   (4, [3, 1, 5]),\n",
       "   (4, [3, 1, 6]),\n",
       "   (4, [9, 1, 2]),\n",
       "   (4, [3, 4, 7, 5]),\n",
       "   (4, [7, 4, 1]),\n",
       "   (4, [6, 5, 4, 7]),\n",
       "   (4, [4, 2, 6]),\n",
       "   (4, [4, 7, 8]),\n",
       "   (4, [5, 4, 3, 7]),\n",
       "   (4, [5, 4, 3, 9]),\n",
       "   (4, [9, 5, 7, 4]),\n",
       "   (4, [7, 9, 3]),\n",
       "   (4, [9, 8, 2]),\n",
       "   (4, [3, 9, 8, 2]),\n",
       "   (4, [3, 9, 1, 2]),\n",
       "   (4, [3, 9, 2]),\n",
       "   (4, [1, 5]),\n",
       "   (4, [1, 6]),\n",
       "   (4, [3, 1, 9, 2]),\n",
       "   (4, [9, 1, 5]),\n",
       "   (4, [8, 6]),\n",
       "   (4, [9, 5, 3]),\n",
       "   (4, [5, 9, 1]),\n",
       "   (4, [7, 1, 3]),\n",
       "   (4, [4, 3, 2]),\n",
       "   (4, [9, 6]),\n",
       "   (4, [3, 2, 5]),\n",
       "   (4, [5, 2]),\n",
       "   (4, [4, 9, 7]),\n",
       "   (3, [3, 6, 9]),\n",
       "   (3, [9, 4, 7]),\n",
       "   (3, [9, 4, 5]),\n",
       "   (3, [7, 5, 8]),\n",
       "   (3, [1, 9, 4]),\n",
       "   (3, [1, 8, 3, 2]),\n",
       "   (3, [7, 1, 5]),\n",
       "   (3, [2, 4, 3]),\n",
       "   (3, [4, 5, 7, 3, 2, 1, 8, 9, 6]),\n",
       "   (3, [2, 6, 3]),\n",
       "   (3, [3, 7, 5, 4]),\n",
       "   (3, [7, 5, 9, 4]),\n",
       "   (3, [9, 7, 3]),\n",
       "   (3, [6, 2, 8, 3]),\n",
       "   (3, [6, 7, 5]),\n",
       "   (3, [6, 7, 2]),\n",
       "   (3, [4, 5, 7, 8, 2]),\n",
       "   (3, [7, 3, 2]),\n",
       "   (3, [4, 5, 1, 7]),\n",
       "   (3, [5, 3, 4]),\n",
       "   (3, [3, 9, 5, 4]),\n",
       "   (3, [5, 6, 4, 7]),\n",
       "   (3, [5, 4, 7, 2, 3]),\n",
       "   (3, [9, 8, 3]),\n",
       "   (3, [1, 7, 9]),\n",
       "   (3, [9, 1, 4, 5, 7]),\n",
       "   (3, [2, 6, 5, 4]),\n",
       "   (3, [8, 5, 3, 2]),\n",
       "   (3, [3, 2, 8, 1]),\n",
       "   (3, [3, 1, 8, 2]),\n",
       "   (3, [5, 2, 8]),\n",
       "   (3, [4, 1, 3]),\n",
       "   (3, [3, 9, 1, 2, 8]),\n",
       "   (3, [1, 7, 4]),\n",
       "   (3, [4, 9, 1]),\n",
       "   (3, [4, 3, 7, 5]),\n",
       "   (3, [1, 3, 7]),\n",
       "   (3, [5, 4, 9]),\n",
       "   (3, [5, 4, 8]),\n",
       "   (3, [5, 4, 6]),\n",
       "   (3, [5, 4, 1]),\n",
       "   (3, [9, 1, 3, 7]),\n",
       "   (3, [5, 7, 4, 8]),\n",
       "   (3, [3, 8, 2, 4]),\n",
       "   (3, [6, 3, 2, 8]),\n",
       "   (3, [6, 8, 3]),\n",
       "   (3, [5, 8, 7]),\n",
       "   (3, [6, 8, 2, 9]),\n",
       "   (3, [5, 6]),\n",
       "   (3, [3, 8, 4]),\n",
       "   (3, [7, 3, 8]),\n",
       "   (3, [9, 3, 7]),\n",
       "   (3, [2, 8, 5]),\n",
       "   (3, [3, 4, 5, 7, 8]),\n",
       "   (3, [4, 8, 2]),\n",
       "   (3, [9, 2]),\n",
       "   (3, [5, 7, 3]),\n",
       "   (3, [5, 9, 3]),\n",
       "   (3, [4, 7, 5, 2]),\n",
       "   (3, [6, 3, 9]),\n",
       "   (3, [1, 5, 9]),\n",
       "   (3, [5, 7, 4, 8, 2]),\n",
       "   (3, [3, 9, 1, 6]),\n",
       "   (3, [9, 4, 7, 5]),\n",
       "   (3, [3, 9, 2, 8]),\n",
       "   (3, [1, 2, 8]),\n",
       "   (3, [9, 3, 5]),\n",
       "   (3, [3, 4, 5, 1]),\n",
       "   (3, [7, 6]),\n",
       "   (3, [4, 9, 5]),\n",
       "   (3, [3, 7, 1]),\n",
       "   (3, [3, 7, 8]),\n",
       "   (3, [9, 5, 4]),\n",
       "   (3, [6, 2, 4]),\n",
       "   (3, [9, 5, 7]),\n",
       "   (3, [3, 6, 4, 2]),\n",
       "   (3, [4, 3, 1]),\n",
       "   (3, [4, 3, 8]),\n",
       "   (3, [4, 5, 7, 3, 2, 6, 8, 1, 9]),\n",
       "   (3, [5, 4, 7, 3, 2]),\n",
       "   (3, [1, 7, 4, 5]),\n",
       "   (3, [4, 5, 2, 7]),\n",
       "   (3, [3, 1, 2, 6]),\n",
       "   (3, [5, 6, 4]),\n",
       "   (3, [3, 2, 8, 6]),\n",
       "   (3, [5, 3, 8]),\n",
       "   (3, [6, 3, 1]),\n",
       "   (3, [6, 5, 1]),\n",
       "   (3, [3, 8, 6, 2]),\n",
       "   (3, [6, 4, 7]),\n",
       "   (3, [3, 7, 4, 5]),\n",
       "   (2, [2, 5, 7, 4]),\n",
       "   (2, [3, 6, 7]),\n",
       "   (2, [3, 6, 4]),\n",
       "   (2, [5, 3, 4, 7]),\n",
       "   (2, [7, 5, 2]),\n",
       "   (2, [7, 5, 3]),\n",
       "   (2, [3, 9, 6, 8]),\n",
       "   (2, [2, 6, 5]),\n",
       "   (2, [9, 4, 1]),\n",
       "   (2, [9, 8, 1]),\n",
       "   (2, [3, 4, 8, 2]),\n",
       "   (2, [5, 2, 7]),\n",
       "   (2, [4, 9, 5, 7]),\n",
       "   (2, [3, 2, 6, 1, 8, 9, 4, 5, 7]),\n",
       "   (2, [6, 2, 8, 4]),\n",
       "   (2, [4, 3, 8, 9]),\n",
       "   (2, [5, 3, 2, 8]),\n",
       "   (2, [6, 5, 7, 4]),\n",
       "   (2, [3, 6, 2, 8, 5, 4, 7, 9, 1]),\n",
       "   (2, [3, 5, 4, 8]),\n",
       "   (2, [8, 5, 4]),\n",
       "   (2, [6, 5, 4, 2]),\n",
       "   (2, [7, 4, 9, 1]),\n",
       "   (2, [8, 2, 1, 9]),\n",
       "   (2, [3, 1, 6, 2, 4, 5, 7, 9, 8]),\n",
       "   (2, [3, 9, 4, 7]),\n",
       "   (2, [4, 7, 1, 9]),\n",
       "   (2, [5, 1, 3]),\n",
       "   (2, [5, 3, 2]),\n",
       "   (2, [5, 1, 4]),\n",
       "   (2, [7, 3, 5]),\n",
       "   (2, [3, 8, 2, 1, 9]),\n",
       "   (2, [1, 9, 5, 4, 7]),\n",
       "   (2, [4, 7, 1, 3]),\n",
       "   (2, [8, 9, 3]),\n",
       "   (2, [9, 1, 3, 5]),\n",
       "   (2, [4, 1, 7, 9]),\n",
       "   (2, [5, 4, 7, 6, 2]),\n",
       "   (2, [2, 4, 5, 7, 8]),\n",
       "   (2, [2, 3, 8, 1]),\n",
       "   (2, [6, 1, 8]),\n",
       "   (2, [9, 2, 3, 4, 7, 8, 6, 1, 5]),\n",
       "   (2, [5, 4, 7, 2, 6]),\n",
       "   (2, [9, 1, 8, 2]),\n",
       "   (2, [4, 5, 7, 2, 3, 8, 6, 9, 1]),\n",
       "   (2, [9, 7, 3, 1]),\n",
       "   (2, [5, 1, 8, 9, 6, 4, 3, 2]),\n",
       "   (2, [6, 4, 5, 7]),\n",
       "   (2, [1, 3, 5, 7]),\n",
       "   (2, [5, 1, 9]),\n",
       "   (2, [5, 4, 3, 8]),\n",
       "   (2, [3, 1, 2, 8, 6, 4, 5, 7, 9]),\n",
       "   (2, [5, 7, 4, 6, 2, 3, 8, 1, 9]),\n",
       "   (2, [9, 8, 3, 1]),\n",
       "   (2, [9, 7, 1, 3]),\n",
       "   (2, [2, 3, 9]),\n",
       "   (2, [2, 3, 1]),\n",
       "   (2, [7, 5, 4, 2, 8]),\n",
       "   (2, [7, 5, 6, 4]),\n",
       "   (2, [3, 1, 7]),\n",
       "   (2, [9, 8, 1, 3]),\n",
       "   (2, [1, 3, 9, 8, 2]),\n",
       "   (2, [3, 9, 8, 2, 1, 6, 5, 4, 7]),\n",
       "   (2, [6, 2, 9]),\n",
       "   (2, [3, 9, 2, 1]),\n",
       "   (2, [2, 9, 4, 5, 7]),\n",
       "   (2, [6, 4, 2]),\n",
       "   (2, [9, 1, 5, 7]),\n",
       "   (2, [6, 1, 3]),\n",
       "   (2, [3, 4, 5, 7, 2, 1, 6, 8, 9]),\n",
       "   (2, [3, 2, 4, 6]),\n",
       "   (2, [6, 8, 1]),\n",
       "   (2, [8, 2, 5]),\n",
       "   (2, [8, 2, 1]),\n",
       "   (2, [5, 1, 3, 7]),\n",
       "   (2, [5, 1, 3, 4]),\n",
       "   (2, [5, 1, 3, 2]),\n",
       "   (2, [8, 6, 2]),\n",
       "   (2, [5, 4, 7, 3, 1, 2, 6, 8, 9]),\n",
       "   (2, [5, 7, 4, 3, 9, 1, 2, 8, 6]),\n",
       "   (2, [7, 5, 4, 1, 9, 8, 6, 3, 2]),\n",
       "   (2, [4, 3, 7, 9]),\n",
       "   (2, [6, 2, 5, 4]),\n",
       "   (2, [6, 2, 5, 7]),\n",
       "   (2, [7, 4, 2]),\n",
       "   (2, [3, 2, 4, 9]),\n",
       "   (2, [2, 5, 4]),\n",
       "   (2, [4, 2, 8]),\n",
       "   (2, [4, 5, 7, 9, 1, 2, 8, 3, 6]),\n",
       "   (2, [1, 3, 5]),\n",
       "   (2, [3, 9, 7, 1]),\n",
       "   (2, [4, 5, 7, 3, 2]),\n",
       "   (2, [4, 5, 7, 3, 1]),\n",
       "   (2, [4, 5, 7, 2, 1, 8, 6, 3, 9]),\n",
       "   (2, [3, 9, 1, 8, 2, 4, 7, 5, 6]),\n",
       "   (2, [1, 3, 2]),\n",
       "   (2, [4, 8, 7, 5]),\n",
       "   (2, [3, 5, 4, 7, 8, 2, 6, 9, 1]),\n",
       "   (2, [1, 3, 2, 8]),\n",
       "   (2, [2, 4, 5, 3]),\n",
       "   (2, [4, 8]),\n",
       "   (2, [8, 3, 6]),\n",
       "   (2, [4, 5, 7, 9, 3]),\n",
       "   (2, [3, 4, 5, 7, 8, 1, 2, 6, 9]),\n",
       "   (2, [7, 5, 4, 3, 2, 8, 9, 6, 1]),\n",
       "   (2, [3, 9, 4, 5, 7, 1, 2, 6, 8]),\n",
       "   (2, [3, 2, 7, 5, 4]),\n",
       "   (2, [3, 2, 8, 9, 1, 6]),\n",
       "   (2, [7, 5, 4, 9, 1]),\n",
       "   (2, [9, 6, 4]),\n",
       "   (2, [3, 8, 5]),\n",
       "   (2, [3, 8, 7]),\n",
       "   (2, [4, 3, 5, 7, 9]),\n",
       "   (2, [2, 8, 1]),\n",
       "   (2, [5, 9, 1, 4, 7]),\n",
       "   (2, [9, 1, 3, 8, 2]),\n",
       "   (2, [1, 2, 4, 7]),\n",
       "   (2, [1, 2, 4, 5]),\n",
       "   (2, [7, 9, 2]),\n",
       "   (2, [7, 5, 2, 8]),\n",
       "   (2, [3, 6, 9, 2, 1, 8, 7, 5, 4]),\n",
       "   (2, [4, 2, 5]),\n",
       "   (2, [3, 4, 6]),\n",
       "   (2, [5, 9, 3, 8]),\n",
       "   (2, [2, 8, 4]),\n",
       "   (2, [7, 4, 5, 1, 9]),\n",
       "   (2, [4, 8, 6]),\n",
       "   (2, [1, 9, 4, 5, 7]),\n",
       "   (2, [4, 2, 7]),\n",
       "   (2, [3, 4, 7, 8, 9]),\n",
       "   (2, [5, 7, 8]),\n",
       "   (2, [5, 7, 6]),\n",
       "   (2, [9, 3, 2, 8]),\n",
       "   (2, [4, 5, 7, 6, 2]),\n",
       "   (2, [3, 9, 8, 4]),\n",
       "   (2, [3, 2, 4, 5, 7, 9, 8, 1, 6]),\n",
       "   (2, [5, 9, 2]),\n",
       "   (2, [4, 7, 5, 6]),\n",
       "   (2, [3, 1, 4]),\n",
       "   (2, [9, 8, 4]),\n",
       "   (2, [9, 5, 4, 3]),\n",
       "   (2, [6, 5, 7]),\n",
       "   (2, [6, 5, 3]),\n",
       "   (2, [8, 1, 5]),\n",
       "   (2, [8, 1, 2]),\n",
       "   (2, [8, 1, 9]),\n",
       "   (2, [7, 8, 9]),\n",
       "   (2, [6, 7]),\n",
       "   (2, [9, 4, 7, 3]),\n",
       "   (2, [9, 4, 7, 1]),\n",
       "   (2, [5, 4, 7, 8, 2]),\n",
       "   (2, [2, 7, 5, 4]),\n",
       "   (2, [3, 5, 2]),\n",
       "   (2, [1, 2, 4]),\n",
       "   (2, [1, 4, 8]),\n",
       "   (2, [7, 1, 9, 4, 5]),\n",
       "   (2, [1, 2, 3]),\n",
       "   (2, [1, 4, 3]),\n",
       "   (2, [4, 5, 7, 9, 8, 6, 3, 2, 1]),\n",
       "   (2, [9, 3, 6]),\n",
       "   (2, [7, 5, 4, 8, 2]),\n",
       "   (2, [3, 8, 7, 4]),\n",
       "   (2, [1, 8, 9]),\n",
       "   (2, [9, 7, 8]),\n",
       "   (2, [8, 4]),\n",
       "   (2, [8, 5]),\n",
       "   (2, [3, 1, 9, 8, 7, 5, 4, 2, 6]),\n",
       "   (2, [1, 5, 3, 7]),\n",
       "   (2, [4, 5, 7, 6, 2, 8, 9, 1, 3]),\n",
       "   (2, [2, 8, 7, 5, 4, 3, 9, 1, 6]),\n",
       "   (2, [9, 3, 1, 8]),\n",
       "   (2, [3, 4, 5, 2]),\n",
       "   (2, [3, 4, 5, 6]),\n",
       "   (2, [1, 4, 5, 6]),\n",
       "   (2, [3, 4, 5, 8]),\n",
       "   (2, [3, 4, 5, 9]),\n",
       "   (2, [6, 9]),\n",
       "   (2, [5, 9, 4, 7]),\n",
       "   (2, [7, 9, 4, 5]),\n",
       "   (2, [6, 7, 2, 4]),\n",
       "   (2, [4, 3, 1, 6]),\n",
       "   (2, [9, 1, 8, 3, 2, 4, 5, 7, 6]),\n",
       "   (2, [7, 2, 9]),\n",
       "   (2, [7, 2, 8]),\n",
       "   (2, [1, 9, 7, 8]),\n",
       "   (2, [7, 8]),\n",
       "   (2, [4, 9, 3]),\n",
       "   (2, [8, 9, 1, 3]),\n",
       "   (2, [1, 9, 5, 3, 4]),\n",
       "   (2, [6, 2, 4, 5, 7, 8, 3, 1, 9]),\n",
       "   (2, [3, 9, 8, 1, 2, 6, 7, 5, 4]),\n",
       "   (2, [9, 5, 1]),\n",
       "   (2, [6, 2, 1]),\n",
       "   (2, [8, 2, 9]),\n",
       "   (2, [9, 1, 7, 5]),\n",
       "   (2, [5, 4, 7, 1, 9]),\n",
       "   (2, [4, 5, 8, 3]),\n",
       "   (2, [8, 3, 1, 9]),\n",
       "   (2, [8, 4, 2]),\n",
       "   (2, [4, 3, 9]),\n",
       "   (2, [3, 6, 8, 2]),\n",
       "   (2, [3, 6, 8, 9]),\n",
       "   (2, [7, 9, 1]),\n",
       "   (2, [3, 1, 2, 9]),\n",
       "   (2, [4, 5, 7, 1, 3]),\n",
       "   (2, [2, 8, 7, 5]),\n",
       "   (2, [1, 3, 4, 5]),\n",
       "   (2, [4, 7, 2, 1]),\n",
       "   (2, [6, 5, 4, 7, 2]),\n",
       "   (2, [7, 4, 5, 3, 2, 8, 6, 1, 9]),\n",
       "   (2, [9, 3, 1, 2]),\n",
       "   (2, [9, 3, 1, 5]),\n",
       "   (2, [5, 7, 4, 9, 1]),\n",
       "   (2, [9, 7, 4, 5]),\n",
       "   (1, [5, 1, 2, 4]),\n",
       "   (1, [3, 2, 5, 4, 6, 7, 8, 1, 9]),\n",
       "   (1, [5, 1, 2, 3]),\n",
       "   (1, [5, 4, 8, 7, 9, 1, 2, 3, 6]),\n",
       "   (1, [4, 2, 3, 9]),\n",
       "   (1, [3, 4, 2, 8]),\n",
       "   (1, [3, 2, 4, 5, 7, 8, 1]),\n",
       "   (1, [1, 4, 5, 7, 6, 3, 2, 8, 9]),\n",
       "   (1, [3, 6, 5]),\n",
       "   (1, [3, 4, 5, 7, 2, 8, 1, 9, 6]),\n",
       "   (1, [6, 3, 9, 8, 7, 2]),\n",
       "   (1, [9, 4, 3]),\n",
       "   (1, [3, 5, 4, 7, 8, 2, 6, 1, 9]),\n",
       "   (1, [2, 3, 4, 8, 5, 7]),\n",
       "   (1, [3, 8, 6, 9, 1]),\n",
       "   (1, [8, 1, 3, 2, 6, 9, 4, 5, 7]),\n",
       "   (1, [5, 4, 7, 9, 8, 1, 2, 6]),\n",
       "   (1, [5, 4, 7, 6, 8, 2, 3, 1, 9]),\n",
       "   (1, [7, 4, 5, 9, 8, 3, 1]),\n",
       "   (1, [3, 1, 5, 4, 7]),\n",
       "   (1, [1, 4, 3, 9, 8, 7, 6, 5, 2]),\n",
       "   (1, [4, 5, 7, 2, 6]),\n",
       "   (1, [3, 5, 4, 7, 9]),\n",
       "   (1, [3, 5, 4, 7, 8]),\n",
       "   (1, [3, 2, 4, 5, 6, 7, 8, 1, 9]),\n",
       "   (1, [5, 7, 4, 1, 9, 3, 2, 6, 8]),\n",
       "   (1, [3, 5, 4, 7, 2]),\n",
       "   (1, [1, 7, 8, 6, 4, 5, 2, 9, 3]),\n",
       "   (1, [2, 8, 5, 3]),\n",
       "   (1, [1, 3, 4, 5, 7, 9, 2, 8, 6]),\n",
       "   (1, [1, 9, 2, 3]),\n",
       "   (1, [5, 4, 8, 2]),\n",
       "   (1, [1, 9, 2, 4]),\n",
       "   (1, [4, 2, 3, 8]),\n",
       "   (1, [1, 9, 6]),\n",
       "   (1, [1, 9, 5]),\n",
       "   (1, [2, 4, 5, 8, 7, 9, 1, 3]),\n",
       "   (1, [3, 4, 5, 2, 1, 8, 7, 9, 6]),\n",
       "   (1, [1, 4, 9, 7, 2, 8, 3, 5]),\n",
       "   (1, [5, 7, 3, 4]),\n",
       "   (1, [4, 7, 5, 8, 9, 1]),\n",
       "   (1, [2, 4, 9]),\n",
       "   (1, [1, 3, 4, 7, 5, 9, 8, 2, 6]),\n",
       "   (1, [2, 4, 6]),\n",
       "   (1, [3, 8, 4, 7]),\n",
       "   (1, [1, 5, 7, 4, 3, 2, 8, 9, 6]),\n",
       "   (1, [7, 4, 5, 3, 9, 2, 1, 8, 6]),\n",
       "   (1, [2, 8, 9, 6, 5, 4, 3, 1]),\n",
       "   (1, [3, 8, 4, 1]),\n",
       "   (1, [5, 4, 8, 1, 3, 2, 7, 9, 6]),\n",
       "   (1, [4, 1, 2, 9]),\n",
       "   (1, [5, 7, 6, 3]),\n",
       "   (1, [4, 1, 2, 7]),\n",
       "   (1, [1, 6, 2, 7]),\n",
       "   (1, [6, 3, 2, 1, 8, 9, 7, 4, 5]),\n",
       "   (1, [5, 7, 9, 8, 4, 1, 2, 3, 6]),\n",
       "   (1, [3, 9, 2, 1, 8, 7, 5, 4, 6]),\n",
       "   (1, [7, 5, 4, 9, 2, 3, 1, 6, 8]),\n",
       "   (1, [7, 3, 4, 5, 9, 1, 8, 2, 6]),\n",
       "   (1, [5, 7, 8, 6, 3, 2]),\n",
       "   (1, [1, 6, 3, 8, 2, 5]),\n",
       "   (1, [5, 7, 4, 2, 8, 1]),\n",
       "   (1, [1, 9, 3, 5, 7, 4, 2, 8, 6]),\n",
       "   (1, [4, 7, 5, 3, 2, 8, 9, 1, 6]),\n",
       "   (1, [5, 7, 4, 2, 8, 6]),\n",
       "   (1, [5, 7, 9, 8, 4, 2, 3, 1]),\n",
       "   (1, [7, 2, 3, 6]),\n",
       "   (1, [2, 8, 3, 6, 1, 4, 5, 7, 9]),\n",
       "   (1, [2, 6, 7]),\n",
       "   (1, [4, 1, 3, 5, 7, 9, 2, 8, 6]),\n",
       "   (1, [5, 9, 1, 3]),\n",
       "   (1, [7, 5, 8, 4]),\n",
       "   (1, [3, 4, 1, 5, 6, 7, 8, 9, 2]),\n",
       "   (1, [8, 3, 6, 4, 5, 7]),\n",
       "   (1, [7, 3, 1, 4, 5, 2, 6, 9, 8]),\n",
       "   (1, [1, 9, 8, 2]),\n",
       "   (1, [4, 5, 7, 2, 3, 8]),\n",
       "   (1, [9, 7, 3, 2]),\n",
       "   (1, [4, 5, 7, 8, 9, 6, 3, 2, 1]),\n",
       "   (1, [9, 8, 7]),\n",
       "   (1, [4, 5, 7, 1, 9, 8, 3, 2, 6]),\n",
       "   (1, [4, 7, 9, 5, 1]),\n",
       "   (1, [3, 5, 7, 2, 6, 8]),\n",
       "   (1, [3, 9, 1, 4, 5, 2, 6, 7, 8]),\n",
       "   (1, [7, 3, 2, 9]),\n",
       "   (1, [7, 3, 2, 4]),\n",
       "   (1, [1, 8, 9, 3, 2]),\n",
       "   (1, [9, 4, 6]),\n",
       "   (1, [2, 1, 3, 9]),\n",
       "   (1, [3, 4, 8, 5]),\n",
       "   (1, [5, 4, 7, 6, 8, 2, 1, 9, 3]),\n",
       "   (1, [4, 9, 5, 1]),\n",
       "   (1, [5, 4, 3, 2, 8]),\n",
       "   (1, [5, 4, 7, 3, 2, 1, 8, 9, 6]),\n",
       "   (1, [2, 4, 5, 3, 7, 8, 6, 1, 9]),\n",
       "   (1, [7, 4, 5, 8, 2, 1, 3, 9, 6]),\n",
       "   (1, [3, 4, 8, 9, 5, 7, 1, 2, 6]),\n",
       "   (1, [6, 9, 3, 2]),\n",
       "   (1, [6, 8, 7, 4, 2, 5, 3]),\n",
       "   (1, [5, 3, 4, 7, 9, 1, 2, 8, 6]),\n",
       "   (1, [7, 5, 4, 1, 9, 3, 6, 2, 8]),\n",
       "   (1, [9, 5, 3, 8]),\n",
       "   (1, [3, 8, 4, 7, 2]),\n",
       "   (1, [9, 1, 3, 2, 4, 8, 5, 7, 6]),\n",
       "   (1, [1, 4, 5, 7, 6, 2, 8, 9, 3]),\n",
       "   (1, [8, 3, 2, 1, 5, 4, 7, 6, 9]),\n",
       "   (1, [1, 8, 9, 4, 5, 7]),\n",
       "   (1, [1, 3, 5, 4, 7, 8, 2, 9, 6]),\n",
       "   (1, [4, 5, 9, 1, 7]),\n",
       "   (1, [8, 2, 3, 9]),\n",
       "   (1, [6, 8, 4, 5, 2, 7, 3, 1, 9]),\n",
       "   (1, [4, 7, 5, 3, 8, 9, 6, 2, 1]),\n",
       "   (1, [1, 8, 9, 4]),\n",
       "   (1, [5, 4, 7, 2, 6, 1, 8]),\n",
       "   (1, [3, 4, 7, 1, 5]),\n",
       "   (1, [3, 8, 7, 5, 4, 2, 6]),\n",
       "   (1, [2, 6, 8, 4]),\n",
       "   (1, [2, 6, 8, 7]),\n",
       "   (1, [8, 9, 5]),\n",
       "   (1, [4, 3, 6, 7]),\n",
       "   (1, [3, 9, 4, 1, 7, 5]),\n",
       "   (1, [2, 4, 5, 7, 8, 1, 3, 6, 9]),\n",
       "   (1, [7, 4, 5, 9, 3]),\n",
       "   (1, [6, 7, 5, 4, 8, 3, 2, 1, 9]),\n",
       "   (1, [8, 1, 3, 2]),\n",
       "   (1, [3, 9, 8, 2, 1]),\n",
       "   (1, [2, 4, 7, 8, 5]),\n",
       "   (1, [7, 4, 5, 2, 6]),\n",
       "   (1, [8, 5, 4, 7, 3, 6, 2, 1, 9]),\n",
       "   (1, [4, 7, 5, 3, 1, 2, 8, 6, 9]),\n",
       "   (1, [5, 4, 6, 7]),\n",
       "   (1, [5, 4, 6, 3]),\n",
       "   (1, [9, 5, 3, 7, 8]),\n",
       "   (1, [9, 3, 5, 4, 7]),\n",
       "   (1, [5, 7, 4, 3, 2, 8, 6]),\n",
       "   (1, [2, 9, 8, 7, 1, 3, 4, 5, 6]),\n",
       "   (1, [4, 6, 8]),\n",
       "   (1, [4, 6, 9]),\n",
       "   (1, [4, 5, 7, 6, 2, 8]),\n",
       "   (1, [6, 9, 8]),\n",
       "   (1, [9, 1, 6, 7, 5, 4, 8, 2, 3]),\n",
       "   (1, [3, 9, 8, 4, 7]),\n",
       "   (1, [5, 6, 2, 4]),\n",
       "   (1, [5, 6, 2, 7]),\n",
       "   (1, [1, 7, 4, 5, 2]),\n",
       "   (1, [4, 7, 5, 9, 3, 1]),\n",
       "   (1, [1, 3, 7, 9, 5, 4, 2, 6, 8]),\n",
       "   (1, [6, 4, 2, 7]),\n",
       "   (1, [3, 1, 5, 6, 7, 8, 9, 4, 2]),\n",
       "   (1, [5, 4, 2, 7]),\n",
       "   (1, [8, 7, 5, 2, 4, 3, 6, 1, 9]),\n",
       "   (1, [7, 9, 2, 1, 5, 3, 4, 6, 8]),\n",
       "   (1, [3, 4, 1, 7, 6, 5, 8, 2, 9]),\n",
       "   (1, [4, 5, 7, 2, 1, 3, 6, 8, 9]),\n",
       "   (1, [4, 9, 3, 7, 2]),\n",
       "   (1, [1, 9, 4, 6]),\n",
       "   (1, [5, 1, 3, 7, 4, 2, 8, 9, 6]),\n",
       "   (1, [9, 3, 2, 7, 4, 5, 6, 8, 1]),\n",
       "   (1, [3, 7, 5, 4, 9, 1, 2, 8, 6]),\n",
       "   (1, [5, 2, 7, 4, 8, 1, 6, 9, 3]),\n",
       "   (1, [5, 7, 4, 9, 2, 3, 6, 8, 1]),\n",
       "   (1, [3, 8, 5, 4]),\n",
       "   (1, [9, 4, 5, 7, 8, 2, 3, 6, 1]),\n",
       "   (1, [5, 3, 2, 6]),\n",
       "   (1, [3, 9, 8, 6, 5, 1, 7, 4, 2]),\n",
       "   (1, [2, 4, 5, 7, 9, 8]),\n",
       "   (1, [3, 2, 9, 8]),\n",
       "   (1, [9, 1, 4, 5]),\n",
       "   (1, [3, 4, 5, 7, 1, 8, 9, 2, 6]),\n",
       "   (1, [9, 1, 4, 7]),\n",
       "   (1, [6, 9, 4]),\n",
       "   (1, [4, 6, 3, 2]),\n",
       "   (1, [8, 3, 9, 2]),\n",
       "   (1, [4, 7, 5, 2, 1, 6, 8, 3, 9]),\n",
       "   (1, [5, 7, 4, 2, 3]),\n",
       "   (1, [7, 4, 3, 5, 1]),\n",
       "   (1, [6, 7, 8]),\n",
       "   (1, [5, 4, 7, 9, 1, 3, 2, 6, 8]),\n",
       "   (1, [6, 7, 3]),\n",
       "   (1, [5, 7, 4, 2, 8]),\n",
       "   (1, [9, 1, 7, 8]),\n",
       "   (1, [3, 1, 8, 2, 9, 6, 7, 5, 4]),\n",
       "   (1, [7, 5, 4, 3, 8, 9, 1]),\n",
       "   (1, [1, 9, 8, 3, 2]),\n",
       "   (1, [6, 5, 7, 2]),\n",
       "   (1, [4, 5, 7, 2, 8, 9, 6, 1, 3]),\n",
       "   (1, [4, 5, 7, 8, 9]),\n",
       "   (1, [8, 4, 5, 2, 1, 7]),\n",
       "   (1, [4, 3, 6, 2]),\n",
       "   (1, [3, 5, 2, 8]),\n",
       "   (1, [3, 9, 8, 2, 1, 6, 7, 5, 4]),\n",
       "   (1, [4, 5, 7, 9, 6, 8, 2, 1, 3]),\n",
       "   (1, [4, 5, 7, 8, 3]),\n",
       "   (1, [3, 4, 7, 5, 6, 1, 2, 8, 9]),\n",
       "   (1, [9, 3, 1, 8, 2]),\n",
       "   (1, [9, 8, 4, 3]),\n",
       "   (1, [9, 7, 4, 5, 3, 2, 8, 6, 1]),\n",
       "   (1, [6, 5, 3, 2]),\n",
       "   (1, [5, 7, 4, 8, 9, 1, 2, 3, 6]),\n",
       "   (1, [4, 9, 1, 5, 7]),\n",
       "   (1, [2, 4, 5, 7, 3, 1, 6, 8, 9]),\n",
       "   (1, [2, 1, 7, 5, 4]),\n",
       "   (1, [3, 1, 9, 2, 7, 5, 4, 8, 6]),\n",
       "   (1, [9, 3, 2, 7, 1]),\n",
       "   (1, [4, 2, 1, 9]),\n",
       "   (1, [5, 9, 7, 8]),\n",
       "   (1, [8, 4, 7, 3]),\n",
       "   (1, [3, 5, 7, 8]),\n",
       "   (1, [2, 3, 4, 5, 6, 7, 8, 1, 9]),\n",
       "   (1, [7, 9, 4, 5, 2, 8, 3, 1, 6]),\n",
       "   (1, [4, 5, 9, 7, 3, 8, 6, 1, 2]),\n",
       "   (1, [3, 9, 1, 2, 6, 8]),\n",
       "   (1, [2, 4, 8, 7]),\n",
       "   (1, [2, 4, 8, 3]),\n",
       "   (1, [7, 4, 5, 9, 2, 6, 8, 1, 3]),\n",
       "   (1, [3, 4, 5, 7, 9, 2]),\n",
       "   (1, [3, 4, 5, 7, 9, 6]),\n",
       "   (1, [8, 3, 6, 2]),\n",
       "   (1, [9, 4, 5, 1, 3, 2, 6, 8, 7]),\n",
       "   (1, [5, 4, 7, 8, 2, 6, 3, 1, 9]),\n",
       "   (1, [6, 9, 7, 3, 2, 4, 1, 5, 8]),\n",
       "   (1, [4, 9, 6, 5, 2, 1, 3, 7, 8]),\n",
       "   (1, [3, 7, 9, 4, 1, 8]),\n",
       "   (1, [8, 3, 9]),\n",
       "   (1, [3, 4, 5, 1, 2, 6, 7, 8, 9]),\n",
       "   (1, [9, 3, 1, 4, 5, 7]),\n",
       "   (1, [3, 9, 1, 2, 8, 7, 4, 5, 6]),\n",
       "   (1, [9, 8, 7, 5, 4, 3, 2, 1, 6]),\n",
       "   (1, [3, 2, 5, 4]),\n",
       "   (1, [7, 3, 1]),\n",
       "   (1, [3, 2, 5, 7]),\n",
       "   (1, [7, 4, 5, 2, 8, 1, 9, 3, 6]),\n",
       "   (1, [8, 5, 2]),\n",
       "   (1, [8, 5, 1]),\n",
       "   (1, [4, 5, 7, 1, 8, 2, 3, 9, 6]),\n",
       "   (1, [8, 5, 7]),\n",
       "   (1, [3, 8, 9, 1, 2]),\n",
       "   (1, [5, 9, 7, 4, 3, 2, 1, 8, 6]),\n",
       "   (1, [4, 5, 7, 6, 2, 3, 9, 8, 1]),\n",
       "   (1, [4, 7, 1, 2, 5]),\n",
       "   (1, [9, 3, 5, 8, 1, 7, 4, 2, 6]),\n",
       "   (1, [3, 2, 1, 9]),\n",
       "   (1, [3, 1, 6, 8, 9, 2, 4, 5, 7]),\n",
       "   (1, [3, 1, 7, 5]),\n",
       "   (1, [3, 1, 7, 4]),\n",
       "   (1, [7, 4, 5, 6, 3, 2, 8, 9, 1]),\n",
       "   (1, [3, 1, 9, 8, 2, 4, 5, 7, 6]),\n",
       "   (1, [6, 4, 5, 3, 2]),\n",
       "   (1, [3, 2, 1, 6]),\n",
       "   (1, [3, 1, 7, 2]),\n",
       "   (1, [7, 4, 9, 5, 1, 2, 3, 8, 6]),\n",
       "   (1, [4, 9, 5, 7, 8, 2, 1, 3, 6]),\n",
       "   (1, [2, 4, 5, 8, 6, 7]),\n",
       "   (1, [6, 2, 4, 8]),\n",
       "   (1, [7, 5, 2, 4, 1, 3, 6, 8, 9]),\n",
       "   (1, [5, 6, 8, 7]),\n",
       "   (1, [7, 4, 9, 5]),\n",
       "   (1, [4, 5, 6, 7, 9]),\n",
       "   (1, [3, 8, 1, 9, 6, 2]),\n",
       "   (1, [3, 1, 6, 2, 9]),\n",
       "   (1, [7, 5, 4, 2, 8, 1, 6, 3, 9]),\n",
       "   (1, [3, 1, 9, 5, 4, 7, 6, 8, 2]),\n",
       "   (1, [9, 4, 3, 1, 2, 5, 6, 7, 8]),\n",
       "   (1, [3, 9, 4, 6]),\n",
       "   (1, [1, 3, 2, 9, 5, 7, 8, 4, 6]),\n",
       "   (1, [1, 4, 5, 7, 9]),\n",
       "   (1, [1, 4, 5, 7, 8]),\n",
       "   (1, [3, 6, 7, 9, 8, 5, 4, 2, 1]),\n",
       "   (1, [3, 2, 1, 8, 4, 5, 7, 9, 6]),\n",
       "   (1, [3, 9, 4, 8]),\n",
       "   (1, [3, 8, 4, 5, 6, 7, 9, 2, 1]),\n",
       "   (1, [1, 5, 9, 4, 7, 3, 2, 8]),\n",
       "   (1, [3, 1, 4, 8]),\n",
       "   (1, [3, 1, 8, 5, 4, 2, 6, 7, 9]),\n",
       "   (1, [5, 4, 8, 9, 7, 1, 2, 3, 6]),\n",
       "   (1, [4, 5, 7, 3, 6, 8, 9, 2, 1]),\n",
       "   (1, [4, 5, 7, 3, 1, 8, 9, 2, 6]),\n",
       "   (1, [3, 4, 9, 8]),\n",
       "   (1, [4, 2, 8, 6]),\n",
       "   (1, [4, 2, 8, 5]),\n",
       "   (1, [9, 5, 6, 3, 2]),\n",
       "   (1, [7, 4, 5, 1, 9, 3, 8, 2, 6]),\n",
       "   (1, [8, 5, 9, 7, 4, 6, 2, 1, 3]),\n",
       "   (1, [1, 8, 5, 3, 9, 4, 2, 6, 7]),\n",
       "   (1, [3, 9, 6, 1]),\n",
       "   (1, [3, 9, 6, 2]),\n",
       "   (1, [1, 3, 8, 4, 5, 7]),\n",
       "   (1, [4, 3, 9, 5, 2, 1, 7, 8, 6]),\n",
       "   (1, [3, 8, 1, 2, 4]),\n",
       "   (1, [9, 8, 2, 4]),\n",
       "   (1, [6, 7, 5, 8, 9, 3, 1, 2, 4]),\n",
       "   (1, [5, 4, 7, 8, 2, 6, 1, 9, 3]),\n",
       "   (1, [3, 9, 6, 2, 4, 5, 7, 8]),\n",
       "   (1, [4, 5, 7, 3, 8, 2, 1, 9, 6]),\n",
       "   (1, [3, 2, 4, 5, 6]),\n",
       "   (1, [3, 2, 4, 5, 7]),\n",
       "   (1, [4, 7, 1, 5]),\n",
       "   (1, [5, 4, 7, 2, 3, 8, 1, 9, 6]),\n",
       "   (1, [4, 7, 9, 5, 8, 1, 3, 2, 6]),\n",
       "   (1, [5, 7, 4, 2, 9, 1, 3, 6, 8]),\n",
       "   (1, [3, 1, 4, 9, 7, 8, 6, 5, 2]),\n",
       "   (1, [3, 4, 5, 2, 1, 7, 6, 9, 8]),\n",
       "   (1, [8, 9, 1]),\n",
       "   (1, [9, 1, 3, 5, 4]),\n",
       "   (1, [4, 6, 5, 7, 2, 8, 3, 1, 9]),\n",
       "   (1, [3, 8, 5, 7]),\n",
       "   (1, [4, 5, 7, 8, 9, 2, 1]),\n",
       "   (1, [4, 7, 5, 9, 1]),\n",
       "   (1, [4, 7, 5, 9, 3]),\n",
       "   (1, [7, 4, 5, 3, 2, 6, 8, 9, 1]),\n",
       "   (1, [4, 5, 9, 7, 6]),\n",
       "   (1, [4, 5, 9, 7, 1]),\n",
       "   (1, [7, 5, 4, 1, 8, 6, 3, 2, 9]),\n",
       "   (1, [7, 5, 4, 1, 3, 2, 8, 6, 9]),\n",
       "   (1, [4, 5, 9, 7, 8]),\n",
       "   (1, [4, 7, 3, 8]),\n",
       "   (1, [4, 1, 3, 5, 6, 2, 7, 8, 9]),\n",
       "   (1, [4, 7, 3, 2]),\n",
       "   (1, [2, 6, 4, 3, 1, 7, 8]),\n",
       "   (1, [1, 3, 4, 5, 8, 7]),\n",
       "   (1, [1, 3, 6, 8, 5, 4]),\n",
       "   (1, [4, 5, 7, 9, 3, 2, 1, 8, 6]),\n",
       "   (1, [9, 1, 3, 4, 5, 2, 8, 6, 7]),\n",
       "   (1, [3, 8, 1, 2, 9]),\n",
       "   (1, [9, 1, 3, 5, 4, 7]),\n",
       "   (1, [3, 7, 9, 1, 4, 5, 8, 6, 2]),\n",
       "   (1, [3, 7, 8, 6, 5, 2, 4, 1, 9]),\n",
       "   (1, [9, 4, 5, 6, 7, 8, 1, 2, 3]),\n",
       "   (1, [3, 1, 4, 5]),\n",
       "   (1, [8, 1, 9, 2]),\n",
       "   (1, [5, 4, 7, 9, 1, 2, 8, 3, 6]),\n",
       "   (1, [7, 4, 1, 9, 5]),\n",
       "   (1, [9, 3, 4, 5, 7]),\n",
       "   (1, [3, 6, 1, 5]),\n",
       "   (1, [9, 4, 7, 8]),\n",
       "   (1, [3, 6, 1, 2]),\n",
       "   (1, [6, 4, 5, 2]),\n",
       "   (1, [6, 1, 2, 8, 9, 5, 7, 4, 3]),\n",
       "   (1, [5, 2, 8, 9]),\n",
       "   (1, [8, 3, 2, 9, 1, 5, 6, 4, 7]),\n",
       "   (1, [5, 2, 8, 7]),\n",
       "   (1, [8, 2, 3, 6, 4, 5, 7]),\n",
       "   (1, [5, 2, 8, 4]),\n",
       "   (1, [2, 3, 8, 4]),\n",
       "   (1, [2, 3, 8, 6]),\n",
       "   (1, [3, 6, 5, 9]),\n",
       "   (1, [1, 8, 3, 9]),\n",
       "   (1, [1, 8, 9, 2, 3]),\n",
       "   (1, [1, 5, 7, 9, 2, 3, 8]),\n",
       "   (1, [2, 4, 5, 7, 6]),\n",
       "   (1, [4, 9, 3, 1, 5, 7, 8]),\n",
       "   (1, [6, 3, 1, 8]),\n",
       "   (1, [2, 4, 5, 7, 3]),\n",
       "   (1, [2, 3, 4, 5]),\n",
       "   (1, [4, 6, 3]),\n",
       "   (1, [3, 6, 2, 8, 9, 5, 4, 1, 7]),\n",
       "   (1, [6, 1, 4]),\n",
       "   (1, [9, 3, 1, 8, 2, 7, 5, 4, 6]),\n",
       "   (1, [9, 3, 4, 5, 7, 8, 2, 1, 6]),\n",
       "   (1, [7, 5, 4, 6, 8, 3, 2, 1, 9]),\n",
       "   (1, [9, 1, 7, 8, 2, 3, 6, 5, 4]),\n",
       "   (1, [2, 5, 9, 1]),\n",
       "   (1, [8, 4, 2, 3, 5, 6, 7, 9, 1]),\n",
       "   (1, [9, 5, 2, 4]),\n",
       "   (1, [3, 4, 5, 7, 1, 2, 8, 6, 9]),\n",
       "   (1, [5, 4, 7, 2, 8]),\n",
       "   (1, [5, 4, 7, 2, 9]),\n",
       "   (1, [4, 5, 7, 8, 3, 2, 1, 6, 9]),\n",
       "   (1, [9, 1, 4, 5, 7, 3, 2, 8]),\n",
       "   (1, [7, 5, 4, 8, 6, 2, 3, 1, 9]),\n",
       "   (1, [5, 6, 4, 2, 3, 7]),\n",
       "   (1, [5, 4, 7, 6, 2, 8, 1, 9]),\n",
       "   (1, [5, 4, 7, 8, 3, 2, 1, 9, 6]),\n",
       "   (1, [7, 4, 8, 2, 5]),\n",
       "   (1, [6, 3, 1, 5]),\n",
       "   (1, [7, 5, 4, 2, 3, 6, 8, 9, 1]),\n",
       "   (1, [6, 5, 2]),\n",
       "   (1, [1, 6, 4, 5, 7]),\n",
       "   (1, [1, 3, 9, 2, 6]),\n",
       "   (1, [2, 9, 1, 3, 4]),\n",
       "   (1, [3, 1, 9, 5, 7, 2, 4, 6, 8]),\n",
       "   (1, [6, 3, 8, 2, 4, 5, 7, 9, 1]),\n",
       "   (1, [3, 2, 4, 5, 7, 9]),\n",
       "   (1, [7, 5, 4, 3, 2, 1, 8]),\n",
       "   (1, [3, 8, 9, 2, 6, 1]),\n",
       "   (1, [3, 2, 1, 6, 8, 9, 7, 4, 5]),\n",
       "   (1, [3, 9, 4, 8, 2, 5, 1, 7, 6]),\n",
       "   (1, [1, 3, 2, 4, 5, 8, 7, 9, 6]),\n",
       "   (1, [7, 5, 2, 8, 3, 4, 9, 1, 6]),\n",
       "   (1, [9, 6, 2, 1]),\n",
       "   (1, [5, 8, 2, 3, 7, 4, 1, 6, 9]),\n",
       "   (1, [4, 5, 7, 9, 1, 2, 3, 6, 8]),\n",
       "   (1, [3, 8, 9, 7]),\n",
       "   (1, [3, 8, 9, 5]),\n",
       "   (1, [2, 5, 6]),\n",
       "   (1, [8, 6, 3]),\n",
       "   (1, [4, 5, 7, 1, 8, 2, 3, 9]),\n",
       "   (1, [9, 3, 8, 1, 2]),\n",
       "   (1, [5, 4, 9, 3, 1, 2, 7, 8, 6]),\n",
       "   (1, [7, 2, 4, 5, 6, 9, 8, 1, 3]),\n",
       "   (1, [3, 1, 4, 7]),\n",
       "   (1, [2, 6, 9, 7, 4]),\n",
       "   (1, [3, 2, 9, 8, 1]),\n",
       "   (1, [3, 5, 7, 4, 9, 8, 2, 1, 6]),\n",
       "   (1, [3, 8, 9, 7, 5, 4, 2, 1, 6]),\n",
       "   (1, [8, 7, 9, 5]),\n",
       "   (1, [7, 2, 4, 5]),\n",
       "   (1, [8, 3, 4, 7, 5, 6]),\n",
       "   (1, [3, 1, 9, 4, 8]),\n",
       "   (1, [5, 4, 7, 8, 1, 2, 3, 6, 9]),\n",
       "   (1, [1, 7, 4, 5, 3, 2, 6, 8, 9]),\n",
       "   (1, [1, 4, 3, 9, 8, 5, 6, 2, 7]),\n",
       "   (1, [3, 2, 8, 6, 9, 1]),\n",
       "   (1, [5, 7, 8, 6, 4, 2]),\n",
       "   ...]),\n",
       " (9334,\n",
       "  defaultdict(int,\n",
       "              {1: 0.20762802656953075,\n",
       "               2: 0.23184058281551317,\n",
       "               3: 0.233983286908078,\n",
       "               4: 0.15856010284979644,\n",
       "               8: 0.10531390614956075,\n",
       "               5: 0.0432826226698093,\n",
       "               6: 0.013284765373901865,\n",
       "               7: 0.006106706663809728,\n",
       "               0: 0}),\n",
       "  [(647, [2]),\n",
       "   (632, [4, 5]),\n",
       "   (358, [7]),\n",
       "   (334, [4]),\n",
       "   (222, [5, 4]),\n",
       "   (195, [1]),\n",
       "   (165, [6]),\n",
       "   (141, [2, 6]),\n",
       "   (121, [5]),\n",
       "   (113, [1, 6]),\n",
       "   (102, [4, 5, 1]),\n",
       "   (100, [6, 1]),\n",
       "   (97, [7, 1]),\n",
       "   (89, [2, 3]),\n",
       "   (85, [3]),\n",
       "   (81, [4, 5, 6]),\n",
       "   (72, [2, 6, 1]),\n",
       "   (71, [2, 1, 6]),\n",
       "   (70, [4, 5, 2]),\n",
       "   (66, [2, 1]),\n",
       "   (55, [1, 7]),\n",
       "   (53, [2, 4]),\n",
       "   (49, [2, 7]),\n",
       "   (48, [7, 1, 6]),\n",
       "   (46, [2, 4, 5]),\n",
       "   (43, [5, 4, 1]),\n",
       "   (42, [2, 3, 6]),\n",
       "   (42, [1, 2]),\n",
       "   (41, [6, 2]),\n",
       "   (40, [1, 2, 6]),\n",
       "   (37, [4, 5, 7]),\n",
       "   (36, [5, 4, 6]),\n",
       "   (34, [7, 6]),\n",
       "   (33, [8]),\n",
       "   (31, [1, 6, 2]),\n",
       "   (30, [7, 1, 2]),\n",
       "   (30, [4, 5, 1, 6]),\n",
       "   (30, [4, 2]),\n",
       "   (30, [2, 7, 1]),\n",
       "   (30, [2, 1, 7]),\n",
       "   (29, [4, 1]),\n",
       "   (29, [1, 4]),\n",
       "   (28, [1, 4, 5]),\n",
       "   (27, [7, 6, 1]),\n",
       "   (27, [4, 5, 3]),\n",
       "   (26, [1, 4, 5, 6]),\n",
       "   (26, [7, 2]),\n",
       "   (25, [4, 5, 2, 1]),\n",
       "   (24, [2, 6, 3]),\n",
       "   (24, [6, 1, 2]),\n",
       "   (24, [5, 4, 2]),\n",
       "   (24, [2, 1, 4]),\n",
       "   (23, [4, 5, 1, 2]),\n",
       "   (23, [4, 5, 7, 1]),\n",
       "   (23, [2, 1, 4, 5]),\n",
       "   (22, [4, 1, 5]),\n",
       "   (21, [4, 5, 6, 1]),\n",
       "   (20, [2, 3, 8]),\n",
       "   (20, [7, 4, 5]),\n",
       "   (20, [1, 6, 7]),\n",
       "   (20, [7, 4]),\n",
       "   (20, [6, 2, 1]),\n",
       "   (19, [2, 4, 5, 1]),\n",
       "   (19, [3, 2]),\n",
       "   (19, [4, 6]),\n",
       "   (19, [2, 8, 6]),\n",
       "   (19, [1, 7, 6]),\n",
       "   (18, [2, 6, 8]),\n",
       "   (18, [3, 6]),\n",
       "   (17, [7, 1, 4]),\n",
       "   (17, [2, 5]),\n",
       "   (17, [1, 5, 4]),\n",
       "   (17, [2, 1, 3]),\n",
       "   (16, [2, 1, 7, 6]),\n",
       "   (16, [3, 8]),\n",
       "   (16, [4, 2, 5]),\n",
       "   (16, [7, 5, 4]),\n",
       "   (15, [2, 8]),\n",
       "   (15, [5, 4, 1, 6]),\n",
       "   (15, [6, 8, 3]),\n",
       "   (15, [3, 8, 6]),\n",
       "   (15, [4, 2, 1]),\n",
       "   (15, [6, 3]),\n",
       "   (15, [2, 3, 6, 1]),\n",
       "   (14, [2, 6, 7]),\n",
       "   (14, [4, 5, 2, 3]),\n",
       "   (14, [4, 7]),\n",
       "   (14, [5, 4, 6, 1]),\n",
       "   (14, [5, 2]),\n",
       "   (14, [5, 6]),\n",
       "   (14, [1, 4, 6]),\n",
       "   (14, [2, 3, 6, 8]),\n",
       "   (13, [4, 5, 1, 7]),\n",
       "   (13, [2, 3, 1]),\n",
       "   (13, [5, 4, 3]),\n",
       "   (13, [6, 1, 4, 5]),\n",
       "   (13, [7, 1, 4, 5]),\n",
       "   (13, [7, 5]),\n",
       "   (13, [4, 5, 8]),\n",
       "   (13, [6, 2, 3]),\n",
       "   (12, [2, 4, 6]),\n",
       "   (12, [2, 6, 4]),\n",
       "   (12, [2, 1, 3, 6]),\n",
       "   (12, [5, 4, 7]),\n",
       "   (12, [1, 2, 4]),\n",
       "   (12, [2, 5, 4]),\n",
       "   (12, [3, 4, 5]),\n",
       "   (12, [6, 8]),\n",
       "   (12, [7, 2, 1]),\n",
       "   (12, [4, 5, 2, 6]),\n",
       "   (11, [3, 6, 8]),\n",
       "   (11, [2, 4, 1]),\n",
       "   (11, [6, 1, 7]),\n",
       "   (11, [4, 1, 6]),\n",
       "   (11, [3, 8, 2]),\n",
       "   (11, [8, 6]),\n",
       "   (11, [2, 8, 3]),\n",
       "   (11, [6, 3, 8]),\n",
       "   (11, [1, 6, 3]),\n",
       "   (11, [1, 6, 4]),\n",
       "   (11, [2, 7, 6]),\n",
       "   (11, [2, 6, 3, 1]),\n",
       "   (11, [4, 5, 6, 3]),\n",
       "   (11, [4, 5, 6, 2]),\n",
       "   (11, [3, 2, 6, 8]),\n",
       "   (11, [1, 7, 4]),\n",
       "   (10, [3, 2, 6]),\n",
       "   (10, [3, 1]),\n",
       "   (10, [6, 7]),\n",
       "   (10, [1, 4, 5, 2]),\n",
       "   (10, [2, 1, 3, 8]),\n",
       "   (10, [6, 5, 4]),\n",
       "   (9, [2, 6, 8, 3]),\n",
       "   (9, [1, 6, 5, 4]),\n",
       "   (9, [2, 1, 6, 7]),\n",
       "   (9, [4, 1, 2]),\n",
       "   (9, [7, 4, 1]),\n",
       "   (9, [4, 5, 7, 6]),\n",
       "   (9, [4, 1, 5, 6]),\n",
       "   (9, [6, 4]),\n",
       "   (9, [6, 1, 5, 4]),\n",
       "   (9, [1, 2, 7]),\n",
       "   (9, [1, 2, 7, 6]),\n",
       "   (9, [2, 6, 1, 7]),\n",
       "   (9, [2, 6, 3, 8]),\n",
       "   (9, [6, 5, 4, 1]),\n",
       "   (9, [3, 2, 8]),\n",
       "   (8, [2, 7, 1, 6]),\n",
       "   (8, [6, 4, 5, 1]),\n",
       "   (8, [2, 3, 8, 6]),\n",
       "   (8, [6, 1, 4]),\n",
       "   (8, [6, 2, 8]),\n",
       "   (8, [2, 1, 6, 8]),\n",
       "   (8, [2, 1, 6, 3]),\n",
       "   (8, [2, 5, 1]),\n",
       "   (8, [2, 6, 4, 5]),\n",
       "   (8, [2, 4, 5, 6]),\n",
       "   (8, [3, 4]),\n",
       "   (8, [4, 2, 6]),\n",
       "   (8, [1, 5]),\n",
       "   (8, [2, 3, 8, 6, 1, 4, 5, 7]),\n",
       "   (8, [1, 6, 5]),\n",
       "   (8, [2, 1, 5]),\n",
       "   (8, [5, 1]),\n",
       "   (8, [1, 7, 2]),\n",
       "   (8, [2, 1, 8]),\n",
       "   (7, [2, 6, 5]),\n",
       "   (7, [5, 4, 2, 1]),\n",
       "   (7, [4, 1, 7]),\n",
       "   (7, [2, 3, 4, 5]),\n",
       "   (7, [6, 1, 3]),\n",
       "   (7, [6, 1, 5]),\n",
       "   (7, [6, 1, 8]),\n",
       "   (7, [2, 3, 4]),\n",
       "   (7, [5, 4, 1, 2]),\n",
       "   (7, [8, 3, 6]),\n",
       "   (7, [8, 6, 3]),\n",
       "   (7, [4, 7, 1]),\n",
       "   (7, [4, 3]),\n",
       "   (7, [1, 4, 6, 5]),\n",
       "   (7, [1, 7, 4, 5]),\n",
       "   (7, [7, 2, 1, 6]),\n",
       "   (7, [7, 4, 5, 1]),\n",
       "   (7, [8, 3, 2]),\n",
       "   (7, [6, 2, 1, 8]),\n",
       "   (7, [2, 4, 6, 1]),\n",
       "   (7, [4, 3, 5]),\n",
       "   (6, [5, 2, 4]),\n",
       "   (6, [1, 6, 4, 5]),\n",
       "   (6, [4, 6, 1]),\n",
       "   (6, [6, 5, 1]),\n",
       "   (6, [5, 4, 1, 7]),\n",
       "   (6, [6, 4, 5]),\n",
       "   (6, [6, 4, 2]),\n",
       "   (6, [8, 6, 2]),\n",
       "   (6, [1, 2, 3]),\n",
       "   (6, [2, 5, 6]),\n",
       "   (6, [1, 2, 8]),\n",
       "   (6, [1, 3, 6]),\n",
       "   (6, [5, 6, 4]),\n",
       "   (6, [2, 8, 1]),\n",
       "   (6, [4, 5, 3, 8]),\n",
       "   (6, [1, 2, 6, 3]),\n",
       "   (6, [1, 6, 8, 2]),\n",
       "   (6, [6, 3, 2]),\n",
       "   (6, [1, 2, 5]),\n",
       "   (6, [8, 3]),\n",
       "   (6, [1, 7, 5, 4]),\n",
       "   (6, [1, 4, 5, 7]),\n",
       "   (6, [1, 6, 8]),\n",
       "   (6, [7, 2, 6]),\n",
       "   (6, [2, 6, 7, 1]),\n",
       "   (6, [3, 8, 6, 2]),\n",
       "   (5, [2, 1, 4, 7]),\n",
       "   (5, [2, 6, 8, 3, 1]),\n",
       "   (5, [5, 2, 3]),\n",
       "   (5, [2, 6, 8, 1]),\n",
       "   (5, [7, 1, 5]),\n",
       "   (5, [6, 2, 8, 3]),\n",
       "   (5, [5, 1, 4]),\n",
       "   (5, [3, 1, 6]),\n",
       "   (5, [7, 6, 2]),\n",
       "   (5, [2, 1, 8, 6]),\n",
       "   (5, [2, 8, 6, 3]),\n",
       "   (5, [1, 3, 4]),\n",
       "   (5, [2, 4, 5, 7]),\n",
       "   (5, [5, 4, 8]),\n",
       "   (5, [3, 5]),\n",
       "   (5, [1, 2, 6, 7]),\n",
       "   (5, [1, 5, 4, 6]),\n",
       "   (5, [4, 5, 1, 7, 6]),\n",
       "   (5, [1, 2, 4, 5]),\n",
       "   (5, [4, 5, 2, 1, 6, 7, 8, 3]),\n",
       "   (5, [7, 1, 5, 4]),\n",
       "   (5, [2, 4, 5, 3]),\n",
       "   (5, [2, 3, 7]),\n",
       "   (5, [1, 3]),\n",
       "   (5, [6, 8, 3, 1]),\n",
       "   (5, [2, 6, 1, 8]),\n",
       "   (5, [6, 8, 1, 2]),\n",
       "   (5, [7, 6, 1, 2]),\n",
       "   (5, [4, 3, 6]),\n",
       "   (5, [4, 5, 6, 8]),\n",
       "   (5, [7, 2, 6, 1]),\n",
       "   (5, [5, 3]),\n",
       "   (5, [3, 2, 1]),\n",
       "   (5, [4, 6, 5, 1]),\n",
       "   (5, [7, 5, 4, 1]),\n",
       "   (5, [2, 5, 4, 6]),\n",
       "   (4, [3, 6, 1]),\n",
       "   (4, [2, 4, 3]),\n",
       "   (4, [2, 4, 7]),\n",
       "   (4, [2, 3, 1, 6]),\n",
       "   (4, [1, 6, 2, 5, 4]),\n",
       "   (4, [2, 1, 7, 4]),\n",
       "   (4, [5, 4, 6, 2]),\n",
       "   (4, [4, 6, 5]),\n",
       "   (4, [6, 7, 1]),\n",
       "   (4, [5, 7, 4]),\n",
       "   (4, [8, 6, 1]),\n",
       "   (4, [8, 2, 6]),\n",
       "   (4, [4, 5, 1, 3]),\n",
       "   (4, [3, 2, 1, 6]),\n",
       "   (4, [4, 1, 5, 7]),\n",
       "   (4, [2, 1, 6, 5]),\n",
       "   (4, [3, 5, 4]),\n",
       "   (4, [3, 2, 8, 6]),\n",
       "   (4, [6, 8, 2]),\n",
       "   (4, [2, 1, 4, 5, 6, 7, 8, 3]),\n",
       "   (4, [1, 3, 8]),\n",
       "   (4, [2, 4, 1, 5]),\n",
       "   (4, [2, 8, 3, 1]),\n",
       "   (4, [4, 7, 5]),\n",
       "   (4, [2, 4, 5, 8]),\n",
       "   (4, [1, 2, 6, 8]),\n",
       "   (4, [1, 5, 4, 2]),\n",
       "   (4, [4, 5, 3, 1]),\n",
       "   (4, [4, 5, 1, 2, 3, 6, 8, 7]),\n",
       "   (4, [4, 5, 3, 1, 2, 6, 8, 7]),\n",
       "   (4, [4, 5, 3, 6]),\n",
       "   (4, [5, 7, 1]),\n",
       "   (4, [6, 5]),\n",
       "   (4, [1, 6, 8, 3]),\n",
       "   (4, [4, 5, 1, 6, 7]),\n",
       "   (4, [7, 1, 6, 5]),\n",
       "   (4, [2, 1, 4, 6]),\n",
       "   (4, [1, 4, 2]),\n",
       "   (4, [8, 2]),\n",
       "   (4, [2, 6, 1, 5]),\n",
       "   (4, [1, 5, 6]),\n",
       "   (4, [1, 2, 3, 8]),\n",
       "   (4, [7, 2, 3, 1]),\n",
       "   (4, [4, 5, 1, 7, 6, 2, 3, 8]),\n",
       "   (4, [2, 7, 4]),\n",
       "   (4, [2, 3, 6, 8, 5]),\n",
       "   (4, [4, 5, 1, 8]),\n",
       "   (4, [2, 7, 3]),\n",
       "   (4, [4, 5, 1, 6, 7, 8, 3, 2]),\n",
       "   (4, [6, 2, 4]),\n",
       "   (4, [7, 5, 1, 4]),\n",
       "   (4, [4, 5, 2, 8]),\n",
       "   (4, [3, 6, 2, 1]),\n",
       "   (4, [4, 6, 1, 5]),\n",
       "   (4, [5, 1, 2]),\n",
       "   (4, [2, 8, 1, 6]),\n",
       "   (4, [1, 5, 2]),\n",
       "   (4, [3, 6, 2, 8]),\n",
       "   (4, [2, 1, 5, 4]),\n",
       "   (3, [3, 6, 2]),\n",
       "   (3, [4, 5, 2, 6, 1]),\n",
       "   (3, [6, 1, 8, 3]),\n",
       "   (3, [4, 7, 2, 1]),\n",
       "   (3, [7, 1, 3]),\n",
       "   (3, [1, 6, 2, 8]),\n",
       "   (3, [1, 6, 2, 7]),\n",
       "   (3, [2, 6, 1, 5, 4]),\n",
       "   (3, [5, 4, 1, 7, 6]),\n",
       "   (3, [1, 6, 4, 2]),\n",
       "   (3, [5, 4, 6, 3]),\n",
       "   (3, [5, 4, 2, 8]),\n",
       "   (3, [7, 1, 6, 4, 5]),\n",
       "   (3, [4, 2, 1, 6]),\n",
       "   (3, [4, 2, 5, 1]),\n",
       "   (3, [4, 2, 5, 6]),\n",
       "   (3, [1, 4, 5, 7, 6]),\n",
       "   (3, [5, 1, 6]),\n",
       "   (3, [6, 2, 4, 5, 1, 3, 8, 7]),\n",
       "   (3, [6, 1, 2, 3]),\n",
       "   (3, [4, 1, 2, 6]),\n",
       "   (3, [2, 4, 3, 5]),\n",
       "   (3, [7, 6, 2, 1]),\n",
       "   (3, [5, 4, 7, 1]),\n",
       "   (3, [5, 4, 3, 2]),\n",
       "   (3, [5, 4, 3, 6]),\n",
       "   (3, [2, 3, 4, 5, 1, 6, 7, 8]),\n",
       "   (3, [2, 6, 1, 4, 5]),\n",
       "   (3, [4, 1, 6, 5]),\n",
       "   (3, [2, 1, 6, 7, 4, 5, 3, 8]),\n",
       "   (3, [2, 3, 5]),\n",
       "   (3, [4, 5, 6, 7, 1]),\n",
       "   (3, [3, 1, 2]),\n",
       "   (3, [2, 1, 6, 4]),\n",
       "   (3, [4, 3, 5, 6]),\n",
       "   (3, [4, 5, 1, 2, 7, 6, 8, 3]),\n",
       "   (3, [2, 6, 5, 4]),\n",
       "   (3, [6, 1, 4, 2]),\n",
       "   (3, [6, 4, 1]),\n",
       "   (3, [6, 3, 8, 2, 1]),\n",
       "   (3, [6, 1, 4, 5, 7, 8, 2, 3]),\n",
       "   (3, [5, 4, 6, 1, 2, 3, 8, 7]),\n",
       "   (3, [7, 6, 3]),\n",
       "   (3, [4, 1, 3]),\n",
       "   (3, [7, 4, 2]),\n",
       "   (3, [5, 1, 7, 4]),\n",
       "   (3, [2, 4, 3, 6]),\n",
       "   (3, [2, 5, 3]),\n",
       "   (3, [4, 5, 6, 7, 8, 3, 2, 1]),\n",
       "   (3, [1, 3, 7]),\n",
       "   (3, [4, 2, 7]),\n",
       "   (3, [8, 3, 2, 6]),\n",
       "   (3, [2, 4, 1, 6]),\n",
       "   (3, [2, 8, 3, 6]),\n",
       "   (3, [3, 8, 2, 6]),\n",
       "   (3, [2, 1, 3, 4, 5, 6, 8, 7]),\n",
       "   (3, [5, 6, 1]),\n",
       "   (3, [1, 5, 4, 7]),\n",
       "   (3, [4, 5, 2, 1, 8, 6, 3, 7]),\n",
       "   (3, [2, 6, 3, 8, 1]),\n",
       "   (3, [5, 7]),\n",
       "   (3, [1, 4, 6, 2]),\n",
       "   (3, [1, 4, 6, 7]),\n",
       "   (3, [1, 7, 2, 6]),\n",
       "   (3, [3, 4, 6]),\n",
       "   (3, [2, 3, 6, 8, 1, 4, 5, 7]),\n",
       "   (3, [1, 2, 4, 6]),\n",
       "   (3, [4, 5, 2, 1, 8, 7, 6, 3]),\n",
       "   (3, [7, 1, 5, 6]),\n",
       "   (3, [4, 7, 6]),\n",
       "   (3, [5, 1, 4, 7]),\n",
       "   (3, [6, 2, 3, 1]),\n",
       "   (3, [6, 1, 2, 5]),\n",
       "   (3, [7, 1, 4, 6]),\n",
       "   (3, [4, 8, 5]),\n",
       "   (3, [5, 7, 2]),\n",
       "   (3, [5, 4, 2, 3]),\n",
       "   (3, [2, 4, 5, 6, 1]),\n",
       "   (3, [7, 4, 5, 6]),\n",
       "   (3, [2, 1, 4, 5, 6, 3, 8, 7]),\n",
       "   (3, [2, 6, 1, 8, 3, 4, 5, 7]),\n",
       "   (3, [6, 3, 1]),\n",
       "   (3, [6, 3, 4]),\n",
       "   (3, [6, 1, 2, 8]),\n",
       "   (3, [1, 4, 7, 2]),\n",
       "   (3, [4, 5, 6, 1, 7]),\n",
       "   (3, [2, 1, 7, 6, 4, 5, 8, 3]),\n",
       "   (3, [1, 4, 7]),\n",
       "   (3, [4, 5, 2, 1, 6]),\n",
       "   (3, [4, 5, 3, 2, 6]),\n",
       "   (3, [2, 6, 1, 4]),\n",
       "   (3, [1, 2, 3, 6]),\n",
       "   (3, [2, 1, 6, 8, 3, 4, 5, 7]),\n",
       "   (3, [5, 2, 1]),\n",
       "   (3, [6, 7, 2, 1]),\n",
       "   (3, [2, 7, 8]),\n",
       "   (3, [2, 1, 6, 3, 8]),\n",
       "   (3, [1, 6, 3, 8]),\n",
       "   (3, [4, 5, 2, 1, 7, 6, 8, 3]),\n",
       "   (3, [5, 4, 2, 6, 8]),\n",
       "   (3, [2, 3, 6, 8, 1]),\n",
       "   (3, [6, 2, 5]),\n",
       "   (3, [6, 3, 8, 1]),\n",
       "   (3, [6, 2, 1, 3]),\n",
       "   (3, [4, 5, 8, 3]),\n",
       "   (3, [4, 5, 6, 1, 2, 3, 8, 7]),\n",
       "   (3, [1, 7, 6, 2]),\n",
       "   (3, [4, 3, 2]),\n",
       "   (3, [1, 7, 6, 4]),\n",
       "   (3, [3, 1, 6, 2]),\n",
       "   (3, [4, 6, 5, 2]),\n",
       "   (3, [4, 5, 7, 1, 6]),\n",
       "   (3, [4, 5, 1, 6, 2, 8, 7, 3]),\n",
       "   (3, [5, 3, 4]),\n",
       "   (3, [5, 3, 6]),\n",
       "   (3, [2, 1, 3, 6, 4, 5, 8, 7]),\n",
       "   (3, [5, 6, 2]),\n",
       "   (3, [4, 5, 6, 1, 7, 2, 3, 8]),\n",
       "   (2, [2, 3, 6, 1, 8]),\n",
       "   (2, [6, 7, 1, 4]),\n",
       "   (2, [5, 1, 6, 4]),\n",
       "   (2, [5, 4, 8, 3]),\n",
       "   (2, [1, 6, 2, 5]),\n",
       "   (2, [1, 6, 2, 4]),\n",
       "   (2, [1, 6, 2, 3]),\n",
       "   (2, [2, 1, 6, 4, 5]),\n",
       "   (2, [2, 3, 1, 7]),\n",
       "   (2, [2, 1, 6, 5, 4, 3, 8, 7]),\n",
       "   (2, [2, 1, 7, 8]),\n",
       "   (2, [2, 1, 7, 5]),\n",
       "   (2, [5, 4, 2, 1, 6]),\n",
       "   (2, [5, 2, 7]),\n",
       "   (2, [2, 3, 6, 7, 1, 8, 5, 4]),\n",
       "   (2, [5, 2, 8]),\n",
       "   (2, [2, 1, 3, 4]),\n",
       "   (2, [1, 7, 5, 4, 6, 2, 8, 3]),\n",
       "   (2, [6, 5, 4, 2, 1]),\n",
       "   (2, [1, 6, 3, 8, 2, 5, 4, 7]),\n",
       "   (2, [2, 4, 5, 6, 1, 8, 7, 3]),\n",
       "   (2, [8, 2, 3]),\n",
       "   (2, [1, 2, 6, 8, 3, 4, 5, 7]),\n",
       "   (2, [5, 4, 6, 8]),\n",
       "   (2, [4, 5, 2, 6, 8, 1, 7, 3]),\n",
       "   (2, [4, 6, 2]),\n",
       "   (2, [4, 6, 3]),\n",
       "   (2, [4, 5, 1, 2, 6, 7, 3, 8]),\n",
       "   (2, [5, 4, 2, 6]),\n",
       "   (2, [2, 8, 3, 1, 6, 7, 4, 5]),\n",
       "   (2, [6, 2, 8, 1]),\n",
       "   (2, [3, 6, 2, 4, 5, 8, 1, 7]),\n",
       "   (2, [4, 3, 8, 6]),\n",
       "   (2, [5, 4, 7, 6, 1, 8, 3, 2]),\n",
       "   (2, [6, 7, 4]),\n",
       "   (2, [6, 2, 8, 3, 1, 5, 4, 7]),\n",
       "   (2, [6, 1, 8, 2]),\n",
       "   (2, [4, 6, 8, 5]),\n",
       "   (2, [4, 5, 7, 6, 2, 3, 1, 8]),\n",
       "   (2, [2, 4, 8, 5]),\n",
       "   (2, [3, 2, 1, 4]),\n",
       "   (2, [6, 2, 4, 5]),\n",
       "   (2, [2, 1, 5, 6, 4, 7]),\n",
       "   (2, [3, 8, 2, 1, 6]),\n",
       "   (2, [8, 3, 5, 4]),\n",
       "   (2, [2, 3, 6, 8, 1, 5, 4, 7]),\n",
       "   (2, [1, 2, 4, 5, 6, 8, 7, 3]),\n",
       "   (2, [3, 6, 1, 8]),\n",
       "   (2, [2, 6, 8, 1, 3, 4, 5, 7]),\n",
       "   (2, [7, 6, 1, 4, 5]),\n",
       "   (2, [2, 8, 4, 3]),\n",
       "   (2, [7, 1, 2, 4, 5, 6, 8, 3]),\n",
       "   (2, [2, 1, 4, 5, 6, 3, 7, 8]),\n",
       "   (2, [4, 1, 7, 6]),\n",
       "   (2, [5, 4, 1, 6, 2]),\n",
       "   (2, [2, 3, 8, 1]),\n",
       "   (2, [4, 1, 2, 5]),\n",
       "   (2, [6, 3, 1, 8]),\n",
       "   (2, [1, 2, 6, 3, 4, 5]),\n",
       "   (2, [2, 3, 8, 4]),\n",
       "   (2, [1, 6, 4, 5, 7]),\n",
       "   (2, [3, 6, 2, 1, 8]),\n",
       "   (2, [4, 5, 1, 8, 6]),\n",
       "   (2, [1, 2, 4, 5, 6, 7, 8, 3]),\n",
       "   (2, [4, 1, 6, 3]),\n",
       "   (2, [2, 8, 6, 1, 3]),\n",
       "   (2, [4, 1, 5, 2]),\n",
       "   (2, [4, 5, 6, 8, 1]),\n",
       "   (2, [2, 3, 4, 5, 8, 1, 6, 7]),\n",
       "   (2, [2, 1, 3, 4, 5, 6, 7, 8]),\n",
       "   (2, [2, 4, 5, 1, 3, 6, 8, 7]),\n",
       "   (2, [1, 6, 5, 4, 7, 2, 8, 3]),\n",
       "   (2, [4, 5, 1, 3, 2, 6, 8, 7]),\n",
       "   (2, [2, 3, 4, 5, 1]),\n",
       "   (2, [2, 3, 4, 5, 8]),\n",
       "   (2, [4, 3, 1, 2, 5, 6, 7, 8]),\n",
       "   (2, [5, 4, 1, 7, 2]),\n",
       "   (2, [2, 1, 3, 6, 8, 7, 4, 5]),\n",
       "   (2, [6, 1, 4, 5, 2]),\n",
       "   (2, [4, 5, 6, 1, 2, 8, 7, 3]),\n",
       "   (2, [4, 5, 2, 3, 8]),\n",
       "   (2, [5, 6, 1, 4]),\n",
       "   (2, [1, 5, 4, 7, 6, 2]),\n",
       "   (2, [6, 4, 3]),\n",
       "   (2, [2, 6, 1, 5, 4, 8, 3, 7]),\n",
       "   (2, [6, 8, 1]),\n",
       "   (2, [7, 6, 4]),\n",
       "   (2, [8, 2, 1]),\n",
       "   (2, [8, 6, 4]),\n",
       "   (2, [7, 4, 6]),\n",
       "   (2, [6, 1, 3, 4]),\n",
       "   (2, [2, 4, 3, 1]),\n",
       "   (2, [2, 8, 4, 5, 6]),\n",
       "   (2, [4, 5, 6, 7, 1, 2, 8, 3]),\n",
       "   (2, [2, 5, 7]),\n",
       "   (2, [1, 7, 2, 4, 5, 8, 6, 3]),\n",
       "   (2, [3, 2, 6, 1, 7, 8, 4, 5]),\n",
       "   (2, [1, 2, 4, 5, 6]),\n",
       "   (2, [1, 2, 4, 5, 8]),\n",
       "   (2, [4, 5, 1, 2, 3, 6, 7, 8]),\n",
       "   (2, [2, 1, 3, 5, 4, 6, 8, 7]),\n",
       "   (2, [1, 3, 2]),\n",
       "   (2, [2, 6, 4, 5, 1]),\n",
       "   (2, [8, 1, 2, 6]),\n",
       "   (2, [4, 7, 3]),\n",
       "   (2, [2, 3, 1, 6, 4, 5, 8, 7]),\n",
       "   (2, [2, 3, 6, 8, 4, 1, 5, 7]),\n",
       "   (2, [2, 7, 6, 1]),\n",
       "   (2, [5, 4, 3, 8]),\n",
       "   (2, [4, 5, 1, 3, 6]),\n",
       "   (2, [3, 8, 2, 4]),\n",
       "   (2, [6, 1, 2, 8, 3, 4, 5, 7]),\n",
       "   (2, [6, 3, 2, 8]),\n",
       "   (2, [6, 3, 2, 1]),\n",
       "   (2, [4, 3, 1]),\n",
       "   (2, [6, 2, 8, 3, 1]),\n",
       "   (2, [2, 4, 8]),\n",
       "   (2, [4, 2, 8]),\n",
       "   (2, [4, 5, 6, 1, 2, 3, 7, 8]),\n",
       "   (2, [2, 6, 1, 7, 5, 4, 8, 3]),\n",
       "   (2, [6, 8, 2, 1]),\n",
       "   (2, [6, 8, 2, 3]),\n",
       "   (2, [6, 4, 2, 1]),\n",
       "   (2, [2, 3, 6, 4, 5, 1, 8, 7]),\n",
       "   (2, [2, 8, 1, 3, 6, 4, 5, 7]),\n",
       "   (2, [2, 1, 5, 6]),\n",
       "   (2, [6, 2, 8, 3, 1, 4, 5, 7]),\n",
       "   (2, [1, 7, 2, 4]),\n",
       "   (2, [1, 2, 6, 4, 5, 3, 7, 8]),\n",
       "   (2, [6, 1, 2, 4, 5, 7, 8, 3]),\n",
       "   (2, [2, 6, 3, 8, 1, 4, 5, 7]),\n",
       "   (2, [5, 8, 2, 1]),\n",
       "   (2, [8, 1]),\n",
       "   (2, [4, 2, 7, 1]),\n",
       "   (2, [2, 4, 5, 1, 6, 8, 7, 3]),\n",
       "   (2, [7, 5, 2, 1]),\n",
       "   (2, [7, 2, 1, 6, 4]),\n",
       "   (2, [2, 6, 5, 4, 3, 1, 8, 7]),\n",
       "   (2, [4, 2, 3]),\n",
       "   (2, [4, 7, 8]),\n",
       "   (2, [1, 2, 6, 5]),\n",
       "   (2, [5, 4, 6, 8, 1]),\n",
       "   (2, [4, 5, 7, 2]),\n",
       "   (2, [6, 7, 5, 4]),\n",
       "   (2, [4, 3, 2, 1]),\n",
       "   (2, [1, 4, 2, 7]),\n",
       "   (2, [2, 1, 6, 8, 5, 4, 3, 7]),\n",
       "   (2, [2, 8, 7]),\n",
       "   (2, [4, 8, 3]),\n",
       "   (2, [7, 5, 4, 1, 2, 6, 8, 3]),\n",
       "   (2, [4, 5, 7, 1, 2, 6, 8, 3]),\n",
       "   (2, [1, 2, 6, 3, 7]),\n",
       "   (2, [2, 3, 6, 1, 8, 7, 4, 5]),\n",
       "   (2, [2, 3, 8, 1, 4, 5, 6, 7]),\n",
       "   (2, [2, 1, 4, 5, 6, 7, 3, 8]),\n",
       "   (2, [2, 4, 5, 1, 6]),\n",
       "   (2, [2, 7, 8, 1]),\n",
       "   (2, [1, 2, 7, 4, 5]),\n",
       "   (2, [2, 1, 6, 8, 7, 3, 4, 5]),\n",
       "   (2, [1, 2, 4, 6, 7, 5, 8, 3]),\n",
       "   (2, [1, 4, 7, 6]),\n",
       "   (2, [2, 5, 7, 4]),\n",
       "   (2, [3, 6, 8, 2, 1]),\n",
       "   (2, [5, 4, 1, 2, 6, 8, 7, 3]),\n",
       "   (2, [5, 4, 1, 2, 6, 8, 3, 7]),\n",
       "   (2, [8, 3, 6, 2, 1]),\n",
       "   (2, [7, 1, 6, 2]),\n",
       "   (2, [5, 4, 2, 1, 6, 8, 3, 7]),\n",
       "   (2, [1, 2, 5, 6]),\n",
       "   (2, [1, 2, 5, 7]),\n",
       "   (2, [5, 2, 4, 6]),\n",
       "   (2, [6, 8, 3, 2]),\n",
       "   (2, [5, 4, 1, 2, 6]),\n",
       "   (2, [2, 1, 7, 4, 5, 6, 8, 3]),\n",
       "   (2, [1, 4, 3, 5]),\n",
       "   (2, [7, 1, 6, 3, 8, 2, 4, 5]),\n",
       "   (2, [2, 5, 4, 1, 6]),\n",
       "   (2, [1, 8, 6]),\n",
       "   (2, [1, 8, 3]),\n",
       "   (2, [4, 5, 1, 2, 6, 8, 3, 7]),\n",
       "   (2, [1, 8, 6, 2]),\n",
       "   (2, [2, 1, 6, 3, 8, 4, 5, 7]),\n",
       "   (2, [1, 7, 6, 4, 5]),\n",
       "   (2, [1, 5, 4, 6, 2]),\n",
       "   (2, [3, 4, 5, 8]),\n",
       "   (2, [4, 5, 6, 7, 8, 3, 1, 2]),\n",
       "   (2, [1, 4, 5, 6, 7]),\n",
       "   (2, [2, 1, 6, 7, 8, 5, 4, 3]),\n",
       "   (2, [1, 4, 5, 6, 2]),\n",
       "   (2, [7, 2, 8]),\n",
       "   (2, [1, 4, 3]),\n",
       "   (2, [2, 3, 6, 5]),\n",
       "   (2, [1, 6, 7, 2]),\n",
       "   (2, [3, 7, 6]),\n",
       "   (2, [1, 6, 4, 3, 2, 8, 7, 5]),\n",
       "   (2, [2, 1, 3, 8, 4, 5, 6, 7]),\n",
       "   (2, [2, 4, 7, 1]),\n",
       "   (2, [5, 4, 1, 2, 6, 7, 8, 3]),\n",
       "   (2, [2, 3, 6, 8, 7, 1, 4, 5]),\n",
       "   (2, [2, 6, 8, 3, 1, 7, 5, 4]),\n",
       "   (2, [2, 6, 3, 7]),\n",
       "   (2, [7, 6, 1, 4]),\n",
       "   (2, [7, 6, 1, 5]),\n",
       "   (2, [6, 5, 4, 1, 2]),\n",
       "   (2, [2, 1, 8, 3]),\n",
       "   (2, [1, 6, 7, 4, 5, 8, 2, 3]),\n",
       "   (2, [4, 5, 8, 2]),\n",
       "   (2, [4, 5, 1, 6, 2, 7, 8, 3]),\n",
       "   (2, [2, 7, 3, 1]),\n",
       "   (2, [1, 6, 4, 5, 2, 8, 3, 7]),\n",
       "   (2, [5, 2, 1, 4]),\n",
       "   (2, [3, 7, 6, 1]),\n",
       "   (2, [2, 1, 6, 4, 5, 3, 7, 8]),\n",
       "   (2, [8, 4, 6]),\n",
       "   (2, [3, 6, 4, 5]),\n",
       "   (2, [2, 3, 8, 6, 4, 5, 1, 7]),\n",
       "   (2, [4, 5, 3, 2]),\n",
       "   (2, [3, 6, 8, 2]),\n",
       "   (2, [1, 5, 2, 6]),\n",
       "   (2, [1, 5, 2, 4]),\n",
       "   (2, [3, 2, 6, 1]),\n",
       "   (2, [6, 2, 3, 8]),\n",
       "   (2, [6, 1, 2, 7]),\n",
       "   (2, [1, 3, 6, 8]),\n",
       "   (2, [1, 6, 7, 4]),\n",
       "   (2, [3, 8, 6, 4]),\n",
       "   (2, [7, 1, 4, 5, 2]),\n",
       "   (2, [6, 1, 3, 2, 4, 5]),\n",
       "   (2, [2, 3, 5, 6]),\n",
       "   (2, [7, 1, 5, 6, 4]),\n",
       "   (2, [2, 3, 5, 8]),\n",
       "   (2, [5, 1, 7]),\n",
       "   (2, [1, 3, 6, 4, 5]),\n",
       "   (2, [5, 6, 3]),\n",
       "   (2, [7, 1, 2, 6]),\n",
       "   (2, [5, 6, 8]),\n",
       "   (2, [7, 2, 1, 4, 5, 6, 8, 3]),\n",
       "   (2, [2, 5, 4, 1]),\n",
       "   (1, [2, 1, 5, 6, 4, 8, 3, 7]),\n",
       "   (1, [4, 2, 3, 5]),\n",
       "   (1, [6, 7, 1, 2]),\n",
       "   (1, [3, 4, 2, 6]),\n",
       "   (1, [3, 4, 2, 1]),\n",
       "   (1, [5, 1, 2, 3]),\n",
       "   (1, [5, 4, 3, 6, 1]),\n",
       "   (1, [2, 7, 5, 4, 1, 3, 8]),\n",
       "   (1, [3, 4, 2, 8]),\n",
       "   (1, [2, 3, 7, 6, 1]),\n",
       "   (1, [1, 5, 3, 2, 4, 6, 8, 7]),\n",
       "   (1, [6, 8, 3, 4, 5, 2, 1, 7]),\n",
       "   (1, [3, 2, 6, 4, 5, 1, 7, 8]),\n",
       "   (1, [2, 8, 4, 3, 5, 6, 7, 1]),\n",
       "   (1, [6, 1, 3, 2, 8, 4, 5, 7]),\n",
       "   (1, [5, 3, 4, 1]),\n",
       "   (1, [7, 6, 5, 4, 3, 1, 2, 8]),\n",
       "   (1, [4, 5, 3, 8, 1, 7, 2, 6]),\n",
       "   (1, [7, 5, 1, 6, 4, 8, 3, 2]),\n",
       "   (1, [1, 4, 5, 7, 6, 2, 3, 8]),\n",
       "   (1, [6, 1, 5, 4, 7, 2, 8, 3]),\n",
       "   (1, [1, 8, 4, 6, 5, 3, 7, 2]),\n",
       "   (1, [2, 5, 8, 1, 3, 4, 6]),\n",
       "   (1, [4, 5, 6, 1, 8, 2, 7, 3]),\n",
       "   (1, [3, 8, 2, 6, 4]),\n",
       "   (1, [5, 2, 3, 4, 8]),\n",
       "   (1, [7, 5, 1]),\n",
       "   (1, [7, 5, 2]),\n",
       "   (1, [7, 5, 3]),\n",
       "   (1, [5, 4, 8, 1]),\n",
       "   (1, [4, 5, 3, 1, 2]),\n",
       "   (1, [2, 1, 3, 8, 4, 5]),\n",
       "   (1, [5, 4, 8, 6]),\n",
       "   (1, [1, 3, 2, 5, 8, 6, 4]),\n",
       "   (1, [7, 1, 2, 4, 5, 3, 8, 6]),\n",
       "   (1, [5, 4, 2, 6, 1, 7, 8, 3]),\n",
       "   (1, [1, 8, 3, 6]),\n",
       "   (1, [6, 1, 4, 5, 2, 7, 8, 3]),\n",
       "   (1, [6, 8, 5]),\n",
       "   (1, [2, 6, 1, 4, 5, 7, 8, 3]),\n",
       "   (1, [4, 2, 5, 1, 6, 7, 3, 8]),\n",
       "   (1, [2, 7, 1, 6, 8, 3]),\n",
       "   (1, [5, 4, 1, 7, 3, 2, 6, 8]),\n",
       "   (1, [2, 8, 3, 1, 4, 5, 7, 6]),\n",
       "   (1, [1, 7, 6, 4, 5, 2, 3, 8]),\n",
       "   (1, [2, 3, 5, 8, 6]),\n",
       "   (1, [7, 2, 1, 8, 3]),\n",
       "   (1, [2, 1, 8, 7, 6, 5, 4, 3]),\n",
       "   (1, [7, 4, 8, 2, 1, 3, 5, 6]),\n",
       "   (1, [4, 1, 2, 3]),\n",
       "   (1, [4, 7, 1, 8]),\n",
       "   (1, [3, 2, 6, 8, 1, 7, 4, 5]),\n",
       "   (1, [6, 8, 3, 1, 4, 2, 5]),\n",
       "   (1, [2, 6, 8, 4, 5, 3, 1, 7]),\n",
       "   (1, [5, 4, 8, 3, 2]),\n",
       "   (1, [7, 1, 2, 6, 3, 5, 4, 8]),\n",
       "   (1, [2, 6, 1, 8, 7, 3, 5, 4]),\n",
       "   (1, [2, 4, 6, 5, 7, 3, 1, 8]),\n",
       "   (1, [4, 1, 6, 8]),\n",
       "   (1, [2, 4, 5, 3, 6, 7, 1, 8]),\n",
       "   (1, [4, 1, 6, 7]),\n",
       "   (1, [4, 5, 8, 1, 3, 6, 2, 7]),\n",
       "   (1, [2, 1, 3, 4, 6, 8, 5]),\n",
       "   (1, [1, 7, 2, 4, 6]),\n",
       "   (1, [2, 6, 8, 1, 7]),\n",
       "   (1, [6, 1, 4, 5, 2, 3, 8, 7]),\n",
       "   (1, [3, 2, 8, 4, 1, 6, 5, 7]),\n",
       "   (1, [5, 4, 2, 1, 3]),\n",
       "   (1, [1, 8, 3, 6, 2]),\n",
       "   (1, [2, 1, 7, 3]),\n",
       "   (1, [4, 5, 6, 1, 8, 3, 7, 2]),\n",
       "   (1, [1, 4, 2, 5, 6, 7]),\n",
       "   (1, [2, 1, 6, 3, 5, 4, 7, 8]),\n",
       "   (1, [5, 2, 4, 6, 3]),\n",
       "   (1, [7, 3, 2, 5]),\n",
       "   (1, [8, 4, 5, 3, 2, 1, 6, 7]),\n",
       "   (1, [4, 5, 3, 2, 1, 6, 8, 7]),\n",
       "   (1, [2, 7, 6, 8, 1, 3, 4, 5]),\n",
       "   (1, [6, 7, 4, 5]),\n",
       "   (1, [3, 1, 4, 5, 6]),\n",
       "   (1, [2, 1, 3, 7]),\n",
       "   (1, [2, 6, 3, 8, 1, 5, 4]),\n",
       "   (1, [3, 1, 2, 4, 5, 6, 8, 7]),\n",
       "   (1, [7, 3, 8, 1]),\n",
       "   (1, [1, 6, 4, 3]),\n",
       "   (1, [1, 6, 4, 7]),\n",
       "   (1, [1, 6, 4, 8]),\n",
       "   (1, [7, 6, 4, 1]),\n",
       "   (1, [4, 5, 2, 1, 3, 6, 7, 8]),\n",
       "   (1, [1, 6, 2, 7, 5, 4, 8, 3]),\n",
       "   (1, [1, 7, 6, 8, 5, 4]),\n",
       "   (1, [2, 6, 8, 5]),\n",
       "   (1, [8, 1, 2, 3, 4, 5, 6, 7]),\n",
       "   (1, [2, 6, 8, 7]),\n",
       "   (1, [7, 4, 3, 1]),\n",
       "   (1, [6, 4, 5, 1, 8, 2, 3, 7]),\n",
       "   (1, [6, 1, 8, 7, 4, 5, 2, 3]),\n",
       "   (1, [2, 3, 4, 1, 5, 8, 6, 7]),\n",
       "   (1, [6, 8, 1, 4, 5]),\n",
       "   (1, [2, 1, 4, 6, 5, 7]),\n",
       "   (1, [7, 1, 8, 5]),\n",
       "   (1, [6, 8, 3, 2, 1]),\n",
       "   (1, [1, 7, 4, 2, 6, 8, 5, 3]),\n",
       "   (1, [6, 2, 3, 4, 5, 1, 8, 7]),\n",
       "   (1, [2, 3, 1, 4, 8]),\n",
       "   (1, [2, 3, 6, 7, 5]),\n",
       "   (1, [5, 4, 2, 1, 6, 3, 8, 7]),\n",
       "   (1, [8, 3, 4, 5, 2, 7, 1, 6]),\n",
       "   (1, [4, 6, 8]),\n",
       "   (1, [1, 3, 2, 5, 4, 6, 8, 7]),\n",
       "   (1, [4, 1, 2, 6, 3, 8, 7, 5]),\n",
       "   (1, [2, 1, 4, 6, 3, 5]),\n",
       "   (1, [3, 4, 6, 5, 7, 8, 1]),\n",
       "   (1, [4, 5, 7, 1, 2, 8, 6, 3]),\n",
       "   (1, [2, 1, 4, 7, 6, 8]),\n",
       "   (1, [4, 5, 3, 1, 7, 8, 6, 2]),\n",
       "   (1, [4, 5, 1, 8, 6, 2, 7, 3]),\n",
       "   (1, [4, 3, 6, 1]),\n",
       "   (1, [4, 5, 3, 7, 1, 2, 6, 8]),\n",
       "   (1, [4, 8, 5, 1, 2, 3, 6]),\n",
       "   (1, [5, 4, 1, 7, 6, 2, 3, 8]),\n",
       "   (1, [5, 1, 6, 3, 2, 4, 8, 7]),\n",
       "   (1, [2, 1, 6, 8, 5]),\n",
       "   (1, [2, 6, 4, 3]),\n",
       "   (1, [4, 2, 3, 6, 1]),\n",
       "   (1, [1, 2, 6, 5, 4, 7, 8, 3]),\n",
       "   (1, [6, 2, 8, 5]),\n",
       "   (1, [5, 3, 2, 6]),\n",
       "   (1, [1, 3, 7, 8, 6, 5, 4]),\n",
       "   (1, [4, 3, 8, 7, 5, 6, 2, 1]),\n",
       "   (1, [4, 6, 2, 8, 1]),\n",
       "   (1, [8, 5, 4, 1, 7, 6]),\n",
       "   (1, [8, 6, 3, 2]),\n",
       "   (1, [2, 3, 6, 4, 5, 8, 1, 7]),\n",
       "   (1, [1, 4, 5, 6, 7, 2, 3, 8]),\n",
       "   (1, [6, 7, 5]),\n",
       "   (1, [6, 7, 2]),\n",
       "   (1, [2, 3, 1, 6, 8, 5, 4, 7]),\n",
       "   (1, [2, 3, 1, 5, 4, 6, 8, 7]),\n",
       "   (1, [2, 4, 5, 6, 1, 7, 8, 3]),\n",
       "   (1, [5, 8, 6, 4]),\n",
       "   (1, [2, 5, 1, 6]),\n",
       "   (1, [1, 6, 7, 4, 5]),\n",
       "   (1, [6, 5, 7, 1]),\n",
       "   (1, [1, 4, 6, 5, 3]),\n",
       "   (1, [1, 7, 6, 5, 4, 2, 8, 3]),\n",
       "   (1, [2, 4, 3, 1, 5, 6, 7, 8]),\n",
       "   (1, [7, 6, 4, 8]),\n",
       "   (1, [6, 1, 8, 2, 3, 7, 4, 5]),\n",
       "   (1, [3, 1, 5, 4, 2, 6, 8, 7]),\n",
       "   (1, [2, 6, 1, 4, 8, 5, 3]),\n",
       "   (1, [5, 4, 2, 1, 3, 8, 7, 6]),\n",
       "   (1, [6, 4, 5, 2, 3, 8, 1, 7]),\n",
       "   (1, [2, 1, 3, 6, 4, 5]),\n",
       "   (1, [4, 3, 2, 1, 6]),\n",
       "   (1, [7, 5, 8]),\n",
       "   (1, [7, 1, 2, 3, 6, 8, 4, 5]),\n",
       "   (1, [4, 3, 2, 1, 5]),\n",
       "   (1, [7, 3, 2]),\n",
       "   (1, [6, 4, 5, 7, 1, 2, 8, 3]),\n",
       "   (1, [4, 5, 1, 2, 6]),\n",
       "   (1, [4, 2, 1, 8]),\n",
       "   (1, [4, 5, 2, 3, 1, 7, 8, 6]),\n",
       "   (1, [4, 2, 1, 5]),\n",
       "   (1, [1, 6, 4, 5, 8, 7, 2, 3]),\n",
       "   (1, [4, 2, 1, 3]),\n",
       "   (1, [2, 6, 3, 1, 8, 7]),\n",
       "   (1, [2, 5, 4, 8, 6, 1, 3, 7]),\n",
       "   (1, [2, 8, 1, 3, 4, 5, 6, 7]),\n",
       "   (1, [6, 8, 1, 2, 3, 5, 4, 7]),\n",
       "   (1, [6, 1, 2, 8, 3, 5, 4, 7]),\n",
       "   (1, [6, 1, 4, 5, 8, 3]),\n",
       "   (1, [2, 1, 8, 6, 3, 4, 5, 7]),\n",
       "   (1, [7, 5, 3, 2, 4, 1, 6, 8]),\n",
       "   (1, [3, 8, 6, 1, 2]),\n",
       "   (1, [4, 1, 5, 2, 3, 8, 6, 7]),\n",
       "   (1, [4, 2, 5, 8]),\n",
       "   (1, [6, 1, 2, 8, 5, 4, 3, 7]),\n",
       "   (1, [2, 6, 4, 5, 1, 8, 3, 7]),\n",
       "   (1, [2, 4, 8, 7]),\n",
       "   (1, [4, 2, 5, 7]),\n",
       "   (1, [1, 7, 4, 2, 5, 6]),\n",
       "   (1, [1, 7, 3, 4, 2, 8, 6, 5]),\n",
       "   (1, [2, 4, 7, 1, 5, 6, 8, 3]),\n",
       "   (1, [3, 6, 7, 8]),\n",
       "   (1, [4, 5, 2, 6, 3, 1, 8, 7]),\n",
       "   (1, [8, 3, 6, 1]),\n",
       "   (1, [8, 3, 6, 2]),\n",
       "   (1, [2, 7, 5, 4, 1, 6, 3, 8]),\n",
       "   (1, [2, 1, 8, 6, 3, 4, 5]),\n",
       "   (1, [1, 8, 2, 4, 6]),\n",
       "   (1, [4, 6, 5, 1, 2, 8, 3, 7]),\n",
       "   (1, [2, 4, 1, 5, 7, 6, 8, 3]),\n",
       "   (1, [3, 5, 4, 8]),\n",
       "   (1, [4, 3, 5, 8, 7]),\n",
       "   (1, [2, 3, 5, 4, 1, 8, 6, 7]),\n",
       "   (1, [3, 5, 6, 1]),\n",
       "   (1, [4, 3, 1, 2, 5]),\n",
       "   (1, [5, 4, 1, 8, 7, 6, 3, 2]),\n",
       "   (1, [8, 2, 1, 3, 4, 5, 6, 7]),\n",
       "   (1, [4, 5, 2, 8, 3, 1, 6, 7]),\n",
       "   (1, [3, 4, 5, 6, 8, 1, 7, 2]),\n",
       "   (1, [5, 4, 1, 8, 3, 2, 6, 7]),\n",
       "   (1, [3, 2, 5, 4]),\n",
       "   (1, [3, 8, 1, 2, 6, 4, 5, 7]),\n",
       "   (1, [6, 4, 3, 1]),\n",
       "   (1, [3, 2, 1, 6, 8, 7]),\n",
       "   (1, [7, 4, 5, 6, 1, 2, 8, 3]),\n",
       "   (1, [3, 4, 6, 1, 8, 5, 2, 7]),\n",
       "   (1, [8, 5, 1]),\n",
       "   (1, [8, 5, 6]),\n",
       "   (1, [2, 4, 1, 6, 7, 8, 3, 5]),\n",
       "   (1, [4, 5, 3, 8, 1]),\n",
       "   (1, [4, 5, 3, 8, 2]),\n",
       "   (1, [4, 5, 3, 8, 6]),\n",
       "   (1, [1, 6, 7, 4, 5, 2, 3, 8]),\n",
       "   (1, [4, 5, 1, 6, 3, 2, 8, 7]),\n",
       "   (1, [3, 2, 4, 5, 6, 1, 8, 7]),\n",
       "   (1, [2, 6, 1, 3, 5]),\n",
       "   (1, [3, 2, 1, 5]),\n",
       "   (1, [3, 2, 1, 7]),\n",
       "   (1, [5, 4, 2, 1, 3, 8, 6, 7]),\n",
       "   (1, [6, 2, 4, 1]),\n",
       "   (1, [6, 2, 4, 3]),\n",
       "   (1, [6, 5, 4, 7, 1, 2, 3, 8]),\n",
       "   (1, [6, 2, 4, 7]),\n",
       "   (1, [1, 5, 4, 8, 2, 6, 3, 7]),\n",
       "   (1, [5, 4, 6, 1, 2, 8, 3, 7]),\n",
       "   (1, [2, 3, 6, 8, 4, 5, 1, 7]),\n",
       "   (1, [2, 7, 1, 5]),\n",
       "   (1, [8, 2, 1, 5]),\n",
       "   (1, [2, 7, 1, 3]),\n",
       "   (1, [4, 5, 1, 7, 6, 3, 8, 2]),\n",
       "   (1, [3, 5, 6, 7, 1, 2, 4, 8]),\n",
       "   (1, [6, 1, 2, 8, 7, 5, 4, 3]),\n",
       "   (1, [3, 2, 6, 7, 8, 1]),\n",
       "   (1, [1, 2, 5, 6, 3, 4, 7, 8]),\n",
       "   (1, [4, 5, 6, 1, 3, 8]),\n",
       "   (1, [6, 1, 2, 4, 5, 8, 3, 7]),\n",
       "   (1, [7, 1, 6, 8]),\n",
       "   (1, [1, 4, 5, 7, 2]),\n",
       "   (1, [3, 1, 6, 4, 5, 2, 7, 8]),\n",
       "   (1, [7, 1, 2, 6, 4, 5, 3, 8]),\n",
       "   (1, [3, 8, 2, 1, 6, 4, 5, 7]),\n",
       "   (1, [5, 6, 4, 2, 1, 3, 7, 8]),\n",
       "   (1, [4, 2, 6, 1, 7, 5]),\n",
       "   (1, [1, 5, 4, 6, 2, 7, 8, 3]),\n",
       "   (1, [2, 5, 8, 1, 6, 4, 3, 7]),\n",
       "   (1, [4, 1, 7, 6, 2, 5, 3, 8]),\n",
       "   (1, [5, 1, 3]),\n",
       "   (1, [5, 3, 2]),\n",
       "   (1, [4, 2, 8, 3]),\n",
       "   (1, [2, 5, 7, 4, 6, 1, 3, 8]),\n",
       "   (1, [3, 8, 6, 4, 5, 1]),\n",
       "   (1, [3, 6, 2, 8, 1, 4, 5, 7]),\n",
       "   (1, [2, 1, 6, 8, 3, 7, 4, 5]),\n",
       "   (1, [8, 5, 2]),\n",
       "   (1, [4, 5, 3, 2, 1, 8, 6, 7]),\n",
       "   (1, [2, 1, 6, 3, 8, 7, 4, 5]),\n",
       "   (1, [2, 1, 3, 4, 7, 5]),\n",
       "   (1, [3, 8, 1, 2, 6]),\n",
       "   (1, [3, 8, 1, 2, 4]),\n",
       "   (1, [7, 1, 5, 4, 6, 2, 8, 3]),\n",
       "   (1, [3, 2, 4, 5, 8]),\n",
       "   (1, [5, 6, 4, 1]),\n",
       "   (1, [2, 5, 6, 7, 1, 8, 3, 4]),\n",
       "   (1, [2, 1, 3, 8, 4, 6, 7, 5]),\n",
       "   (1, [1, 7, 5, 4, 6]),\n",
       "   (1, [6, 7, 8, 5, 4, 3, 2, 1]),\n",
       "   (1, [5, 4, 1, 7, 6, 2]),\n",
       "   (1, [5, 2, 4, 1, 6]),\n",
       "   (1, [6, 7, 1, 4, 5, 2, 8, 3]),\n",
       "   (1, [2, 1, 5, 4, 7, 6, 8, 3]),\n",
       "   (1, [4, 5, 8, 1, 3, 2, 6, 7]),\n",
       "   (1, [8, 3, 6, 2, 4, 5, 1]),\n",
       "   (1, [5, 4, 3, 1, 8, 6, 7, 2]),\n",
       "   (1, [2, 6, 8, 3, 1, 4, 7, 5]),\n",
       "   (1, [7, 1, 4, 5, 2, 6, 3, 8]),\n",
       "   (1, [2, 3, 4, 6, 8, 1, 5]),\n",
       "   (1, [2, 8, 6, 1, 3, 7, 4, 5]),\n",
       "   (1, [3, 6, 2, 1, 8, 7, 5, 4]),\n",
       "   (1, [2, 3, 1, 4, 5, 6, 7, 8]),\n",
       "   (1, [2, 1, 7, 8, 4, 5, 6, 3]),\n",
       "   (1, [2, 4, 6, 1, 5, 3, 7, 8]),\n",
       "   (1, [1, 2, 7, 8, 6, 3, 4, 5]),\n",
       "   (1, [4, 5, 1, 2, 8, 6, 3]),\n",
       "   (1, [1, 6, 8, 2, 3]),\n",
       "   (1, [5, 4, 2, 1, 8, 6, 3, 7]),\n",
       "   (1, [5, 4, 1, 6, 7]),\n",
       "   (1, [4, 1, 7, 5]),\n",
       "   (1, [4, 5, 7, 1, 2, 3, 6, 8]),\n",
       "   (1, [6, 4, 5, 3]),\n",
       "   (1, [4, 1, 2, 7]),\n",
       "   (1, [3, 6, 1, 2]),\n",
       "   (1, [6, 4, 5, 2]),\n",
       "   (1, [4, 1, 5, 7, 8, 3, 2, 6]),\n",
       "   (1, [4, 5, 1, 8, 7, 2, 3, 6]),\n",
       "   (1, [8, 4]),\n",
       "   (1, [8, 4, 5, 3, 1, 6]),\n",
       "   (1, [5, 4, 2, 6, 1, 7, 3, 8]),\n",
       "   (1, [6, 8, 1, 3, 2, 4, 5, 7]),\n",
       "   (1, [6, 3, 1, 2]),\n",
       "   (1, [1, 3, 5, 6]),\n",
       "   (1, [6, 8, 2, 3, 1, 5, 4, 7]),\n",
       "   (1, [2, 3, 4, 6]),\n",
       "   (1, [2, 3, 8, 7, 6, 1]),\n",
       "   (1, [3, 2, 4, 1, 5, 6, 8, 7]),\n",
       "   (1, [2, 8, 3, 4, 5, 6, 1, 7]),\n",
       "   (1, [2, 3, 6, 5, 4, 1, 8, 7]),\n",
       "   (1, [2, 1, 8, 7, 6, 3, 4, 5]),\n",
       "   (1, [2, 7, 8, 3, 5, 4, 1, 6]),\n",
       "   (1, [6, 1, 4, 5, 2, 7, 3, 8]),\n",
       "   (1, [1, 4, 2, 6, 8, 3, 7, 5]),\n",
       "   (1, [3, 6, 8, 1, 5, 4, 7, 2]),\n",
       "   (1, [7, 6, 4, 3, 5, 2, 1, 8]),\n",
       "   (1, [5, 4, 6, 3, 2, 8, 7, 1]),\n",
       "   (1, [4, 8, 5, 3, 6, 1, 7, 2]),\n",
       "   (1, [2, 8, 4, 5, 3, 6, 1, 7]),\n",
       "   (1, [4, 5, 1, 6, 3, 8]),\n",
       "   (1, [7, 1, 2, 5, 8]),\n",
       "   (1, [2, 1, 6, 7, 4, 5]),\n",
       "   (1, [3, 8, 5, 1, 2, 6, 4, 7]),\n",
       "   (1, [4, 5, 1, 6, 3, 2]),\n",
       "   (1, [5, 6, 4, 1, 7, 2]),\n",
       "   (1, [4, 5, 1, 7, 8, 6, 3, 2]),\n",
       "   (1, [6, 3, 1, 4]),\n",
       "   (1, [7, 1, 2, 6, 3, 4, 5, 8]),\n",
       "   (1, [2, 6, 1, 3, 5, 4, 8, 7]),\n",
       "   (1, [7, 6, 5, 4]),\n",
       "   (1, [1, 6, 2, 4, 5]),\n",
       "   (1, [6, 2, 1, 7, 3, 8, 4, 5]),\n",
       "   (1, [1, 6, 2, 8, 3, 7, 4, 5]),\n",
       "   (1, [1, 6, 4, 5, 2]),\n",
       "   (1, [2, 8, 3, 1, 6, 4, 5, 7]),\n",
       "   (1, [5, 2, 1, 3, 6]),\n",
       "   (1, [4, 3, 2, 5, 8, 1]),\n",
       "   ...]),\n",
       " (9901,\n",
       "  defaultdict(int,\n",
       "              {1: 0.2435107564892435,\n",
       "               2: 0.23886476113523886,\n",
       "               3: 0.22361377638622362,\n",
       "               4: 0.14897485102514899,\n",
       "               8: 0.10130289869710131,\n",
       "               5: 0.02918897081102919,\n",
       "               6: 0.007675992324007676,\n",
       "               7: 0.006867993132006868,\n",
       "               0: 0}),\n",
       "  [(723, [1]),\n",
       "   (551, [8, 5]),\n",
       "   (403, [8]),\n",
       "   (400, [5]),\n",
       "   (373, [5, 8]),\n",
       "   (369, [4]),\n",
       "   (296, [3]),\n",
       "   (120, [4, 3]),\n",
       "   (109, [3, 6]),\n",
       "   (105, [3, 4]),\n",
       "   (98, [6]),\n",
       "   (93, [1, 6]),\n",
       "   (76, [8, 5, 3]),\n",
       "   (76, [1, 3]),\n",
       "   (73, [1, 7]),\n",
       "   (69, [7]),\n",
       "   (56, [1, 2]),\n",
       "   (54, [8, 5, 1]),\n",
       "   (54, [1, 8]),\n",
       "   (53, [3, 1, 6]),\n",
       "   (53, [3, 1]),\n",
       "   (53, [2]),\n",
       "   (52, [3, 4, 6]),\n",
       "   (48, [3, 8]),\n",
       "   (45, [1, 6, 3]),\n",
       "   (44, [5, 8, 3]),\n",
       "   (43, [6, 3]),\n",
       "   (41, [3, 6, 1]),\n",
       "   (41, [1, 3, 6]),\n",
       "   (41, [1, 5]),\n",
       "   (40, [1, 2, 7]),\n",
       "   (39, [4, 6]),\n",
       "   (39, [1, 6, 7]),\n",
       "   (38, [1, 7, 2]),\n",
       "   (37, [5, 8, 1]),\n",
       "   (36, [1, 4]),\n",
       "   (35, [8, 3]),\n",
       "   (34, [3, 6, 8]),\n",
       "   (34, [8, 5, 6]),\n",
       "   (34, [8, 1]),\n",
       "   (34, [1, 5, 8]),\n",
       "   (33, [4, 1]),\n",
       "   (30, [4, 3, 6]),\n",
       "   (28, [4, 8]),\n",
       "   (28, [5, 1]),\n",
       "   (27, [5, 1, 8]),\n",
       "   (27, [6, 3, 1]),\n",
       "   (26, [5, 8, 6]),\n",
       "   (26, [1, 3, 8]),\n",
       "   (25, [3, 4, 8]),\n",
       "   (25, [8, 5, 7]),\n",
       "   (24, [1, 8, 5]),\n",
       "   (24, [6, 3, 8]),\n",
       "   (23, [1, 3, 4]),\n",
       "   (23, [6, 1]),\n",
       "   (23, [8, 1, 5]),\n",
       "   (22, [5, 3]),\n",
       "   (22, [1, 4, 3]),\n",
       "   (21, [8, 5, 4]),\n",
       "   (21, [8, 3, 5]),\n",
       "   (20, [3, 6, 4]),\n",
       "   (20, [3, 5]),\n",
       "   (20, [4, 3, 1]),\n",
       "   (20, [3, 8, 6]),\n",
       "   (20, [3, 5, 8]),\n",
       "   (19, [1, 3, 7]),\n",
       "   (19, [5, 8, 7]),\n",
       "   (19, [2, 1]),\n",
       "   (19, [8, 1, 3]),\n",
       "   (18, [4, 6, 3]),\n",
       "   (18, [6, 1, 3]),\n",
       "   (18, [8, 5, 3, 6]),\n",
       "   (18, [1, 8, 3]),\n",
       "   (18, [7, 1]),\n",
       "   (17, [3, 1, 4]),\n",
       "   (17, [6, 8]),\n",
       "   (17, [5, 8, 3, 6]),\n",
       "   (17, [1, 2, 6]),\n",
       "   (16, [5, 8, 6, 3]),\n",
       "   (16, [1, 7, 6]),\n",
       "   (16, [1, 6, 2]),\n",
       "   (15, [3, 1, 8]),\n",
       "   (15, [3, 8, 5]),\n",
       "   (15, [1, 2, 3]),\n",
       "   (15, [3, 4, 1]),\n",
       "   (15, [1, 7, 3]),\n",
       "   (14, [3, 6, 5]),\n",
       "   (14, [8, 4]),\n",
       "   (14, [8, 5, 6, 3]),\n",
       "   (14, [1, 3, 5]),\n",
       "   (14, [4, 3, 8]),\n",
       "   (14, [3, 8, 1]),\n",
       "   (14, [8, 5, 2]),\n",
       "   (14, [3, 6, 8, 5]),\n",
       "   (13, [5, 3, 8]),\n",
       "   (13, [8, 5, 1, 3]),\n",
       "   (13, [6, 4, 3]),\n",
       "   (13, [5, 7]),\n",
       "   (13, [3, 8, 4]),\n",
       "   (13, [8, 6]),\n",
       "   (13, [7, 2]),\n",
       "   (13, [8, 5, 3, 1]),\n",
       "   (12, [6, 1, 7]),\n",
       "   (12, [3, 6, 5, 8]),\n",
       "   (12, [4, 1, 3]),\n",
       "   (12, [1, 4, 6]),\n",
       "   (12, [1, 3, 2]),\n",
       "   (12, [1, 2, 6, 7]),\n",
       "   (12, [5, 2]),\n",
       "   (12, [7, 8]),\n",
       "   (12, [5, 8, 2]),\n",
       "   (12, [6, 5, 8]),\n",
       "   (11, [3, 1, 4, 6]),\n",
       "   (11, [2, 6]),\n",
       "   (11, [1, 7, 2, 6]),\n",
       "   (11, [5, 7, 8]),\n",
       "   (11, [6, 4]),\n",
       "   (11, [5, 8, 3, 4]),\n",
       "   (11, [3, 5, 6]),\n",
       "   (11, [8, 7]),\n",
       "   (11, [1, 5, 7]),\n",
       "   (11, [7, 3]),\n",
       "   (11, [1, 5, 3]),\n",
       "   (11, [5, 6, 8]),\n",
       "   (10, [3, 6, 7]),\n",
       "   (10, [6, 8, 3]),\n",
       "   (10, [6, 3, 8, 5]),\n",
       "   (10, [5, 6]),\n",
       "   (10, [3, 4, 5]),\n",
       "   (10, [6, 3, 2]),\n",
       "   (10, [1, 6, 8]),\n",
       "   (10, [1, 6, 7, 2]),\n",
       "   (10, [4, 3, 1, 6]),\n",
       "   (9, [5, 1, 3]),\n",
       "   (9, [8, 5, 1, 2]),\n",
       "   (9, [6, 8, 5]),\n",
       "   (9, [1, 3, 2, 6]),\n",
       "   (9, [5, 8, 1, 3]),\n",
       "   (9, [4, 8, 3]),\n",
       "   (9, [6, 5]),\n",
       "   (9, [8, 3, 1]),\n",
       "   (9, [3, 5, 1]),\n",
       "   (9, [1, 2, 7, 6]),\n",
       "   (9, [8, 2]),\n",
       "   (9, [1, 2, 7, 3]),\n",
       "   (9, [5, 8, 1, 6]),\n",
       "   (9, [7, 6]),\n",
       "   (9, [3, 6, 1, 4]),\n",
       "   (9, [1, 3, 6, 4]),\n",
       "   (9, [6, 2]),\n",
       "   (8, [1, 5, 8, 3]),\n",
       "   (8, [1, 6, 2, 7]),\n",
       "   (8, [2, 6, 1]),\n",
       "   (8, [6, 7, 2]),\n",
       "   (8, [8, 6, 3]),\n",
       "   (8, [3, 1, 5]),\n",
       "   (8, [8, 5, 3, 4]),\n",
       "   (8, [4, 1, 8]),\n",
       "   (8, [1, 3, 8, 6]),\n",
       "   (8, [6, 3, 7]),\n",
       "   (8, [1, 4, 3, 6]),\n",
       "   (8, [7, 2, 1]),\n",
       "   (8, [1, 6, 7, 3]),\n",
       "   (8, [8, 5, 1, 6]),\n",
       "   (8, [4, 3, 5]),\n",
       "   (8, [6, 3, 5, 8]),\n",
       "   (8, [2, 1, 6]),\n",
       "   (8, [1, 7, 8]),\n",
       "   (8, [5, 3, 1]),\n",
       "   (7, [3, 6, 2]),\n",
       "   (7, [1, 8, 5, 3]),\n",
       "   (7, [3, 8, 5, 6]),\n",
       "   (7, [3, 6, 1, 8]),\n",
       "   (7, [5, 3, 6, 8]),\n",
       "   (7, [4, 5]),\n",
       "   (7, [4, 7]),\n",
       "   (7, [8, 5, 4, 3]),\n",
       "   (7, [6, 3, 4]),\n",
       "   (7, [6, 8, 5, 3]),\n",
       "   (7, [7, 5]),\n",
       "   (7, [1, 6, 3, 8]),\n",
       "   (7, [8, 3, 6]),\n",
       "   (7, [8, 1, 5, 3]),\n",
       "   (7, [3, 8, 6, 5]),\n",
       "   (6, [5, 1, 2, 7]),\n",
       "   (6, [2, 6, 7]),\n",
       "   (6, [3, 5, 6, 8]),\n",
       "   (6, [1, 3, 5, 8]),\n",
       "   (6, [7, 6, 1]),\n",
       "   (6, [3, 4, 6, 1]),\n",
       "   (6, [1, 2, 5]),\n",
       "   (6, [1, 7, 2, 3]),\n",
       "   (6, [3, 4, 6, 8]),\n",
       "   (6, [5, 7, 1]),\n",
       "   (6, [8, 5, 2, 3]),\n",
       "   (6, [8, 1, 6]),\n",
       "   (6, [3, 5, 8, 6]),\n",
       "   (6, [8, 3, 4]),\n",
       "   (6, [8, 7, 2]),\n",
       "   (6, [1, 2, 5, 8]),\n",
       "   (6, [1, 8, 6]),\n",
       "   (6, [1, 2, 3, 4]),\n",
       "   (6, [2, 1, 7]),\n",
       "   (6, [4, 5, 3]),\n",
       "   (6, [6, 2, 7]),\n",
       "   (6, [3, 1, 6, 4]),\n",
       "   (6, [5, 6, 8, 3]),\n",
       "   (6, [1, 3, 6, 7]),\n",
       "   (6, [1, 3, 6, 2]),\n",
       "   (5, [8, 1, 7]),\n",
       "   (5, [7, 1, 2]),\n",
       "   (5, [7, 1, 6]),\n",
       "   (5, [1, 5, 8, 6]),\n",
       "   (5, [1, 4, 6, 3]),\n",
       "   (5, [4, 1, 6, 3]),\n",
       "   (5, [3, 4, 8, 5]),\n",
       "   (5, [6, 7, 1]),\n",
       "   (5, [2, 7, 1, 6]),\n",
       "   (5, [5, 8, 1, 7]),\n",
       "   (5, [3, 8, 5, 4]),\n",
       "   (5, [1, 3, 5, 6]),\n",
       "   (5, [6, 1, 2]),\n",
       "   (5, [1, 2, 3, 4, 5, 6, 7, 8]),\n",
       "   (5, [2, 7]),\n",
       "   (5, [1, 7, 3, 2]),\n",
       "   (5, [3, 1, 7]),\n",
       "   (5, [3, 1, 2]),\n",
       "   (5, [8, 3, 5, 6]),\n",
       "   (5, [3, 1, 8, 6]),\n",
       "   (5, [7, 6, 2]),\n",
       "   (5, [8, 6, 5]),\n",
       "   (5, [1, 4, 8]),\n",
       "   (5, [4, 1, 3, 6]),\n",
       "   (5, [1, 3, 8, 5]),\n",
       "   (5, [8, 3, 1, 6]),\n",
       "   (5, [8, 3, 1, 5]),\n",
       "   (5, [5, 8, 2, 7]),\n",
       "   (5, [1, 7, 6, 2]),\n",
       "   (5, [5, 8, 4]),\n",
       "   (5, [5, 8, 1, 2]),\n",
       "   (5, [4, 8, 5]),\n",
       "   (5, [5, 1, 8, 3]),\n",
       "   (5, [8, 7, 5]),\n",
       "   (5, [3, 5, 4]),\n",
       "   (5, [1, 4, 7]),\n",
       "   (5, [1, 5, 3, 6]),\n",
       "   (5, [5, 8, 7, 3]),\n",
       "   (5, [3, 7, 6]),\n",
       "   (5, [8, 5, 2, 1]),\n",
       "   (5, [1, 3, 6, 8]),\n",
       "   (4, [7, 1, 3]),\n",
       "   (4, [5, 3, 8, 6]),\n",
       "   (4, [3, 4, 8, 1]),\n",
       "   (4, [6, 3, 1, 4]),\n",
       "   (4, [8, 2, 5]),\n",
       "   (4, [4, 3, 8, 5]),\n",
       "   (4, [5, 8, 6, 1]),\n",
       "   (4, [5, 8, 6, 7]),\n",
       "   (4, [8, 3, 6, 5]),\n",
       "   (4, [5, 3, 4]),\n",
       "   (4, [5, 1, 4]),\n",
       "   (4, [3, 8, 4, 6]),\n",
       "   (4, [3, 6, 1, 5]),\n",
       "   (4, [6, 3, 1, 2]),\n",
       "   (4, [6, 3, 1, 8]),\n",
       "   (4, [6, 1, 5, 8]),\n",
       "   (4, [3, 1, 8, 5]),\n",
       "   (4, [8, 5, 7, 2]),\n",
       "   (4, [5, 3, 1, 8]),\n",
       "   (4, [8, 6, 1]),\n",
       "   (4, [1, 7, 5]),\n",
       "   (4, [1, 4, 5]),\n",
       "   (4, [1, 3, 8, 4]),\n",
       "   (4, [8, 7, 1]),\n",
       "   (4, [3, 7]),\n",
       "   (4, [8, 4, 3]),\n",
       "   (4, [2, 5]),\n",
       "   (4, [5, 4]),\n",
       "   (4, [1, 2, 3, 5, 6, 7, 8, 4]),\n",
       "   (4, [6, 1, 3, 4]),\n",
       "   (4, [6, 5, 8, 3]),\n",
       "   (4, [7, 2, 1, 6]),\n",
       "   (4, [6, 7]),\n",
       "   (4, [3, 2]),\n",
       "   (4, [1, 6, 8, 3]),\n",
       "   (4, [8, 4, 3, 1]),\n",
       "   (4, [6, 5, 3, 8]),\n",
       "   (4, [6, 5, 3]),\n",
       "   (4, [3, 5, 8, 1]),\n",
       "   (4, [7, 1, 2, 3]),\n",
       "   (4, [1, 2, 5, 7]),\n",
       "   (4, [6, 1, 7, 3]),\n",
       "   (4, [1, 8, 4]),\n",
       "   (4, [1, 2, 3, 6]),\n",
       "   (4, [3, 4, 1, 8]),\n",
       "   (4, [1, 5, 7, 8]),\n",
       "   (4, [1, 6, 4]),\n",
       "   (4, [8, 5, 7, 6]),\n",
       "   (4, [7, 2, 6]),\n",
       "   (4, [1, 6, 3, 4]),\n",
       "   (4, [1, 6, 3, 2]),\n",
       "   (4, [1, 3, 4, 6]),\n",
       "   (4, [1, 3, 4, 8]),\n",
       "   (4, [8, 5, 2, 6]),\n",
       "   (4, [3, 1, 6, 8]),\n",
       "   (4, [3, 1, 6, 7]),\n",
       "   (4, [3, 8, 6, 4]),\n",
       "   (4, [3, 8, 6, 1]),\n",
       "   (4, [7, 2, 5, 8]),\n",
       "   (4, [5, 3, 6]),\n",
       "   (4, [8, 1, 2]),\n",
       "   (4, [5, 6, 3]),\n",
       "   (3, [6, 7, 1, 5]),\n",
       "   (3, [5, 1, 6, 8]),\n",
       "   (3, [1, 8, 5, 6]),\n",
       "   (3, [1, 8, 5, 2]),\n",
       "   (3, [7, 5, 1]),\n",
       "   (3, [1, 5, 8, 7]),\n",
       "   (3, [2, 6, 3]),\n",
       "   (3, [1, 2, 7, 6, 3, 8, 5, 4]),\n",
       "   (3, [7, 8, 5, 2]),\n",
       "   (3, [4, 3, 8, 1]),\n",
       "   (3, [6, 7, 5]),\n",
       "   (3, [6, 7, 3]),\n",
       "   (3, [8, 1, 5, 7]),\n",
       "   (3, [3, 6, 7, 8]),\n",
       "   (3, [3, 6, 7, 1]),\n",
       "   (3, [5, 1, 2]),\n",
       "   (3, [5, 1, 7]),\n",
       "   (3, [1, 3, 4, 6, 5, 8, 2, 7]),\n",
       "   (3, [3, 5, 8, 4, 6, 1, 2, 7]),\n",
       "   (3, [8, 5, 3, 6, 4, 2, 1, 7]),\n",
       "   (3, [3, 6, 1, 2]),\n",
       "   (3, [5, 2, 8, 7]),\n",
       "   (3, [7, 6, 3, 2]),\n",
       "   (3, [1, 6, 5, 3]),\n",
       "   (3, [5, 8, 2, 6]),\n",
       "   (3, [3, 1, 5, 8]),\n",
       "   (3, [1, 7, 3, 8]),\n",
       "   (3, [6, 1, 3, 2]),\n",
       "   (3, [8, 5, 4, 3, 6]),\n",
       "   (3, [6, 4, 3, 1]),\n",
       "   (3, [6, 4, 8]),\n",
       "   (3, [8, 5, 1, 2, 7, 3, 6, 4]),\n",
       "   (3, [1, 5, 4, 8]),\n",
       "   (3, [6, 8, 7]),\n",
       "   (3, [5, 2, 8]),\n",
       "   (3, [1, 7, 4, 2]),\n",
       "   (3, [8, 5, 7, 2, 3, 6, 1, 4]),\n",
       "   (3, [3, 1, 4, 8]),\n",
       "   (3, [6, 1, 8]),\n",
       "   (3, [3, 6, 1, 2, 7]),\n",
       "   (3, [1, 8, 7]),\n",
       "   (3, [7, 5, 2, 8]),\n",
       "   (3, [1, 3, 2, 7]),\n",
       "   (3, [1, 2, 6, 7, 3]),\n",
       "   (3, [1, 7, 6, 3]),\n",
       "   (3, [1, 7, 5, 8]),\n",
       "   (3, [7, 3, 2]),\n",
       "   (3, [7, 3, 1]),\n",
       "   (3, [6, 1, 4, 3]),\n",
       "   (3, [1, 2, 4, 3]),\n",
       "   (3, [5, 8, 4, 3]),\n",
       "   (3, [8, 6, 3, 5]),\n",
       "   (3, [2, 8, 6]),\n",
       "   (3, [6, 2, 7, 8]),\n",
       "   (3, [1, 2, 6, 3]),\n",
       "   (3, [5, 8, 6, 4]),\n",
       "   (3, [6, 3, 5]),\n",
       "   (3, [3, 8, 1, 6]),\n",
       "   (3, [3, 8, 1, 5]),\n",
       "   (3, [8, 3, 7]),\n",
       "   (3, [5, 3, 8, 6, 4, 7, 2, 1]),\n",
       "   (3, [6, 8, 3, 1]),\n",
       "   (3, [5, 8, 3, 2]),\n",
       "   (3, [5, 8, 3, 1]),\n",
       "   (3, [6, 7, 2, 3]),\n",
       "   (3, [1, 4, 2]),\n",
       "   (3, [1, 2, 8]),\n",
       "   (3, [3, 6, 7, 2]),\n",
       "   (3, [1, 2, 3, 7]),\n",
       "   (3, [1, 5, 3, 8]),\n",
       "   (3, [3, 4, 1, 6]),\n",
       "   (3, [5, 8, 7, 1]),\n",
       "   (3, [3, 4, 5, 8]),\n",
       "   (3, [1, 6, 5]),\n",
       "   (3, [4, 3, 6, 1, 8, 5, 2, 7]),\n",
       "   (3, [2, 7, 6]),\n",
       "   (3, [2, 6, 7, 8]),\n",
       "   (3, [1, 6, 3, 7]),\n",
       "   (3, [8, 5, 6, 4]),\n",
       "   (3, [1, 5, 2]),\n",
       "   (3, [8, 5, 1, 7]),\n",
       "   (3, [4, 5, 8]),\n",
       "   (3, [6, 2, 3]),\n",
       "   (3, [6, 2, 1]),\n",
       "   (3, [8, 5, 1, 3, 6]),\n",
       "   (3, [8, 5, 3, 1, 7, 2, 6, 4]),\n",
       "   (3, [7, 1, 3, 6]),\n",
       "   (3, [8, 7, 5, 2]),\n",
       "   (3, [7, 8, 2]),\n",
       "   (3, [5, 7, 8, 2]),\n",
       "   (3, [5, 1, 3, 4]),\n",
       "   (3, [1, 3, 8, 2]),\n",
       "   (3, [1, 6, 3, 2, 7]),\n",
       "   (3, [3, 6, 8, 1]),\n",
       "   (3, [8, 1, 6, 3]),\n",
       "   (3, [6, 2, 7, 1]),\n",
       "   (3, [3, 2, 6]),\n",
       "   (3, [3, 2, 1]),\n",
       "   (3, [2, 5, 7, 8]),\n",
       "   (3, [2, 8, 5]),\n",
       "   (3, [1, 5, 6]),\n",
       "   (3, [1, 7, 2, 6, 3]),\n",
       "   (3, [5, 8, 1, 2, 3, 6, 4, 7]),\n",
       "   (3, [5, 6, 7]),\n",
       "   (3, [4, 3, 1, 8, 6, 5, 2, 7]),\n",
       "   (2, [5, 1, 2, 6]),\n",
       "   (2, [5, 1, 2, 8]),\n",
       "   (2, [4, 8, 6, 3]),\n",
       "   (2, [5, 8, 7, 6, 2]),\n",
       "   (2, [1, 8, 5, 7]),\n",
       "   (2, [7, 5, 8]),\n",
       "   (2, [5, 8, 1, 3, 6, 2, 7, 4]),\n",
       "   (2, [1, 8, 3, 4]),\n",
       "   (2, [3, 8, 4, 7]),\n",
       "   (2, [3, 8, 4, 1]),\n",
       "   (2, [4, 8, 5, 3]),\n",
       "   (2, [8, 1, 3, 4, 5, 2, 7, 6]),\n",
       "   (2, [5, 7, 6, 8]),\n",
       "   (2, [1, 6, 2, 3]),\n",
       "   (2, [1, 6, 2, 3, 8]),\n",
       "   (2, [7, 5, 8, 2]),\n",
       "   (2, [8, 5, 6, 3, 4, 1, 2, 7]),\n",
       "   (2, [1, 5, 8, 3, 6, 7, 2, 4]),\n",
       "   (2, [7, 3, 2, 8]),\n",
       "   (2, [3, 7, 5, 6]),\n",
       "   (2, [5, 3, 4, 8]),\n",
       "   (2, [6, 3, 8, 5, 4]),\n",
       "   (2, [4, 3, 6, 1]),\n",
       "   (2, [4, 3, 6, 2]),\n",
       "   (2, [1, 5, 8, 6, 2]),\n",
       "   (2, [5, 8, 1, 3, 4, 2, 6, 7]),\n",
       "   (2, [6, 2, 8, 5]),\n",
       "   (2, [5, 8, 6, 2]),\n",
       "   (2, [7, 1, 8]),\n",
       "   (2, [8, 5, 3, 4, 6]),\n",
       "   (2, [1, 8, 2, 3]),\n",
       "   (2, [6, 3, 5, 8, 1, 2, 7, 4]),\n",
       "   (2, [1, 8, 3, 2]),\n",
       "   (2, [3, 5, 4, 8]),\n",
       "   (2, [4, 3, 5, 8, 6]),\n",
       "   (2, [3, 5, 4, 6]),\n",
       "   (2, [8, 5, 6, 7]),\n",
       "   (2, [4, 6, 3, 8]),\n",
       "   (2, [3, 1, 7, 8]),\n",
       "   (2, [8, 1, 5, 6]),\n",
       "   (2, [3, 1, 7, 6]),\n",
       "   (2, [3, 1, 5, 8, 2, 7, 6, 4]),\n",
       "   (2, [1, 3, 7, 2, 6, 4, 8, 5]),\n",
       "   (2, [3, 5, 6, 8, 7, 4, 2, 1]),\n",
       "   (2, [5, 8, 7, 6, 2, 3, 1, 4]),\n",
       "   (2, [6, 7, 2, 1, 3, 4, 5, 8]),\n",
       "   (2, [8, 3, 5, 4]),\n",
       "   (2, [5, 1, 8, 3, 2, 4, 6, 7]),\n",
       "   (2, [8, 5, 1, 4]),\n",
       "   (2, [1, 3, 7, 4]),\n",
       "   (2, [1, 3, 7, 6]),\n",
       "   (2, [5, 8, 4, 3, 1, 2, 7, 6]),\n",
       "   (2, [3, 4, 6, 8, 5, 1, 2, 7]),\n",
       "   (2, [1, 3, 5, 8, 6, 2, 7, 4]),\n",
       "   (2, [1, 3, 4, 5, 6, 7, 8, 2]),\n",
       "   (2, [3, 6, 1, 7]),\n",
       "   (2, [5, 2, 8, 1]),\n",
       "   (2, [6, 3, 1, 5]),\n",
       "   (2, [6, 1, 4]),\n",
       "   (2, [1, 6, 5, 8]),\n",
       "   (2, [3, 6, 5, 1]),\n",
       "   (2, [2, 5, 8]),\n",
       "   (2, [5, 4, 3, 8]),\n",
       "   (2, [1, 7, 3, 6]),\n",
       "   (2, [2, 8]),\n",
       "   (2, [8, 5, 7, 1, 6, 2, 3, 4]),\n",
       "   (2, [8, 5, 3, 6, 2, 1, 7, 4]),\n",
       "   (2, [8, 5, 6, 1, 2, 3, 4, 7]),\n",
       "   (2, [8, 6, 7]),\n",
       "   (2, [1, 2, 7, 6, 3, 8, 4, 5]),\n",
       "   (2, [2, 1, 6, 3]),\n",
       "   (2, [4, 3, 5, 1]),\n",
       "   (2, [8, 5, 3, 2]),\n",
       "   (2, [7, 6, 1, 3]),\n",
       "   (2, [6, 4, 5]),\n",
       "   (2, [6, 4, 1]),\n",
       "   (2, [8, 5, 6, 2]),\n",
       "   (2, [6, 8, 4]),\n",
       "   (2, [1, 2, 7, 6, 8]),\n",
       "   (2, [5, 1, 3, 6]),\n",
       "   (2, [2, 1, 3, 6]),\n",
       "   (2, [4, 1, 6]),\n",
       "   (2, [5, 1, 7, 8]),\n",
       "   (2, [1, 8, 2, 5]),\n",
       "   (2, [1, 3, 2, 4, 5, 7, 6, 8]),\n",
       "   (2, [1, 3, 4, 6, 5, 8, 7, 2]),\n",
       "   (2, [1, 7, 2, 3, 6, 8, 4, 5]),\n",
       "   (2, [7, 8, 3]),\n",
       "   (2, [1, 3, 6, 2, 4, 7, 5, 8]),\n",
       "   (2, [3, 6, 8, 5, 1]),\n",
       "   (2, [8, 6, 7, 5]),\n",
       "   (2, [4, 3, 6, 8]),\n",
       "   (2, [5, 6, 3, 8]),\n",
       "   (2, [3, 5, 6, 8, 7, 4]),\n",
       "   (2, [1, 3, 6, 5, 8, 4, 2, 7]),\n",
       "   (2, [1, 2, 7, 6, 3, 5, 8, 4]),\n",
       "   (2, [1, 5, 2, 7, 8]),\n",
       "   (2, [8, 1, 2, 7]),\n",
       "   (2, [8, 1, 2, 5]),\n",
       "   (2, [8, 1, 2, 4]),\n",
       "   (2, [3, 8, 5, 1, 2, 6, 7, 4]),\n",
       "   (2, [1, 3, 2, 4]),\n",
       "   (2, [3, 8, 7, 1]),\n",
       "   (2, [1, 5, 8, 3, 6]),\n",
       "   (2, [1, 7, 6, 2, 3, 4, 5, 8]),\n",
       "   (2, [7, 2, 1, 6, 3, 4, 8, 5]),\n",
       "   (2, [5, 4, 8]),\n",
       "   (2, [5, 4, 6]),\n",
       "   (2, [5, 4, 1]),\n",
       "   (2, [4, 2]),\n",
       "   (2, [8, 3, 4, 5]),\n",
       "   (2, [3, 6, 1, 4, 8]),\n",
       "   (2, [6, 3, 2, 1]),\n",
       "   (2, [8, 3, 6, 7]),\n",
       "   (2, [5, 8, 2, 1, 6, 7, 3, 4]),\n",
       "   (2, [4, 3, 2]),\n",
       "   (2, [1, 3, 5, 6, 8, 7, 4]),\n",
       "   (2, [1, 7, 6, 5]),\n",
       "   (2, [8, 1, 5, 4]),\n",
       "   (2, [1, 4, 3, 2, 6, 7, 5, 8]),\n",
       "   (2, [8, 5, 4, 3, 6, 7, 2, 1]),\n",
       "   (2, [1, 4, 6, 2]),\n",
       "   (2, [1, 2, 8, 5]),\n",
       "   (2, [8, 5, 7, 3, 4]),\n",
       "   (2, [3, 8, 7]),\n",
       "   (2, [1, 5, 8, 3, 4, 2, 6, 7]),\n",
       "   (2, [8, 5, 3, 6, 1, 7, 2, 4]),\n",
       "   (2, [6, 8, 3, 4]),\n",
       "   (2, [3, 4, 1, 6, 7, 2, 8, 5]),\n",
       "   (2, [8, 1, 7, 5]),\n",
       "   (2, [7, 8, 5]),\n",
       "   (2, [1, 3, 6, 8, 4]),\n",
       "   (2, [1, 6, 2, 7, 3, 4, 5, 8]),\n",
       "   (2, [5, 8, 2, 3]),\n",
       "   (2, [1, 3, 2, 7, 6]),\n",
       "   (2, [3, 4, 6, 5]),\n",
       "   (2, [1, 2, 6, 7, 3, 4, 5, 8]),\n",
       "   (2, [1, 4, 2, 7]),\n",
       "   (2, [7, 1, 2, 8, 6, 5, 3, 4]),\n",
       "   (2, [4, 2, 1]),\n",
       "   (2, [5, 1, 8, 2, 3, 4, 6, 7]),\n",
       "   (2, [7, 3, 6]),\n",
       "   (2, [8, 7, 2, 5]),\n",
       "   (2, [8, 5, 7, 1, 2, 3, 4, 6]),\n",
       "   (2, [4, 8, 6]),\n",
       "   (2, [8, 5, 6, 3, 7, 1, 2, 4]),\n",
       "   (2, [5, 1, 8, 6]),\n",
       "   (2, [5, 1, 8, 2]),\n",
       "   (2, [6, 7, 3, 2]),\n",
       "   (2, [6, 3, 5, 8, 2]),\n",
       "   (2, [5, 8, 1, 2, 4, 3, 7, 6]),\n",
       "   (2, [2, 7, 6, 3]),\n",
       "   (2, [4, 3, 8, 5, 6, 1, 7, 2]),\n",
       "   (2, [1, 5, 8, 4, 3, 6, 2, 7]),\n",
       "   (2, [6, 3, 4, 5, 8, 2, 1, 7]),\n",
       "   (2, [6, 3, 1, 8, 5, 4]),\n",
       "   (2, [8, 4, 5, 3]),\n",
       "   (2, [8, 5, 2, 7]),\n",
       "   (2, [8, 7, 1, 5]),\n",
       "   (2, [6, 5, 1]),\n",
       "   (2, [6, 1, 2, 7]),\n",
       "   (2, [1, 2, 3, 4, 7, 6, 5, 8]),\n",
       "   (2, [3, 5, 8, 7]),\n",
       "   (2, [3, 5, 8, 4]),\n",
       "   (2, [3, 5, 8, 2]),\n",
       "   (2, [7, 1, 2, 5]),\n",
       "   (2, [7, 1, 2, 6]),\n",
       "   (2, [7, 1, 6, 2]),\n",
       "   (2, [7, 1, 6, 4]),\n",
       "   (2, [8, 5, 2, 7, 1]),\n",
       "   (2, [8, 3, 2]),\n",
       "   (2, [8, 5, 1, 3, 2, 6, 4, 7]),\n",
       "   (2, [8, 7, 6, 5]),\n",
       "   (2, [3, 1, 5, 6]),\n",
       "   (2, [6, 8, 3, 5]),\n",
       "   (2, [6, 1, 5, 3]),\n",
       "   (2, [8, 5, 2, 3, 6, 7, 4, 1]),\n",
       "   (2, [6, 3, 2, 7]),\n",
       "   (2, [1, 4, 3, 5]),\n",
       "   (2, [6, 4, 1, 3]),\n",
       "   (2, [6, 1, 7, 2]),\n",
       "   (2, [6, 3, 4, 1]),\n",
       "   (2, [8, 5, 1, 2, 4, 3, 6, 7]),\n",
       "   (2, [1, 8, 2]),\n",
       "   (2, [6, 8, 5, 4]),\n",
       "   (2, [5, 8, 4, 3, 6, 7, 2, 1]),\n",
       "   (2, [4, 8, 2]),\n",
       "   (2, [1, 8, 2, 5, 3, 4, 6, 7]),\n",
       "   (2, [8, 5, 3, 1, 6, 7, 2]),\n",
       "   (2, [8, 6, 5, 3]),\n",
       "   (2, [1, 6, 3, 7, 2]),\n",
       "   (2, [1, 5, 3, 7]),\n",
       "   (2, [1, 2, 7, 8]),\n",
       "   (2, [5, 1, 2, 3, 4, 6, 7, 8]),\n",
       "   (2, [6, 7, 8, 5]),\n",
       "   (2, [8, 5, 3, 6, 7, 4, 2, 1]),\n",
       "   (2, [6, 8, 1, 7]),\n",
       "   (2, [6, 8, 1, 5]),\n",
       "   (2, [1, 7, 2, 3, 6, 5, 8, 4]),\n",
       "   (2, [5, 2, 1]),\n",
       "   (2, [5, 8, 7, 2]),\n",
       "   (2, [5, 8, 7, 6]),\n",
       "   (2, [1, 7, 5, 3]),\n",
       "   (2, [3, 5, 6, 1]),\n",
       "   (2, [1, 8, 6, 5]),\n",
       "   (2, [8, 5, 7, 1]),\n",
       "   (2, [8, 7, 3]),\n",
       "   (2, [7, 2, 8]),\n",
       "   (2, [1, 2, 5, 7, 8, 6, 3, 4]),\n",
       "   (2, [6, 2, 1, 7]),\n",
       "   (2, [2, 1, 6, 3, 7]),\n",
       "   (2, [1, 4, 3, 5, 8, 6, 7, 2]),\n",
       "   (2, [1, 6, 3, 5]),\n",
       "   (2, [3, 5, 6, 8, 4]),\n",
       "   (2, [2, 1, 5]),\n",
       "   (2, [2, 1, 8]),\n",
       "   (2, [7, 1, 8, 6, 2, 3, 4, 5]),\n",
       "   (2, [1, 4, 6, 7]),\n",
       "   (2, [6, 5, 8, 2]),\n",
       "   (2, [1, 3, 4, 5]),\n",
       "   (2, [4, 5, 6]),\n",
       "   (2, [2, 6, 3, 1]),\n",
       "   (2, [5, 1, 3, 8]),\n",
       "   (2, [6, 3, 8, 2]),\n",
       "   (2, [1, 6, 7, 3, 2]),\n",
       "   (2, [8, 1, 3, 5]),\n",
       "   (2, [8, 4, 3, 6]),\n",
       "   (2, [6, 3, 4, 5]),\n",
       "   (2, [8, 3, 1, 7]),\n",
       "   (2, [7, 2, 8, 1]),\n",
       "   (2, [1, 7, 8, 3]),\n",
       "   (2, [3, 6, 4, 1]),\n",
       "   (2, [3, 6, 4, 5]),\n",
       "   (2, [8, 5, 3, 2, 1, 6, 7, 4]),\n",
       "   (2, [1, 8, 7, 2, 5]),\n",
       "   (2, [3, 2, 6, 7]),\n",
       "   (2, [1, 6, 5, 3, 8]),\n",
       "   (2, [3, 1, 6, 2]),\n",
       "   (2, [1, 5, 6, 3]),\n",
       "   (2, [1, 5, 6, 8]),\n",
       "   (2, [5, 1, 8, 7, 3]),\n",
       "   (2, [3, 2, 7]),\n",
       "   (2, [3, 2, 4]),\n",
       "   (2, [6, 7, 3, 1]),\n",
       "   (2, [6, 8, 1, 3, 5, 4, 7, 2]),\n",
       "   (2, [1, 2, 7, 6, 3, 4, 5, 8]),\n",
       "   (2, [5, 3, 8, 4, 6, 1, 2, 7]),\n",
       "   (2, [1, 8, 7, 3]),\n",
       "   (2, [3, 5, 1, 7]),\n",
       "   (2, [3, 7, 2]),\n",
       "   (2, [1, 4, 5, 3]),\n",
       "   (2, [5, 7, 8, 3]),\n",
       "   (2, [5, 1, 6]),\n",
       "   (2, [1, 3, 8, 6, 4, 5, 2, 7]),\n",
       "   (2, [8, 1, 4]),\n",
       "   (2, [3, 6, 2, 1]),\n",
       "   (2, [3, 6, 2, 8]),\n",
       "   (2, [5, 6, 1]),\n",
       "   (2, [4, 2, 6, 3]),\n",
       "   (1, [3, 4, 2, 6]),\n",
       "   (1, [5, 3, 8, 2, 6, 1]),\n",
       "   (1, [1, 2, 3, 4, 6, 7, 5, 8]),\n",
       "   (1, [7, 1, 8, 5, 6, 2, 3, 4]),\n",
       "   (1, [8, 1, 5, 7, 2]),\n",
       "   (1, [1, 8, 7, 5, 3, 2, 4, 6]),\n",
       "   (1, [3, 1, 6, 2, 4, 7]),\n",
       "   (1, [5, 3, 4, 2]),\n",
       "   (1, [5, 1, 6, 3]),\n",
       "   (1, [6, 2, 8, 5, 1, 3, 7, 4]),\n",
       "   (1, [5, 3, 4, 1]),\n",
       "   (1, [8, 7, 6, 2, 1, 3, 5, 4]),\n",
       "   (1, [4, 6, 2, 3, 7, 1, 8, 5]),\n",
       "   (1, [3, 6, 5, 8, 2, 1, 4]),\n",
       "   (1, [1, 3, 6, 2, 5, 8, 7, 4]),\n",
       "   (1, [4, 3, 8, 5, 6, 1, 2, 7]),\n",
       "   (1, [6, 1, 3, 7, 2]),\n",
       "   (1, [5, 8, 1, 4, 6, 7, 2, 3]),\n",
       "   (1, [7, 2, 1, 6, 3, 8, 5, 4]),\n",
       "   (1, [8, 1, 5, 3, 2]),\n",
       "   (1, [1, 3, 4, 6, 2, 5, 7, 8]),\n",
       "   (1, [8, 5, 7, 1, 3, 4, 2, 6]),\n",
       "   (1, [2, 8, 5, 6]),\n",
       "   (1, [7, 5, 6]),\n",
       "   (1, [7, 5, 2]),\n",
       "   (1, [6, 8, 1]),\n",
       "   (1, [3, 1, 6, 4, 8, 2, 7, 5]),\n",
       "   (1, [6, 3, 2, 1, 7, 5, 8, 4]),\n",
       "   (1, [8, 5, 3, 2, 6]),\n",
       "   (1, [5, 8, 1, 4]),\n",
       "   (1, [4, 3, 8, 5, 6, 7, 2, 1]),\n",
       "   (1, [4, 3, 6, 1, 8, 5]),\n",
       "   (1, [1, 8, 3, 6]),\n",
       "   (1, [6, 3, 5, 8, 7]),\n",
       "   (1, [1, 8, 3, 5]),\n",
       "   (1, [3, 1, 4, 5, 6, 7, 8, 2]),\n",
       "   (1, [7, 1, 5]),\n",
       "   (1, [1, 5, 8, 4]),\n",
       "   (1, [2, 4, 1]),\n",
       "   (1, [8, 2, 7, 1]),\n",
       "   (1, [1, 5, 8, 2]),\n",
       "   (1, [4, 8, 5, 1]),\n",
       "   (1, [3, 8, 4, 5]),\n",
       "   (1, [1, 2, 4, 5, 6, 3, 7, 8]),\n",
       "   (1, [5, 8, 7, 1, 3, 2, 4, 6]),\n",
       "   (1, [1, 6, 7, 2, 3, 5, 8, 4]),\n",
       "   (1, [1, 8, 6, 3, 5, 4, 2, 7]),\n",
       "   (1, [8, 5, 7, 2, 1, 3, 6, 4]),\n",
       "   (1, [1, 6, 2, 8]),\n",
       "   (1, [5, 7, 6, 3]),\n",
       "   (1, [3, 4, 6, 5, 8, 1, 2, 7]),\n",
       "   (1, [6, 5, 1, 3]),\n",
       "   (1, [6, 3, 5, 2]),\n",
       "   (1, [8, 5, 4, 3, 6, 2, 1, 7]),\n",
       "   (1, [6, 1, 3, 8, 5, 4]),\n",
       "   (1, [1, 6, 2, 8, 3, 7, 4, 5]),\n",
       "   (1, [1, 5, 8, 3, 2, 4, 6, 7]),\n",
       "   (1, [2, 6, 8]),\n",
       "   (1, [8, 3, 2, 7]),\n",
       "   (1, [8, 5, 3, 7, 2, 1, 6, 4]),\n",
       "   (1, [7, 2, 3, 1]),\n",
       "   (1, [4, 3, 1, 2, 5, 8, 7, 6]),\n",
       "   (1, [8, 6, 5, 3, 1]),\n",
       "   (1, [7, 5, 8, 4]),\n",
       "   (1, [6, 2, 3, 1, 8, 5, 4, 7]),\n",
       "   (1, [6, 7, 2, 3, 1, 8, 5, 4]),\n",
       "   (1, [8, 7, 5, 6, 4, 3, 2, 1]),\n",
       "   (1, [8, 1, 3, 2]),\n",
       "   (1, [8, 5, 2, 1, 3, 6, 7, 4]),\n",
       "   (1, [1, 3, 8, 5, 4, 6, 2, 7]),\n",
       "   (1, [1, 6, 7, 8, 5]),\n",
       "   (1, [5, 3, 8, 7]),\n",
       "   (1, [2, 1, 7, 6]),\n",
       "   (1, [1, 5, 6, 3, 8]),\n",
       "   (1, [1, 4, 8, 3, 6, 5, 7, 2]),\n",
       "   (1, [5, 8, 4, 6, 2, 7, 1, 3]),\n",
       "   (1, [8, 5, 7, 1, 2, 6, 3, 4]),\n",
       "   (1, [8, 6, 5, 4, 3, 2, 1, 7]),\n",
       "   (1, [8, 5, 1, 6, 2, 7]),\n",
       "   (1, [8, 5, 4, 3, 7, 6, 1, 2]),\n",
       "   (1, [7, 8, 2, 1]),\n",
       "   (1, [1, 3, 2, 5, 6, 8, 7, 4]),\n",
       "   (1, [1, 3, 6, 8, 4, 2, 7, 5]),\n",
       "   (1, [8, 5, 3, 4, 6, 1, 2, 7]),\n",
       "   (1, [3, 1, 4, 5, 6]),\n",
       "   (1, [4, 6, 3, 8, 5, 2, 1, 7]),\n",
       "   (1, [2, 5, 8, 6]),\n",
       "   (1, [8, 5, 1, 3, 7, 6, 2, 4]),\n",
       "   (1, [6, 1, 2, 3, 7, 8, 5, 4]),\n",
       "   (1, [2, 5, 8, 1]),\n",
       "   (1, [2, 1, 3, 7]),\n",
       "   (1, [7, 4, 1, 6, 2, 3, 8, 5]),\n",
       "   (1, [1, 4, 6, 3, 2, 7, 5, 8]),\n",
       "   (1, [6, 1, 5, 8, 3]),\n",
       "   (1, [8, 5, 3, 7, 6, 2, 1, 4]),\n",
       "   (1, [5, 2, 8, 7, 3, 6, 1, 4]),\n",
       "   (1, [1, 6, 4, 3]),\n",
       "   (1, [1, 6, 7, 3, 2, 5, 8, 4]),\n",
       "   (1, [6, 1, 3, 8, 7, 5]),\n",
       "   (1, [1, 6, 4, 8]),\n",
       "   (1, [1, 4, 6, 3, 5, 8, 2, 7]),\n",
       "   (1, [2, 1, 6, 4, 3, 7, 5, 8]),\n",
       "   (1, [5, 8, 7, 6, 1, 2, 3]),\n",
       "   (1, [1, 3, 5, 7, 8, 6, 4, 2]),\n",
       "   (1, [2, 1, 3, 5, 6, 7, 8]),\n",
       "   (1, [1, 4, 2, 6, 7]),\n",
       "   (1, [5, 8, 3, 6, 4, 2, 7, 1]),\n",
       "   (1, [3, 1, 7, 8, 6, 5, 4, 2]),\n",
       "   (1, [3, 6, 8, 2, 5, 1]),\n",
       "   (1, [1, 3, 5, 8, 6, 4, 2, 7]),\n",
       "   (1, [7, 1, 3, 4, 2, 6]),\n",
       "   (1, [5, 1, 3, 8, 6, 2, 4, 7]),\n",
       "   (1, [1, 3, 5, 8, 6]),\n",
       "   (1, [8, 1, 3, 6]),\n",
       "   (1, [6, 3, 4, 5, 8, 2, 7]),\n",
       "   (1, [1, 6, 3, 7, 2, 8, 5, 4]),\n",
       "   (1, [3, 1, 5, 7, 8, 6, 2, 4]),\n",
       "   (1, [6, 5, 3, 8, 2, 1, 4, 7]),\n",
       "   (1, [5, 8, 6, 1, 3]),\n",
       "   (1, [7, 1, 8, 3]),\n",
       "   (1, [8, 5, 3, 6, 4, 1, 2, 7]),\n",
       "   (1, [5, 7, 8, 6, 3, 2, 4, 1]),\n",
       "   (1, [6, 1, 4, 3, 7, 2, 8, 5]),\n",
       "   (1, [1, 5, 8, 6, 7]),\n",
       "   (1, [4, 6, 3, 7, 2, 1, 8, 5]),\n",
       "   (1, [1, 5, 8, 3, 6, 4, 2, 7]),\n",
       "   (1, [1, 5, 3, 6, 8]),\n",
       "   (1, [4, 6, 8]),\n",
       "   (1, [5, 3, 1, 2, 8]),\n",
       "   (1, [2, 8, 6, 1, 4, 3, 5, 7]),\n",
       "   (1, [4, 6, 5]),\n",
       "   (1, [5, 3, 8, 4, 6, 7, 2, 1]),\n",
       "   (1, [4, 6, 7]),\n",
       "   (1, [7, 5, 2, 6, 8, 1, 3, 4]),\n",
       "   (1, [4, 6, 1]),\n",
       "   (1, [5, 4, 3, 8, 1, 6, 2, 7]),\n",
       "   (1, [1, 8, 5, 4]),\n",
       "   (1, [8, 5, 2, 1, 7]),\n",
       "   (1, [8, 5, 1, 2, 3, 4, 6, 7]),\n",
       "   (1, [7, 3, 2, 6]),\n",
       "   (1, [4, 1, 2, 3, 7, 6, 5, 8]),\n",
       "   (1, [1, 4, 6, 3, 7, 2, 5, 8]),\n",
       "   (1, [7, 6, 1, 2, 3, 5, 8]),\n",
       "   (1, [8, 5, 7, 1, 3, 2, 4, 6]),\n",
       "   (1, [1, 7, 2, 5, 3, 8, 6, 4]),\n",
       "   (1, [5, 7, 8, 6, 2, 1, 3, 4]),\n",
       "   (1, [8, 4, 1, 2]),\n",
       "   (1, [3, 1, 4, 5, 8, 6, 2, 7]),\n",
       "   (1, [6, 1, 8, 5, 7, 3, 2]),\n",
       "   (1, [1, 4, 2, 7, 3, 6, 8, 5]),\n",
       "   (1, [8, 5, 3, 1, 6, 2, 7, 4]),\n",
       "   (1, [5, 3, 2, 6]),\n",
       "   (1, [7, 6, 1, 2, 5, 8, 3, 4]),\n",
       "   (1, [1, 7, 2, 5]),\n",
       "   (1, [6, 7, 8, 1, 3, 5, 4, 2]),\n",
       "   (1, [5, 3, 2, 8]),\n",
       "   (1, [1, 5, 6, 4, 2, 3, 8, 7]),\n",
       "   (1, [2, 1, 7, 6, 3, 5, 8, 4]),\n",
       "   (1, [6, 7, 8]),\n",
       "   (1, [5, 8, 6, 3, 1, 7, 4, 2]),\n",
       "   (1, [3, 1, 8, 2, 4, 5, 6, 7]),\n",
       "   (1, [5, 8, 3, 6, 4, 7, 2]),\n",
       "   (1, [1, 3, 7, 2, 6, 8, 5, 4]),\n",
       "   (1, [2, 1, 6, 5, 3, 8]),\n",
       "   (1, [7, 1, 2, 4]),\n",
       "   (1, [1, 5, 8, 2, 7, 3, 6, 4]),\n",
       "   (1, [1, 5, 8, 6, 2, 7, 3, 4]),\n",
       "   (1, [5, 8, 3, 6, 2, 4, 7, 1]),\n",
       "   (1, [7, 3, 2, 5]),\n",
       "   (1, [1, 3, 6, 4, 2, 5, 7, 8]),\n",
       "   (1, [8, 5, 3, 1, 4, 2, 6, 7]),\n",
       "   (1, [4, 3, 1, 5, 8, 2, 6, 7]),\n",
       "   (1, [5, 8, 4, 6, 7, 3, 2, 1]),\n",
       "   (1, [5, 3, 8, 7, 6]),\n",
       "   (1, [8, 5, 3, 6, 2, 7, 1, 4]),\n",
       "   (1, [2, 1, 7, 5, 8, 6, 3, 4]),\n",
       "   (1, [8, 6, 7, 2, 3]),\n",
       "   (1, [8, 3, 2, 1, 4, 5, 6, 7]),\n",
       "   (1, [8, 6, 7, 2, 5]),\n",
       "   (1, [7, 3, 1, 6, 8, 5, 4, 2]),\n",
       "   (1, [4, 2, 1, 6]),\n",
       "   (1, [3, 5, 7, 8]),\n",
       "   (1, [3, 5, 1, 8, 6, 4, 2, 7]),\n",
       "   (1, [7, 2, 3, 1, 8, 4, 6, 5]),\n",
       "   (1, [6, 3, 7, 1]),\n",
       "   (1, [4, 1, 3, 7, 6]),\n",
       "   (1, [5, 8, 7, 6, 2, 3]),\n",
       "   (1, [1, 4, 6, 7, 2, 3, 5, 8]),\n",
       "   (1, [1, 8, 2, 4]),\n",
       "   (1, [3, 6, 7, 5, 8]),\n",
       "   (1, [3, 7, 2, 1, 4, 6]),\n",
       "   (1, [6, 3, 7, 2, 1]),\n",
       "   (1, [8, 5, 1, 3, 2, 4, 7, 6]),\n",
       "   (1, [1, 7, 3, 8, 6, 5, 4, 2]),\n",
       "   (1, [1, 6, 3, 8, 4, 5, 2, 7]),\n",
       "   (1, [1, 5, 8, 3, 4, 6, 2, 7]),\n",
       "   (1, [6, 8, 5, 7, 3, 2, 1]),\n",
       "   (1, [1, 7, 8, 2, 3, 5, 4, 6]),\n",
       "   (1, [1, 5, 3, 4, 8]),\n",
       "   (1, [7, 8, 2, 6]),\n",
       "   (1, [5, 8, 3, 6, 7, 2, 4, 1]),\n",
       "   (1, [3, 5, 6, 7]),\n",
       "   (1, [6, 3, 5, 8, 4, 1, 7, 2]),\n",
       "   (1, [4, 1, 8, 6]),\n",
       "   (1, [3, 2, 5, 8]),\n",
       "   (1, [1, 2, 5, 8, 7, 6, 4, 3]),\n",
       "   (1, [3, 2, 5, 7]),\n",
       "   (1, [3, 6, 2, 5, 8]),\n",
       "   (1, [5, 8, 1, 2, 4, 3, 6, 7]),\n",
       "   (1, [8, 5, 3, 4, 6, 7, 2, 1]),\n",
       "   (1, [8, 5, 1, 2, 7]),\n",
       "   (1, [5, 7, 1, 3, 6, 2, 4, 8]),\n",
       "   (1, [8, 5, 6, 4, 3]),\n",
       "   (1, [8, 5, 1, 2, 3]),\n",
       "   (1, [4, 5, 8, 3, 6, 1, 7, 2]),\n",
       "   (1, [5, 6, 7, 8, 2, 1, 3, 4]),\n",
       "   (1, [4, 5, 1, 3]),\n",
       "   (1, [1, 3, 7, 2, 6, 4, 5, 8]),\n",
       "   (1, [3, 2, 1, 4]),\n",
       "   (1, [3, 1, 7, 2]),\n",
       "   (1, [2, 7, 5, 8, 1, 3, 6, 4]),\n",
       "   (1, [8, 5, 3, 1, 7, 2, 4, 6]),\n",
       "   (1, [5, 1, 3, 2, 4, 6, 8, 7]),\n",
       "   (1, [5, 6, 8, 1]),\n",
       "   (1, [5, 6, 8, 7]),\n",
       "   (1, [8, 7, 5, 3, 6, 2, 4, 1]),\n",
       "   (1, [2, 7, 1, 5]),\n",
       "   (1, [8, 2, 1, 6]),\n",
       "   (1, [2, 7, 1, 3]),\n",
       "   (1, [4, 3, 8, 5, 1, 6, 7, 2]),\n",
       "   (1, [6, 2, 3, 1, 7]),\n",
       "   (1, [5, 1, 3, 8, 2]),\n",
       "   (1, [3, 1, 2, 5, 4, 6, 7, 8]),\n",
       "   (1, [4, 1, 7, 2, 3, 6, 8, 5]),\n",
       "   (1, [8, 5, 1, 7, 3, 6, 2, 4]),\n",
       "   (1, [2, 1, 3, 6, 7, 4, 8]),\n",
       "   (1, [2, 1, 5, 7]),\n",
       "   (1, [1, 6, 3, 2, 4, 7, 8, 5]),\n",
       "   (1, [3, 7, 6, 2, 4, 1, 5]),\n",
       "   (1, [3, 6, 5, 8, 7, 2, 4, 1]),\n",
       "   (1, [1, 2, 3, 6, 5, 7, 8, 4]),\n",
       "   (1, [2, 4, 3]),\n",
       "   (1, [8, 2, 3]),\n",
       "   (1, [7, 6, 8, 5, 2, 1, 3, 4]),\n",
       "   (1, [4, 1, 5, 8, 6, 3]),\n",
       "   (1, [5, 7, 8, 3, 1]),\n",
       "   (1, [7, 1, 2, 5, 8, 3, 4, 6]),\n",
       "   (1, [5, 8, 3, 4, 6, 2, 1, 7]),\n",
       "   (1, [1, 6, 3, 4, 2, 7, 8, 5]),\n",
       "   (1, [8, 5, 1, 4, 3, 6, 2, 7]),\n",
       "   (1, [5, 3, 2]),\n",
       "   (1, [2, 6, 1, 3, 5, 4, 7, 8]),\n",
       "   (1, [8, 1, 3, 6, 4, 5]),\n",
       "   (1, [7, 3, 5]),\n",
       "   (1, [8, 1, 6, 5, 3]),\n",
       "   (1, [8, 7, 2, 1]),\n",
       "   (1, [7, 1, 4, 2, 3, 8, 6, 5]),\n",
       "   (1, [7, 8, 6, 1]),\n",
       "   (1, [7, 8, 5, 2, 3, 6, 1, 4]),\n",
       "   (1, [1, 5, 6, 7, 8, 4, 3, 2]),\n",
       "   (1, [5, 3, 8, 1, 6]),\n",
       "   (1, [8, 5, 2, 7, 6, 4, 3, 1]),\n",
       "   (1, [6, 2, 5, 8]),\n",
       "   (1, [8, 6, 1, 7, 3, 4, 5, 2]),\n",
       "   (1, [3, 5, 1, 8, 6, 2, 7, 4]),\n",
       "   (1, [1, 4, 3, 8, 5, 6, 2, 7]),\n",
       "   (1, [3, 6, 8, 5, 7, 4, 2, 1]),\n",
       "   (1, [6, 8, 7, 2]),\n",
       "   (1, [5, 6, 4, 8]),\n",
       "   (1, [1, 3, 6, 8, 5, 2, 4, 7]),\n",
       "   (1, [3, 6, 7, 1, 2]),\n",
       "   (1, [1, 6, 2, 7, 8, 3, 5, 4]),\n",
       "   (1, [3, 1, 2, 6, 8, 4]),\n",
       "   (1, [6, 8, 7, 2, 3, 5, 1]),\n",
       "   (1, [4, 6, 8, 5, 3]),\n",
       "   (1, [3, 8, 5, 7]),\n",
       "   (1, [3, 8, 5, 1]),\n",
       "   (1, [6, 2, 5, 1, 7, 8, 3, 4]),\n",
       "   (1, [1, 3, 7, 2]),\n",
       "   (1, [3, 8, 5, 1, 6, 2, 4, 7]),\n",
       "   (1, [1, 3, 7, 8]),\n",
       "   (1, [8, 5, 3, 6, 4]),\n",
       "   (1, [1, 7, 6, 5, 8, 3, 2, 4]),\n",
       "   (1, [6, 3, 2, 7, 1, 5, 4, 8]),\n",
       "   (1, [4, 1, 3, 5]),\n",
       "   (1, [2, 1, 7, 3, 5, 6, 8, 4]),\n",
       "   (1, [5, 2, 8, 1, 3, 4, 6, 7]),\n",
       "   (1, [4, 1, 3, 8]),\n",
       "   (1, [1, 4, 6, 8, 3, 2, 7, 5]),\n",
       "   (1, [1, 3, 8, 5, 6, 7, 2, 4]),\n",
       "   (1, [4, 3, 5, 8, 1, 6, 2, 7]),\n",
       "   (1, [7, 4, 3, 6]),\n",
       "   (1, [4, 1, 8, 5]),\n",
       "   (1, [6, 5, 7, 8, 3, 1]),\n",
       "   (1, [1, 7, 4, 2, 3, 6, 8, 5]),\n",
       "   (1, [3, 1, 2, 5, 6, 7, 8, 4]),\n",
       "   (1, [5, 6, 8, 2, 4, 1, 3, 7]),\n",
       "   (1, [2, 3, 8, 5]),\n",
       "   (1, [4, 6, 3, 5, 1, 8, 7, 2]),\n",
       "   (1, [1, 7, 6, 2, 8, 5]),\n",
       "   (1, [1, 3, 5, 7]),\n",
       "   (1, [3, 7, 1, 6]),\n",
       "   (1, [1, 3, 5, 2]),\n",
       "   (1, [1, 3, 2, 4, 6, 7, 5, 8]),\n",
       "   (1, [6, 1, 5]),\n",
       "   (1, [5, 8, 1, 4, 3, 6, 2, 7]),\n",
       "   (1, [1, 5, 8, 7, 6, 3, 2, 4]),\n",
       "   (1, [5, 8, 2, 6, 7, 3, 4, 1]),\n",
       "   (1, [5, 7, 1, 3]),\n",
       "   (1, [1, 7, 8, 3, 6, 5, 2, 4]),\n",
       "   (1, [1, 8, 3, 6, 5, 4, 2, 7]),\n",
       "   (1, [1, 2, 4, 6, 7]),\n",
       "   (1, [3, 4, 5, 1, 8, 6, 2, 7]),\n",
       "   (1, [5, 2, 8, 3]),\n",
       "   (1, [8, 1, 4, 5, 6]),\n",
       "   (1, [3, 1, 4, 6, 7]),\n",
       "   (1, [7, 8, 5, 1, 3, 2, 6, 4]),\n",
       "   (1, [3, 1, 5, 6, 2, 4, 7, 8]),\n",
       "   (1, [3, 2, 5, 1, 7, 4]),\n",
       "   (1, [6, 8, 5, 3, 1]),\n",
       "   (1, [3, 2, 5, 7, 6]),\n",
       "   (1, [3, 8, 5, 1, 2, 6, 4, 7]),\n",
       "   (1, [7, 1, 8, 2, 3, 5, 6, 4]),\n",
       "   (1, [8, 1, 7, 6, 5]),\n",
       "   ...]),\n",
       " (8654,\n",
       "  defaultdict(int,\n",
       "              {2: 0.25583545181419,\n",
       "               1: 0.24000462214005083,\n",
       "               3: 0.223249364455743,\n",
       "               4: 0.1509128726600416,\n",
       "               5: 0.03547492489022418,\n",
       "               9: 0.06655881673214699,\n",
       "               6: 0.014097527155072799,\n",
       "               8: 0.008319852091518374,\n",
       "               7: 0.005546568061012249,\n",
       "               0: 0}),\n",
       "  [(643, [4, 7]),\n",
       "   (558, [5]),\n",
       "   (501, [6]),\n",
       "   (433, [7, 4]),\n",
       "   (345, [4]),\n",
       "   (276, [7]),\n",
       "   (121, [9]),\n",
       "   (103, [6, 9]),\n",
       "   (99, [1]),\n",
       "   (87, [5, 1]),\n",
       "   (80, [8]),\n",
       "   (66, [4, 7, 9]),\n",
       "   (60, [5, 9]),\n",
       "   (58, [2]),\n",
       "   (57, [5, 8]),\n",
       "   (55, [6, 9, 1]),\n",
       "   (50, [7, 4, 9]),\n",
       "   (48, [4, 7, 5]),\n",
       "   (48, [5, 6]),\n",
       "   (47, [9, 1]),\n",
       "   (47, [5, 1, 9]),\n",
       "   (46, [5, 7]),\n",
       "   (45, [6, 1]),\n",
       "   (43, [5, 9, 1]),\n",
       "   (42, [6, 1, 9]),\n",
       "   (40, [4, 7, 1]),\n",
       "   (39, [7, 4, 1]),\n",
       "   (39, [3]),\n",
       "   (37, [1, 9]),\n",
       "   (36, [5, 4]),\n",
       "   (36, [6, 7]),\n",
       "   (34, [7, 4, 5]),\n",
       "   (32, [4, 7, 8]),\n",
       "   (32, [4, 5]),\n",
       "   (32, [6, 2]),\n",
       "   (29, [4, 5, 7]),\n",
       "   (29, [5, 4, 7]),\n",
       "   (28, [5, 7, 4]),\n",
       "   (27, [1, 5]),\n",
       "   (26, [2, 6]),\n",
       "   (26, [6, 5, 9]),\n",
       "   (25, [5, 3]),\n",
       "   (25, [6, 5]),\n",
       "   (23, [5, 8, 3]),\n",
       "   (23, [9, 6]),\n",
       "   (22, [6, 7, 9]),\n",
       "   (22, [5, 3, 8]),\n",
       "   (22, [7, 4, 9, 1]),\n",
       "   (21, [7, 4, 6]),\n",
       "   (21, [5, 8, 1]),\n",
       "   (21, [7, 5]),\n",
       "   (20, [4, 7, 9, 1]),\n",
       "   (19, [9, 7, 4]),\n",
       "   (19, [7, 6]),\n",
       "   (18, [9, 6, 1]),\n",
       "   (18, [1, 5, 9]),\n",
       "   (17, [7, 4, 8]),\n",
       "   (17, [4, 1]),\n",
       "   (17, [4, 8]),\n",
       "   (17, [6, 9, 5]),\n",
       "   (17, [9, 5]),\n",
       "   (17, [7, 9]),\n",
       "   (16, [7, 5, 4]),\n",
       "   (16, [8, 5]),\n",
       "   (16, [1, 9, 5]),\n",
       "   (15, [4, 7, 1, 9]),\n",
       "   (15, [4, 7, 3]),\n",
       "   (15, [4, 9]),\n",
       "   (15, [5, 6, 1]),\n",
       "   (15, [5, 6, 9]),\n",
       "   (14, [4, 7, 6]),\n",
       "   (14, [6, 9, 7]),\n",
       "   (14, [8, 3]),\n",
       "   (14, [5, 6, 9, 1]),\n",
       "   (14, [9, 5, 1]),\n",
       "   (13, [6, 7, 4]),\n",
       "   (13, [5, 1, 3]),\n",
       "   (13, [4, 7, 9, 5]),\n",
       "   (13, [6, 5, 1]),\n",
       "   (13, [6, 9, 4]),\n",
       "   (13, [1, 6]),\n",
       "   (13, [9, 1, 5]),\n",
       "   (12, [4, 6, 7]),\n",
       "   (12, [5, 1, 8]),\n",
       "   (12, [6, 9, 5, 1]),\n",
       "   (12, [5, 7, 1]),\n",
       "   (12, [7, 4, 1, 9]),\n",
       "   (12, [1, 4, 7]),\n",
       "   (11, [6, 4, 7]),\n",
       "   (11, [1, 7, 4]),\n",
       "   (11, [2, 5]),\n",
       "   (11, [7, 9, 4]),\n",
       "   (11, [5, 9, 7]),\n",
       "   (11, [7, 6, 9]),\n",
       "   (10, [7, 4, 3]),\n",
       "   (10, [5, 8, 4]),\n",
       "   (10, [5, 8, 9]),\n",
       "   (10, [9, 6, 7]),\n",
       "   (10, [5, 3, 1]),\n",
       "   (9, [6, 5, 1, 9]),\n",
       "   (9, [5, 9, 7, 4]),\n",
       "   (9, [4, 1, 7]),\n",
       "   (9, [3, 5]),\n",
       "   (9, [4, 6]),\n",
       "   (9, [9, 7, 4, 1]),\n",
       "   (9, [5, 7, 8]),\n",
       "   (9, [5, 7, 9]),\n",
       "   (9, [7, 4, 5, 9]),\n",
       "   (9, [1, 7]),\n",
       "   (9, [8, 7, 4]),\n",
       "   (9, [5, 8, 1, 3]),\n",
       "   (9, [7, 8]),\n",
       "   (9, [7, 8, 4]),\n",
       "   (8, [7, 5, 9]),\n",
       "   (8, [5, 1, 6]),\n",
       "   (8, [4, 7, 1, 5]),\n",
       "   (8, [5, 4, 7, 8]),\n",
       "   (8, [5, 4, 7, 9]),\n",
       "   (8, [9, 7, 1]),\n",
       "   (8, [6, 4]),\n",
       "   (8, [6, 8]),\n",
       "   (8, [7, 4, 1, 5]),\n",
       "   (8, [5, 9, 6]),\n",
       "   (8, [7, 6, 4]),\n",
       "   (8, [6, 9, 1, 5]),\n",
       "   (8, [5, 2]),\n",
       "   (8, [4, 7, 8, 9]),\n",
       "   (8, [5, 3, 8, 1]),\n",
       "   (8, [5, 6, 7]),\n",
       "   (7, [5, 9, 1, 3]),\n",
       "   (7, [8, 5, 3]),\n",
       "   (7, [8, 5, 1]),\n",
       "   (7, [5, 3, 9]),\n",
       "   (7, [6, 1, 5]),\n",
       "   (7, [5, 4, 7, 1]),\n",
       "   (7, [9, 1, 6]),\n",
       "   (7, [6, 2, 9]),\n",
       "   (7, [5, 4, 9]),\n",
       "   (7, [6, 9, 1, 4]),\n",
       "   (7, [9, 6, 5]),\n",
       "   (7, [9, 7]),\n",
       "   (7, [5, 1, 8, 9]),\n",
       "   (7, [4, 7, 5, 8]),\n",
       "   (7, [1, 8]),\n",
       "   (7, [8, 3, 5]),\n",
       "   (7, [5, 8, 1, 9]),\n",
       "   (7, [5, 9, 8]),\n",
       "   (7, [7, 1]),\n",
       "   (7, [9, 1, 4]),\n",
       "   (7, [4, 5, 6]),\n",
       "   (7, [1, 5, 4]),\n",
       "   (7, [6, 5, 7]),\n",
       "   (6, [1, 9, 6]),\n",
       "   (6, [9, 7, 5]),\n",
       "   (6, [5, 1, 3, 8]),\n",
       "   (6, [5, 9, 6, 1]),\n",
       "   (6, [6, 7, 1]),\n",
       "   (6, [7, 1, 9]),\n",
       "   (6, [4, 5, 9]),\n",
       "   (6, [6, 1, 7]),\n",
       "   (6, [5, 7, 4, 9]),\n",
       "   (6, [6, 9, 7, 4]),\n",
       "   (6, [7, 9, 1]),\n",
       "   (6, [4, 5, 7, 1]),\n",
       "   (6, [4, 7, 5, 9]),\n",
       "   (6, [7, 4, 5, 1]),\n",
       "   (6, [4, 7, 9, 6]),\n",
       "   (6, [8, 1, 3]),\n",
       "   (6, [1, 3]),\n",
       "   (6, [8, 3, 1]),\n",
       "   (6, [5, 8, 3, 1]),\n",
       "   (6, [1, 4, 9]),\n",
       "   (6, [1, 8, 3]),\n",
       "   (6, [8, 4]),\n",
       "   (6, [5, 9, 4]),\n",
       "   (6, [5, 1, 9, 8]),\n",
       "   (6, [4, 5, 8]),\n",
       "   (6, [7, 4, 6, 9]),\n",
       "   (6, [1, 4, 7, 9]),\n",
       "   (6, [6, 5, 2]),\n",
       "   (5, [7, 5, 1]),\n",
       "   (5, [1, 9, 3]),\n",
       "   (5, [5, 2, 6]),\n",
       "   (5, [7, 1, 4]),\n",
       "   (5, [5, 1, 4]),\n",
       "   (5, [4, 7, 1, 8]),\n",
       "   (5, [6, 1, 4]),\n",
       "   (5, [9, 8]),\n",
       "   (5, [5, 4, 3, 1]),\n",
       "   (5, [2, 8]),\n",
       "   (5, [5, 1, 3, 9]),\n",
       "   (5, [7, 4, 2]),\n",
       "   (5, [1, 3, 5]),\n",
       "   (5, [4, 7, 6, 9]),\n",
       "   (5, [5, 4, 8]),\n",
       "   (5, [5, 4, 3]),\n",
       "   (5, [3, 8]),\n",
       "   (5, [7, 4, 8, 3]),\n",
       "   (5, [6, 9, 1, 2]),\n",
       "   (5, [7, 4, 9, 5]),\n",
       "   (5, [6, 5, 9, 1]),\n",
       "   (5, [4, 5, 7, 9]),\n",
       "   (5, [6, 7, 4, 9]),\n",
       "   (5, [5, 7, 3]),\n",
       "   (5, [4, 7, 5, 3]),\n",
       "   (5, [7, 4, 5, 8]),\n",
       "   (5, [7, 4, 1, 8]),\n",
       "   (5, [8, 1, 5]),\n",
       "   (5, [3, 5, 8]),\n",
       "   (5, [1, 4, 5]),\n",
       "   (5, [5, 9, 3]),\n",
       "   (5, [1, 6, 9]),\n",
       "   (5, [7, 3]),\n",
       "   (5, [4, 9, 1]),\n",
       "   (5, [4, 6, 7, 9]),\n",
       "   (5, [7, 4, 9, 6]),\n",
       "   (5, [9, 4]),\n",
       "   (5, [1, 5, 3]),\n",
       "   (5, [5, 6, 8]),\n",
       "   (5, [4, 7, 8, 5]),\n",
       "   (4, [1, 9, 7]),\n",
       "   (4, [2, 4, 5]),\n",
       "   (4, [9, 1, 4, 7]),\n",
       "   (4, [8, 5, 4]),\n",
       "   (4, [4, 3, 7, 8]),\n",
       "   (4, [5, 1, 7]),\n",
       "   (4, [5, 3, 7]),\n",
       "   (4, [4, 7, 3, 1]),\n",
       "   (4, [4, 7, 3, 2]),\n",
       "   (4, [4, 1, 7, 5]),\n",
       "   (4, [7, 1, 5]),\n",
       "   (4, [5, 4, 7, 6]),\n",
       "   (4, [5, 4, 7, 3]),\n",
       "   (4, [4, 1, 5, 7]),\n",
       "   (4, [3, 1, 5]),\n",
       "   (4, [9, 1, 7]),\n",
       "   (4, [5, 8, 9, 1]),\n",
       "   (4, [5, 8, 9, 3]),\n",
       "   (4, [5, 3, 1, 8]),\n",
       "   (4, [5, 1, 7, 4]),\n",
       "   (4, [5, 4, 6]),\n",
       "   (4, [5, 4, 1]),\n",
       "   (4, [3, 9]),\n",
       "   (4, [5, 7, 4, 3]),\n",
       "   (4, [1, 7, 4, 9]),\n",
       "   (4, [6, 2, 5]),\n",
       "   (4, [5, 8, 7]),\n",
       "   (4, [3, 8, 1]),\n",
       "   (4, [1, 9, 4, 7]),\n",
       "   (4, [8, 1]),\n",
       "   (4, [5, 4, 7, 9, 1]),\n",
       "   (4, [4, 8, 9]),\n",
       "   (4, [4, 5, 7, 8]),\n",
       "   (4, [9, 6, 8]),\n",
       "   (4, [4, 8, 5]),\n",
       "   (4, [9, 1, 6, 5]),\n",
       "   (4, [4, 7, 5, 1]),\n",
       "   (4, [1, 4]),\n",
       "   (4, [5, 8, 3, 9]),\n",
       "   (4, [3, 5, 7]),\n",
       "   (4, [1, 8, 9]),\n",
       "   (4, [9, 3]),\n",
       "   (4, [6, 4, 7, 9]),\n",
       "   (4, [1, 5, 3, 8]),\n",
       "   (4, [9, 5, 7]),\n",
       "   (4, [9, 5, 8]),\n",
       "   (4, [4, 7, 9, 6, 1]),\n",
       "   (4, [5, 1, 9, 7]),\n",
       "   (4, [9, 1, 7, 4]),\n",
       "   (4, [9, 1, 8]),\n",
       "   (4, [7, 4, 8, 9]),\n",
       "   (4, [7, 6, 5, 9]),\n",
       "   (4, [5, 9, 1, 6]),\n",
       "   (4, [7, 8, 5]),\n",
       "   (4, [8, 3, 1, 9]),\n",
       "   (4, [8, 4, 7]),\n",
       "   (4, [5, 3, 8, 9]),\n",
       "   (4, [1, 5, 6, 9]),\n",
       "   (4, [1, 7, 9]),\n",
       "   (4, [1, 5, 8]),\n",
       "   (4, [1, 5, 7]),\n",
       "   (4, [5, 7, 8, 4]),\n",
       "   (4, [5, 1, 2]),\n",
       "   (3, [3, 1, 8, 5]),\n",
       "   (3, [5, 1, 6, 9]),\n",
       "   (3, [7, 5, 6]),\n",
       "   (3, [1, 8, 3, 9]),\n",
       "   (3, [5, 4, 1, 7, 9]),\n",
       "   (3, [9, 8, 1]),\n",
       "   (3, [5, 6, 4, 9]),\n",
       "   (3, [3, 1, 7]),\n",
       "   (3, [5, 3, 1, 9]),\n",
       "   (3, [8, 1, 3, 5]),\n",
       "   (3, [4, 6, 9]),\n",
       "   (3, [1, 9, 4, 5]),\n",
       "   (3, [6, 7, 5]),\n",
       "   (3, [6, 5, 7, 9]),\n",
       "   (3, [2, 5, 1, 9]),\n",
       "   (3, [5, 9, 7, 6]),\n",
       "   (3, [4, 5, 9, 1]),\n",
       "   (3, [5, 2, 6, 9]),\n",
       "   (3, [4, 5, 1, 7]),\n",
       "   (3, [6, 1, 9, 7]),\n",
       "   (3, [8, 9, 1]),\n",
       "   (3, [8, 7, 1]),\n",
       "   (3, [6, 1, 5, 9]),\n",
       "   (3, [7, 9, 4, 1]),\n",
       "   (3, [5, 4, 1, 7]),\n",
       "   (3, [6, 7, 9, 4]),\n",
       "   (3, [6, 4, 9]),\n",
       "   (3, [5, 9, 7, 8]),\n",
       "   (3, [9, 5, 1, 6]),\n",
       "   (3, [5, 9, 8, 1]),\n",
       "   (3, [5, 8, 9, 7]),\n",
       "   (3, [5, 4, 9, 6]),\n",
       "   (3, [4, 1, 8]),\n",
       "   (3, [6, 4, 9, 1]),\n",
       "   (3, [7, 4, 9, 3]),\n",
       "   (3, [7, 4, 9, 8]),\n",
       "   (3, [5, 6, 7, 9]),\n",
       "   (3, [6, 2, 1]),\n",
       "   (3, [4, 7, 2]),\n",
       "   (3, [7, 8, 4, 3]),\n",
       "   (3, [3, 1]),\n",
       "   (3, [7, 4, 2, 1]),\n",
       "   (3, [5, 7, 4, 8]),\n",
       "   (3, [7, 4, 1, 2, 3, 5, 6, 8, 9]),\n",
       "   (3, [5, 8, 6]),\n",
       "   (3, [7, 5, 4, 9]),\n",
       "   (3, [7, 5, 4, 1]),\n",
       "   (3, [7, 4, 3, 8, 1]),\n",
       "   (3, [8, 9]),\n",
       "   (3, [7, 9, 8]),\n",
       "   (3, [7, 9, 6]),\n",
       "   (3, [5, 9, 3, 1]),\n",
       "   (3, [6, 7, 4, 1]),\n",
       "   (3, [2, 8, 5]),\n",
       "   (3, [7, 1, 4, 5]),\n",
       "   (3, [3, 4]),\n",
       "   (3, [5, 1, 8, 3]),\n",
       "   (3, [6, 3]),\n",
       "   (3, [2, 1, 6]),\n",
       "   (3, [7, 4, 5, 6]),\n",
       "   (3, [1, 4, 7, 5]),\n",
       "   (3, [6, 5, 8]),\n",
       "   (3, [9, 5, 6, 1]),\n",
       "   (3, [4, 3]),\n",
       "   (3, [8, 3, 9]),\n",
       "   (3, [8, 7, 3]),\n",
       "   (3, [9, 3, 1]),\n",
       "   (3, [1, 8, 5]),\n",
       "   (3, [1, 5, 3, 9]),\n",
       "   (3, [5, 9, 2]),\n",
       "   (3, [3, 4, 5, 7]),\n",
       "   (3, [7, 4, 5, 3]),\n",
       "   (3, [5, 9, 4, 7]),\n",
       "   (3, [5, 3, 7, 9]),\n",
       "   (3, [9, 4, 7, 1]),\n",
       "   (3, [7, 2]),\n",
       "   (3, [7, 4, 1, 8, 3]),\n",
       "   (3, [2, 7, 4]),\n",
       "   (3, [8, 4, 5]),\n",
       "   (3, [5, 2, 9, 1]),\n",
       "   (3, [4, 5, 1]),\n",
       "   (3, [5, 6, 8, 9]),\n",
       "   (3, [4, 9, 7, 1]),\n",
       "   (3, [4, 6, 7, 5]),\n",
       "   (3, [6, 9, 1, 4, 7]),\n",
       "   (3, [8, 3, 1, 5]),\n",
       "   (3, [7, 5, 4, 8]),\n",
       "   (3, [7, 4, 9, 1, 8]),\n",
       "   (3, [6, 5, 4]),\n",
       "   (3, [5, 7, 4, 1]),\n",
       "   (3, [5, 6, 4]),\n",
       "   (3, [5, 6, 3]),\n",
       "   (3, [5, 6, 2]),\n",
       "   (3, [4, 7, 8, 3]),\n",
       "   (3, [4, 7, 8, 1]),\n",
       "   (2, [7, 4, 6, 5, 9, 1, 8, 3, 2]),\n",
       "   (2, [5, 4, 8, 1, 2, 3, 6, 7, 9]),\n",
       "   (2, [5, 9, 1, 8]),\n",
       "   (2, [5, 9, 1, 4]),\n",
       "   (2, [5, 9, 1, 7]),\n",
       "   (2, [5, 1, 6, 8]),\n",
       "   (2, [4, 7, 2, 6, 9, 1]),\n",
       "   (2, [1, 8, 5, 9]),\n",
       "   (2, [1, 8, 5, 7]),\n",
       "   (2, [5, 4, 8, 7]),\n",
       "   (2, [1, 9, 4]),\n",
       "   (2, [7, 1, 6]),\n",
       "   (2, [3, 8, 4, 7]),\n",
       "   (2, [6, 1, 9, 5]),\n",
       "   (2, [2, 6, 7]),\n",
       "   (2, [5, 4, 1, 7, 8]),\n",
       "   (2, [9, 4, 1]),\n",
       "   (2, [4, 7, 8, 9, 6, 5, 3, 2, 1]),\n",
       "   (2, [5, 2, 1]),\n",
       "   (2, [5, 2, 3]),\n",
       "   (2, [5, 2, 7]),\n",
       "   (2, [9, 5, 3, 1]),\n",
       "   (2, [8, 5, 3, 2]),\n",
       "   (2, [1, 6, 2]),\n",
       "   (2, [5, 6, 1, 3, 8, 9, 7, 4]),\n",
       "   (2, [8, 9, 3]),\n",
       "   (2, [5, 1, 3, 2, 9]),\n",
       "   (2, [4, 3, 8, 7]),\n",
       "   (2, [4, 7, 5, 9, 1]),\n",
       "   (2, [1, 3, 8]),\n",
       "   (2, [7, 6, 4, 1]),\n",
       "   (2, [7, 4, 5, 9, 1, 3, 6, 8, 2]),\n",
       "   (2, [4, 7, 2, 1, 3, 5, 6, 8, 9]),\n",
       "   (2, [4, 7, 3, 9]),\n",
       "   (2, [4, 5, 9, 7]),\n",
       "   (2, [1, 5, 9, 8]),\n",
       "   (2, [2, 4, 7]),\n",
       "   (2, [7, 3, 5]),\n",
       "   (2, [5, 3, 8, 1, 9]),\n",
       "   (2, [6, 1, 9, 8]),\n",
       "   (2, [4, 7, 3, 8]),\n",
       "   (2, [4, 7, 3, 5]),\n",
       "   (2, [7, 4, 3, 8]),\n",
       "   (2, [4, 1, 7, 9]),\n",
       "   (2, [8, 5, 7]),\n",
       "   (2, [1, 3, 5, 7]),\n",
       "   (2, [6, 1, 8]),\n",
       "   (2, [7, 6, 9, 1]),\n",
       "   (2, [7, 4, 8, 3, 1]),\n",
       "   (2, [8, 7]),\n",
       "   (2, [8, 7, 4, 9]),\n",
       "   (2, [8, 7, 4, 1]),\n",
       "   (2, [2, 5, 3]),\n",
       "   (2, [1, 6, 5, 8]),\n",
       "   (2, [6, 1, 7, 4, 9]),\n",
       "   (2, [2, 4]),\n",
       "   (2, [2, 3]),\n",
       "   (2, [2, 1]),\n",
       "   (2, [1, 9, 5, 4]),\n",
       "   (2, [1, 9, 5, 6]),\n",
       "   (2, [1, 9, 5, 3]),\n",
       "   (2, [3, 4, 7, 5]),\n",
       "   (2, [2, 3, 5]),\n",
       "   (2, [7, 5, 6, 9]),\n",
       "   (2, [3, 1, 9]),\n",
       "   (2, [1, 6, 9, 7]),\n",
       "   (2, [5, 8, 9, 6, 3]),\n",
       "   (2, [2, 6, 9, 1]),\n",
       "   (2, [9, 5, 1, 6, 7, 4, 3, 8, 2]),\n",
       "   (2, [5, 7, 1, 9]),\n",
       "   (2, [5, 4, 7, 9, 3]),\n",
       "   (2, [4, 1, 5]),\n",
       "   (2, [4, 7, 9, 5, 6]),\n",
       "   (2, [7, 9, 5]),\n",
       "   (2, [6, 2, 9, 1]),\n",
       "   (2, [8, 5, 3, 1]),\n",
       "   (2, [7, 5, 1, 9]),\n",
       "   (2, [5, 6, 1, 4]),\n",
       "   (2, [6, 4, 8]),\n",
       "   (2, [6, 4, 1]),\n",
       "   (2, [1, 6, 5, 9]),\n",
       "   (2, [7, 9, 4, 2]),\n",
       "   (2, [7, 9, 4, 5]),\n",
       "   (2, [5, 9, 8, 3]),\n",
       "   (2, [5, 8, 1, 3, 9]),\n",
       "   (2, [6, 8, 9]),\n",
       "   (2, [5, 4, 9, 8]),\n",
       "   (2, [5, 2, 9]),\n",
       "   (2, [7, 6, 5]),\n",
       "   (2, [4, 7, 6, 3, 9, 8]),\n",
       "   (2, [5, 3, 1, 7]),\n",
       "   (2, [5, 3, 1, 2]),\n",
       "   (2, [6, 9, 8, 1]),\n",
       "   (2, [1, 9, 3, 8]),\n",
       "   (2, [6, 4, 9, 7]),\n",
       "   (2, [1, 4, 3]),\n",
       "   (2, [2, 5, 4]),\n",
       "   (2, [2, 5, 6]),\n",
       "   (2, [9, 6, 1, 5]),\n",
       "   (2, [5, 1, 4, 9]),\n",
       "   (2, [7, 8, 1]),\n",
       "   (2, [5, 3, 9, 7]),\n",
       "   (2, [7, 5, 8]),\n",
       "   (2, [7, 4, 2, 6]),\n",
       "   (2, [5, 3, 4, 9]),\n",
       "   (2, [4, 7, 6, 3]),\n",
       "   (2, [7, 4, 5, 9, 1]),\n",
       "   (2, [2, 4, 5, 7]),\n",
       "   (2, [5, 4, 2]),\n",
       "   (2, [3, 7]),\n",
       "   (2, [5, 2, 1, 9, 3, 8, 7, 4, 6]),\n",
       "   (2, [7, 8, 4, 9]),\n",
       "   (2, [8, 5, 3, 1, 9]),\n",
       "   (2, [8, 3, 7, 4]),\n",
       "   (2, [7, 4, 8, 5]),\n",
       "   (2, [6, 9, 1, 7]),\n",
       "   (2, [5, 8, 2]),\n",
       "   (2, [9, 7, 4, 5]),\n",
       "   (2, [6, 9, 5, 2]),\n",
       "   (2, [4, 6, 3]),\n",
       "   (2, [9, 6, 5, 1]),\n",
       "   (2, [7, 1, 9, 4]),\n",
       "   (2, [9, 6, 4]),\n",
       "   (2, [4, 1, 5, 8]),\n",
       "   (2, [3, 8, 7]),\n",
       "   (2, [3, 8, 9]),\n",
       "   (2, [9, 2, 6, 5]),\n",
       "   (2, [5, 8, 4, 3]),\n",
       "   (2, [9, 6, 5, 8]),\n",
       "   (2, [5, 1, 4, 2]),\n",
       "   (2, [5, 1, 4, 8]),\n",
       "   (2, [9, 6, 7, 5]),\n",
       "   (2, [4, 5, 7, 3]),\n",
       "   (2, [4, 5, 3, 7]),\n",
       "   (2, [8, 1, 7, 5]),\n",
       "   (2, [4, 8, 7]),\n",
       "   (2, [9, 3, 7]),\n",
       "   (2, [5, 9, 1, 3, 8]),\n",
       "   (2, [5, 1, 8, 2]),\n",
       "   (2, [7, 5, 1, 4, 9]),\n",
       "   (2, [4, 7, 5, 6]),\n",
       "   (2, [4, 7, 5, 2]),\n",
       "   (2, [9, 8, 4]),\n",
       "   (2, [6, 3, 4]),\n",
       "   (2, [8, 4, 3, 2, 1, 5, 6, 7, 9]),\n",
       "   (2, [3, 8, 1, 5]),\n",
       "   (2, [8, 1, 9]),\n",
       "   (2, [6, 9, 2]),\n",
       "   (2, [7, 1, 6, 9]),\n",
       "   (2, [4, 7, 5, 1, 9]),\n",
       "   (2, [8, 3, 4]),\n",
       "   (2, [8, 3, 7]),\n",
       "   (2, [8, 3, 2]),\n",
       "   (2, [1, 5, 9, 7]),\n",
       "   (2, [1, 5, 9, 3]),\n",
       "   (2, [5, 9, 7, 4, 1]),\n",
       "   (2, [7, 4, 5, 3, 9, 1, 6, 8, 2]),\n",
       "   (2, [3, 5, 1]),\n",
       "   (2, [1, 8, 4, 9]),\n",
       "   (2, [1, 8, 7]),\n",
       "   (2, [9, 7, 6]),\n",
       "   (2, [9, 7, 8]),\n",
       "   (2, [6, 4, 7, 5]),\n",
       "   (2, [4, 1, 9, 7]),\n",
       "   (2, [8, 2]),\n",
       "   (2, [1, 7, 5]),\n",
       "   (2, [9, 5, 6]),\n",
       "   (2, [5, 9, 6, 7]),\n",
       "   (2, [5, 8, 7, 4]),\n",
       "   (2, [1, 4, 5, 7]),\n",
       "   (2, [1, 6, 8]),\n",
       "   (2, [6, 7, 2, 4]),\n",
       "   (2, [7, 4, 5, 8, 3, 1, 6, 9, 2]),\n",
       "   (2, [9, 5, 4]),\n",
       "   (2, [5, 2, 7, 3, 8, 4]),\n",
       "   (2, [5, 6, 9, 7]),\n",
       "   (2, [5, 8, 3, 4, 1, 2, 7, 6, 9]),\n",
       "   (2, [5, 7, 9, 1]),\n",
       "   (2, [5, 7, 9, 4]),\n",
       "   (2, [5, 7, 9, 8]),\n",
       "   (2, [8, 3, 5, 1]),\n",
       "   (2, [2, 1, 5]),\n",
       "   (2, [4, 9, 7]),\n",
       "   (2, [7, 4, 1, 6]),\n",
       "   (2, [4, 5, 7, 8, 9, 3, 2, 1, 6]),\n",
       "   (2, [7, 1, 3]),\n",
       "   (2, [5, 1, 9, 4]),\n",
       "   (2, [5, 1, 9, 6]),\n",
       "   (2, [3, 7, 9]),\n",
       "   (2, [5, 8, 1, 3, 9, 6, 7, 4, 2]),\n",
       "   (2, [5, 6, 1, 9]),\n",
       "   (2, [4, 7, 6, 9, 1]),\n",
       "   (2, [1, 3, 4, 9]),\n",
       "   (2, [4, 5, 3]),\n",
       "   (2, [1, 7, 5, 9, 4]),\n",
       "   (2, [7, 6, 5, 8]),\n",
       "   (2, [6, 2, 7]),\n",
       "   (2, [1, 3, 8, 9]),\n",
       "   (2, [5, 4, 6, 9, 7, 8, 3, 2, 1]),\n",
       "   (2, [7, 6, 1, 5]),\n",
       "   (2, [4, 7, 1, 8, 9]),\n",
       "   (2, [4, 5, 8, 3]),\n",
       "   (2, [6, 5, 4, 9]),\n",
       "   (2, [1, 9, 7, 5]),\n",
       "   (2, [1, 9, 7, 4]),\n",
       "   (2, [5, 2, 1, 3]),\n",
       "   (2, [9, 4, 5, 7]),\n",
       "   (2, [4, 3, 7]),\n",
       "   (2, [4, 5, 6, 7]),\n",
       "   (2, [5, 8, 4, 3, 9, 7, 6]),\n",
       "   (2, [7, 9, 4, 6]),\n",
       "   (2, [2, 5, 1]),\n",
       "   (2, [5, 4, 7, 9, 6, 3, 1, 8, 2]),\n",
       "   (2, [3, 2, 5]),\n",
       "   (2, [4, 6, 5, 7]),\n",
       "   (2, [2, 5, 4, 7]),\n",
       "   (2, [4, 9, 3]),\n",
       "   (2, [7, 9, 5, 1]),\n",
       "   (2, [4, 7, 2, 5]),\n",
       "   (2, [4, 7, 2, 6]),\n",
       "   (2, [7, 8, 3, 4]),\n",
       "   (2, [9, 1, 5, 3]),\n",
       "   (2, [7, 5, 9, 8]),\n",
       "   (2, [3, 9, 5]),\n",
       "   (2, [9, 5, 1, 4]),\n",
       "   (1, [5, 1, 2, 7]),\n",
       "   (1, [5, 1, 4, 9, 8]),\n",
       "   (1, [6, 7, 1, 4]),\n",
       "   (1, [6, 7, 1, 5]),\n",
       "   (1, [6, 7, 1, 9]),\n",
       "   (1, [8, 7, 5, 1]),\n",
       "   (1, [8, 1, 5, 7, 9]),\n",
       "   (1, [3, 6, 8]),\n",
       "   (1, [5, 8, 4, 9, 7, 6]),\n",
       "   (1, [5, 1, 6, 2]),\n",
       "   (1, [8, 7, 1, 5, 4, 3, 9, 6, 2]),\n",
       "   (1, [8, 4, 5, 2]),\n",
       "   (1, [1, 7, 6, 8, 9, 5, 4, 3, 2]),\n",
       "   (1, [1, 3, 4, 9, 7, 8]),\n",
       "   (1, [5, 1, 6, 7]),\n",
       "   (1, [9, 4, 5]),\n",
       "   (1, [5, 4, 1, 7, 9, 3]),\n",
       "   (1, [6, 1, 5, 9, 4]),\n",
       "   (1, [7, 4, 9, 5, 1, 8]),\n",
       "   (1, [1, 4, 6, 7, 9, 2, 8, 5, 3]),\n",
       "   (1, [3, 5, 4, 7, 8]),\n",
       "   (1, [1, 8, 5, 4]),\n",
       "   (1, [7, 4, 9, 5, 1, 6]),\n",
       "   (1, [1, 8, 5, 3]),\n",
       "   (1, [5, 3, 4, 7, 6, 9, 1, 8, 2]),\n",
       "   (1, [7, 5, 3]),\n",
       "   (1, [4, 9, 7, 1, 5, 8, 6, 3, 2]),\n",
       "   (1, [6, 1, 8, 3]),\n",
       "   (1, [1, 9, 2]),\n",
       "   (1, [6, 4, 7, 9, 8]),\n",
       "   (1, [7, 4, 2, 8, 3, 5, 6, 9, 1]),\n",
       "   (1, [1, 2, 5, 4, 7, 8, 3, 9, 6]),\n",
       "   (1, [4, 7, 8, 5, 3, 2, 1, 9, 6]),\n",
       "   (1, [1, 8, 3, 7]),\n",
       "   (1, [1, 8, 3, 4]),\n",
       "   (1, [1, 8, 3, 5]),\n",
       "   (1, [7, 4, 5, 9, 6, 1, 3, 8, 2]),\n",
       "   (1, [4, 7, 5, 9, 1, 3, 8, 6, 2]),\n",
       "   (1, [1, 5, 9, 4, 7, 6]),\n",
       "   (1, [6, 9, 7, 4, 3]),\n",
       "   (1, [8, 6, 3, 1, 9, 5, 2, 4, 7]),\n",
       "   (1, [1, 5, 8, 4]),\n",
       "   (1, [1, 9, 5, 4, 7]),\n",
       "   (1, [4, 5, 3, 1, 7, 8, 9]),\n",
       "   (1, [1, 5, 8, 3]),\n",
       "   (1, [3, 8, 4, 5]),\n",
       "   (1, [1, 5, 3, 4, 9, 8, 6, 7, 2]),\n",
       "   (1, [6, 7, 3, 9, 4]),\n",
       "   (1, [1, 5, 7, 9, 4, 6, 3, 2, 8]),\n",
       "   (1, [3, 7, 8]),\n",
       "   (1, [8, 1, 5, 7, 4]),\n",
       "   (1, [3, 7, 5, 9]),\n",
       "   (1, [6, 5, 1, 8]),\n",
       "   (1, [4, 1, 2, 7]),\n",
       "   (1, [6, 5, 1, 3]),\n",
       "   (1, [2, 5, 7, 6, 4, 9, 1, 3, 8]),\n",
       "   (1, [7, 9, 1, 4, 6, 5]),\n",
       "   (1, [6, 1, 9, 4]),\n",
       "   (1, [6, 9, 5, 1, 8]),\n",
       "   (1, [8, 3, 5, 1, 9, 4]),\n",
       "   (1, [1, 7, 9, 4]),\n",
       "   (1, [9, 6, 5, 4, 7]),\n",
       "   (1, [5, 2, 8, 3, 9, 6, 1, 7, 4]),\n",
       "   (1, [2, 6, 1]),\n",
       "   (1, [3, 5, 4, 7, 6, 1, 2, 8, 9]),\n",
       "   (1, [2, 6, 4]),\n",
       "   (1, [2, 6, 5]),\n",
       "   (1, [7, 5, 8, 1]),\n",
       "   (1, [9, 4, 1, 7]),\n",
       "   (1, [7, 4, 1, 9, 8, 5, 3, 6]),\n",
       "   (1, [6, 2, 8, 3, 4, 7, 9, 5, 1]),\n",
       "   (1, [1, 9, 8, 4]),\n",
       "   (1, [7, 6, 1]),\n",
       "   (1, [1, 8, 3, 6, 9]),\n",
       "   (1, [4, 9, 1, 5]),\n",
       "   (1, [4, 9, 1, 6]),\n",
       "   (1, [2, 9, 1, 3, 5, 8, 7, 6, 4]),\n",
       "   (1, [1, 9, 7, 6, 5, 4]),\n",
       "   (1, [4, 9, 1, 3]),\n",
       "   (1, [2, 6, 8, 3, 7, 9, 4, 5, 1]),\n",
       "   (1, [4, 7, 9, 1, 5, 6]),\n",
       "   (1, [7, 3, 2, 6]),\n",
       "   (1, [9, 4, 7]),\n",
       "   (1, [9, 4, 6]),\n",
       "   (1, [5, 3, 4, 7]),\n",
       "   (1, [5, 7, 4, 3, 8, 1, 9, 6, 2]),\n",
       "   (1, [7, 4, 1, 3, 9, 8, 2, 5, 6]),\n",
       "   (1, [2, 5, 8, 7]),\n",
       "   (1, [5, 2, 8]),\n",
       "   (1, [1, 8, 7, 9, 3, 4, 6, 5, 2]),\n",
       "   (1, [7, 5, 4, 1, 9]),\n",
       "   (1, [4, 9, 5, 7]),\n",
       "   (1, [6, 9, 3, 4]),\n",
       "   (1, [9, 5, 3, 6]),\n",
       "   (1, [8, 3, 7, 4, 1]),\n",
       "   (1, [5, 8, 6, 3, 1, 2, 9, 7, 4]),\n",
       "   (1, [5, 3, 4, 8]),\n",
       "   (1, [8, 6, 4, 9, 7]),\n",
       "   (1, [1, 9, 2, 4, 5, 7, 6, 3, 8]),\n",
       "   (1, [3, 8, 9, 5, 6, 1]),\n",
       "   (1, [1, 6, 4, 5]),\n",
       "   (1, [7, 5, 6, 2, 8, 9, 1, 3, 4]),\n",
       "   (1, [7, 4, 9, 6, 3, 5, 1, 8, 2]),\n",
       "   (1, [7, 4, 1, 9, 3, 8, 5, 6, 2]),\n",
       "   (1, [5, 7, 8, 3, 1, 4]),\n",
       "   (1, [9, 4, 7, 8, 6, 1, 5, 3, 2]),\n",
       "   (1, [5, 8, 9, 6, 3, 1, 4, 7, 2]),\n",
       "   (1, [1, 8, 9, 4]),\n",
       "   (1, [1, 8, 9, 5]),\n",
       "   (1, [5, 4, 3, 2, 1, 6, 7, 8]),\n",
       "   (1, [7, 4, 8, 9, 1, 2, 5, 3, 6]),\n",
       "   (1, [7, 4, 3, 1, 6, 5, 8, 9, 2]),\n",
       "   (1, [5, 7, 9, 4, 6, 3, 8, 1, 2]),\n",
       "   (1, [1, 3, 5, 8, 9]),\n",
       "   (1, [6, 7, 5, 1, 9]),\n",
       "   (1, [6, 5, 8, 2, 1]),\n",
       "   (1, [5, 1, 9, 7, 3, 4, 6, 2, 8]),\n",
       "   (1, [7, 9, 1, 4, 8]),\n",
       "   (1, [5, 4, 6, 7]),\n",
       "   (1, [4, 7, 9, 1, 3, 8, 5, 6, 2]),\n",
       "   (1, [9, 5, 3, 6, 8]),\n",
       "   (1, [5, 4, 6, 3]),\n",
       "   (1, [7, 8, 1, 4]),\n",
       "   (1, [7, 4, 5, 3, 2, 8, 9, 1]),\n",
       "   (1, [5, 6, 2, 9]),\n",
       "   (1, [6, 2, 9, 7, 4, 5, 8, 3, 1]),\n",
       "   (1, [5, 6, 7, 1, 4]),\n",
       "   (1, [4, 6, 5]),\n",
       "   (1, [4, 6, 1]),\n",
       "   (1, [4, 5, 6, 7, 9, 8, 3, 2, 1]),\n",
       "   (1, [9, 1, 7, 6, 5, 4, 3, 8, 2]),\n",
       "   (1, [7, 5, 3, 8, 1, 9]),\n",
       "   (1, [5, 9, 7, 4, 8, 3, 1, 6, 2]),\n",
       "   (1, [5, 6, 9, 1, 4, 7]),\n",
       "   (1, [5, 7, 4, 1, 2, 3, 6, 9, 8]),\n",
       "   (1, [6, 1, 5, 9, 8, 4, 3, 7, 2]),\n",
       "   (1, [6, 9, 2, 1, 4, 8, 5, 7, 3]),\n",
       "   (1, [4, 6, 7, 2, 1, 9, 8, 3, 5]),\n",
       "   (1, [1, 9, 4, 6]),\n",
       "   (1, [4, 7, 6, 2, 3, 9, 1, 8, 5]),\n",
       "   (1, [4, 7, 5, 1, 9, 8, 2, 3, 6]),\n",
       "   (1, [5, 1, 6, 9, 7]),\n",
       "   (1, [6, 7, 9, 5, 4, 1, 2, 3, 8]),\n",
       "   (1, [7, 6, 8, 9]),\n",
       "   (1, [6, 9, 1, 7, 3, 4, 2, 5, 8]),\n",
       "   (1, [5, 3, 2, 4]),\n",
       "   (1, [5, 3, 2, 7]),\n",
       "   (1, [4, 3, 8, 5]),\n",
       "   (1, [5, 4, 7, 9, 6, 1]),\n",
       "   (1, [1, 9, 5, 6, 4, 3, 7, 8, 2]),\n",
       "   (1, [9, 1, 4, 3]),\n",
       "   (1, [3, 6, 4, 5, 2, 7, 9, 1, 8]),\n",
       "   (1, [9, 7, 4, 1, 6]),\n",
       "   (1, [5, 1, 2, 4, 9]),\n",
       "   (1, [7, 4, 9, 3, 6, 8, 1, 2]),\n",
       "   (1, [6, 7, 2]),\n",
       "   (1, [4, 7, 6, 5, 2, 9, 8, 1, 3]),\n",
       "   (1, [6, 5, 7, 8]),\n",
       "   (1, [6, 5, 7, 4]),\n",
       "   (1, [5, 8, 6, 9]),\n",
       "   (1, [2, 5, 7, 8, 3, 1, 4, 6, 9]),\n",
       "   (1, [1, 3, 9]),\n",
       "   (1, [8, 4, 7, 9, 6, 5, 3, 1, 2]),\n",
       "   (1, [4, 5, 7, 3, 2, 1, 6, 8, 9]),\n",
       "   (1, [7, 6, 4, 9]),\n",
       "   (1, [4, 5, 7, 8, 9]),\n",
       "   (1, [7, 6, 4, 5]),\n",
       "   (1, [7, 1, 8]),\n",
       "   (1, [4, 5, 7, 9, 1, 6, 3, 2, 8]),\n",
       "   (1, [7, 9, 4, 1, 6]),\n",
       "   (1, [4, 7, 8, 9, 3, 5, 1]),\n",
       "   (1, [4, 7, 3, 1, 8, 9]),\n",
       "   (1, [7, 4, 8, 9, 3, 5, 1]),\n",
       "   (1, [8, 3, 5, 4, 9, 7, 1, 6, 2]),\n",
       "   (1, [8, 4, 9, 7, 1, 3, 5, 6, 2]),\n",
       "   (1, [4, 7, 5, 3, 6]),\n",
       "   (1, [4, 5, 1, 7, 9, 8, 3, 2, 6]),\n",
       "   (1, [9, 5, 7, 4, 6, 1, 3, 8, 2]),\n",
       "   (1, [9, 7, 4, 5, 6, 8, 1, 2, 3]),\n",
       "   (1, [5, 9, 7, 1]),\n",
       "   (1, [2, 8, 9, 6, 4, 5, 1, 3, 7]),\n",
       "   (1, [5, 1, 9, 8, 3]),\n",
       "   (1, [4, 7, 5, 1, 9, 2, 3, 8, 6]),\n",
       "   (1, [7, 4, 2, 5, 3, 1, 9, 8, 6]),\n",
       "   (1, [4, 5, 7, 9, 8, 1, 6, 3, 2]),\n",
       "   (1, [4, 1, 3, 7, 5]),\n",
       "   (1, [7, 1, 6, 4]),\n",
       "   (1, [6, 3, 7, 9]),\n",
       "   (1, [1, 3, 7, 8]),\n",
       "   (1, [4, 1, 7, 9, 8, 3, 5, 6, 2]),\n",
       "   (1, [9, 6, 7, 1, 4, 2, 3, 5, 8]),\n",
       "   (1, [7, 5, 3, 6, 9, 8, 4]),\n",
       "   (1, [9, 1, 6, 7, 4, 2, 8, 3, 5]),\n",
       "   (1, [5, 1, 9, 4, 7]),\n",
       "   (1, [6, 5, 8, 1]),\n",
       "   (1, [9, 5, 4, 7, 3, 1, 8, 6, 2]),\n",
       "   (1, [6, 9, 5, 7, 1]),\n",
       "   (1, [3, 5, 6, 8]),\n",
       "   (1, [1, 4, 3, 2, 5, 6, 7, 9, 8]),\n",
       "   (1, [9, 1, 4, 5, 7, 6]),\n",
       "   (1, [4, 7, 1, 2, 3, 6, 8, 9, 5]),\n",
       "   (1, [8, 3, 2, 9, 1]),\n",
       "   (1, [7, 8, 9, 4, 3, 5, 2, 6, 1]),\n",
       "   (1, [7, 6, 4, 1, 9]),\n",
       "   (1, [7, 3, 1]),\n",
       "   (1, [5, 6, 8, 9, 3, 2, 1]),\n",
       "   (1, [3, 2, 5, 1]),\n",
       "   (1, [5, 2, 4, 7, 3, 1, 9, 8]),\n",
       "   (1, [5, 3, 9, 8, 6, 2, 1, 4, 7]),\n",
       "   (1, [8, 5, 6, 7]),\n",
       "   (1, [1, 9, 8]),\n",
       "   (1, [5, 4, 7, 3, 1, 8, 9, 6, 2]),\n",
       "   (1, [8, 5, 4, 7, 9, 3, 1]),\n",
       "   (1, [5, 7, 1, 2, 3, 4, 8, 9, 6]),\n",
       "   (1, [7, 4, 9, 5, 2, 1, 3, 6, 8]),\n",
       "   (1, [4, 5, 1, 3]),\n",
       "   (1, [6, 5, 9, 8, 3]),\n",
       "   (1, [4, 5, 1, 8]),\n",
       "   (1, [6, 4, 5, 3, 7]),\n",
       "   (1, [5, 2, 6, 3]),\n",
       "   (1, [6, 9, 1, 2, 3, 4]),\n",
       "   (1, [2, 6, 9, 8, 1, 3, 5, 4, 7]),\n",
       "   (1, [4, 9, 3, 1, 7]),\n",
       "   (1, [4, 7, 8, 3, 5, 9, 1, 6, 2]),\n",
       "   (1, [6, 2, 4, 5]),\n",
       "   (1, [6, 2, 4, 7]),\n",
       "   (1, [6, 2, 4, 8]),\n",
       "   (1, [6, 2, 4, 9]),\n",
       "   (1, [5, 6, 8, 1]),\n",
       "   (1, [1, 4, 7, 9, 3, 6, 8, 5, 2]),\n",
       "   (1, [7, 4, 5, 3, 2, 1, 6, 9, 8]),\n",
       "   (1, [5, 6, 8, 4]),\n",
       "   (1, [3, 1, 8, 6, 2, 4, 7, 5, 9]),\n",
       "   (1, [5, 1, 3, 8, 9]),\n",
       "   (1, [7, 4, 3, 1]),\n",
       "   (1, [6, 2, 3, 1, 4]),\n",
       "   (1, [8, 4, 1, 3, 7, 9, 2, 6, 5]),\n",
       "   (1, [5, 1, 3, 8, 7]),\n",
       "   (1, [2, 1, 6, 5, 4, 7, 9, 3, 8]),\n",
       "   (1, [9, 8, 5, 1]),\n",
       "   (1, [6, 9, 2, 1, 7, 4]),\n",
       "   (1, [5, 8, 9, 1, 3, 7, 6, 4]),\n",
       "   (1, [1, 4, 5, 7, 9]),\n",
       "   (1, [5, 8, 4, 3, 2, 1, 7]),\n",
       "   (1, [9, 7, 4, 1, 5, 8, 6]),\n",
       "   (1, [2, 4, 3]),\n",
       "   (1, [1, 3, 4, 7]),\n",
       "   (1, [7, 4, 1, 8, 5, 2]),\n",
       "   (1, [5, 4, 7, 6, 2, 8, 9, 3, 1]),\n",
       "   (1, [8, 7, 9, 4, 6, 5]),\n",
       "   (1, [4, 6, 8, 9, 1]),\n",
       "   (1, [5, 3, 2]),\n",
       "   (1, [6, 5, 4, 7, 9, 2, 1, 3, 8]),\n",
       "   (1, [5, 1, 9, 8, 3, 4, 7, 6, 2]),\n",
       "   (1, [4, 2, 7, 6, 9]),\n",
       "   (1, [8, 3, 9, 1, 7, 4, 5, 6]),\n",
       "   (1, [5, 8, 2, 3, 1, 9, 7, 4]),\n",
       "   (1, [5, 3, 8, 9, 7, 1, 4, 6]),\n",
       "   (1, [5, 9, 7, 4, 1, 6, 8, 3, 2]),\n",
       "   (1, [6, 9, 8, 3, 2, 1, 5, 4]),\n",
       "   (1, [5, 1, 7, 9, 4, 3, 2, 6, 8]),\n",
       "   (1, [5, 6, 4, 7]),\n",
       "   (1, [4, 7, 9, 1, 5, 6, 8, 3, 2]),\n",
       "   (1, [6, 1, 9, 2]),\n",
       "   (1, [4, 7, 1, 2]),\n",
       "   (1, [4, 7, 1, 3]),\n",
       "   (1, [4, 7, 5, 1, 3, 9, 6, 8, 2]),\n",
       "   (1, [4, 7, 1, 6]),\n",
       "   (1, [7, 4, 5, 3, 9, 2, 6, 1]),\n",
       "   (1, [2, 9, 6, 5, 1]),\n",
       "   (1, [8, 9, 7]),\n",
       "   (1, [8, 9, 5]),\n",
       "   (1, [5, 8, 3, 1, 9, 6, 4, 7, 2]),\n",
       "   (1, [1, 5, 7, 4, 9]),\n",
       "   (1, [7, 3, 8, 9]),\n",
       "   (1, [5, 8, 3, 9, 6, 1, 2, 7, 4]),\n",
       "   (1, [3, 8, 5, 9]),\n",
       "   (1, [3, 8, 5, 2]),\n",
       "   (1, [3, 8, 5, 1]),\n",
       "   (1, [7, 1, 2, 8, 3, 4, 9, 6, 5]),\n",
       "   (1, [9, 5, 7, 1, 4, 6]),\n",
       "   (1, [4, 5, 9, 7, 3]),\n",
       "   (1, [7, 6, 1, 4, 9]),\n",
       "   (1, [7, 9, 4, 5, 1, 6, 3, 8, 2]),\n",
       "   (1, [3, 5, 2, 1]),\n",
       "   (1, [4, 1, 3, 7]),\n",
       "   (1, [5, 6, 9, 8, 4, 7, 3, 1, 2]),\n",
       "   (1, [4, 1, 3, 9]),\n",
       "   (1, [4, 7, 5, 9, 6, 3, 8, 1, 2]),\n",
       "   (1, [7, 6, 2, 4]),\n",
       "   (1, [7, 4, 8, 6]),\n",
       "   (1, [7, 3, 9]),\n",
       "   (1, [2, 3, 4, 1, 5, 9, 6, 7, 8]),\n",
       "   (1, [6, 9, 1, 7, 4, 5]),\n",
       "   (1, [8, 1, 3, 9, 2, 7, 4, 6, 5]),\n",
       "   (1, [4, 1, 7, 3]),\n",
       "   (1, [4, 7, 9, 5, 3, 1, 8, 6, 2]),\n",
       "   (1, [4, 1, 7, 8]),\n",
       "   (1, [5, 3, 2, 9, 6, 1]),\n",
       "   (1, [5, 4, 2, 7, 9, 8, 3, 1, 6]),\n",
       "   (1, [5, 7, 4, 8, 3, 2, 1, 6, 9]),\n",
       "   (1, [9, 1, 8, 6, 5, 3, 7, 4, 2]),\n",
       "   (1, [5, 2, 8, 9, 3, 4, 6, 7, 1]),\n",
       "   (1, [2, 3, 8, 1]),\n",
       "   (1, [6, 9, 7, 4, 3, 8, 1, 2, 5]),\n",
       "   (1, [5, 6, 1, 9, 3, 2, 8, 7, 4]),\n",
       "   (1, [5, 2, 8, 3]),\n",
       "   (1, [8, 5, 1, 7]),\n",
       "   (1, [5, 4, 7, 6, 9]),\n",
       "   (1, [3, 6, 5, 7]),\n",
       "   (1, [5, 8, 4, 7, 1, 9]),\n",
       "   (1, [2, 1, 4, 6, 3, 5, 7, 8, 9]),\n",
       "   (1, [6, 8, 3, 9, 5, 1, 2, 7, 4]),\n",
       "   (1, [4, 7, 9, 1, 5, 3, 2, 8, 6]),\n",
       "   (1, [5, 7, 2, 3, 4, 8, 9, 6]),\n",
       "   (1, [7, 4, 8, 3, 9, 1, 2, 5, 6]),\n",
       "   (1, [7, 1, 3, 4, 9, 8, 2, 6, 5]),\n",
       "   (1, [9, 5, 2, 8]),\n",
       "   (1, [9, 1, 8, 7]),\n",
       "   (1, [5, 7, 8, 9, 1]),\n",
       "   (1, [7, 9, 4, 5, 1, 6, 8, 3, 2]),\n",
       "   (1, [8, 4, 7, 9]),\n",
       "   (1, [5, 9, 1, 6, 4, 3, 7, 8, 2]),\n",
       "   (1, [9, 1, 3, 5, 8]),\n",
       "   (1, [1, 8, 9, 5, 6, 7, 2, 3, 4]),\n",
       "   (1, [5, 6, 4, 1, 7, 9]),\n",
       "   (1, [5, 9, 8, 1, 2, 4]),\n",
       "   (1, [9, 1, 4, 6, 7, 5]),\n",
       "   (1, [6, 1, 9, 8, 4, 7, 5, 3, 2]),\n",
       "   (1, [8, 9, 3, 4, 1, 2, 5, 6, 7]),\n",
       "   (1, [2, 9, 8, 6, 1, 3, 5, 4, 7]),\n",
       "   (1, [1, 9, 7, 8, 4, 3, 6]),\n",
       "   (1, [5, 1, 2, 4, 7, 3, 6, 9, 8]),\n",
       "   (1, [9, 6, 2, 5]),\n",
       "   (1, [1, 4, 5, 7, 9, 8, 3, 6, 2]),\n",
       "   (1, [8, 7, 4, 5]),\n",
       "   (1, [4, 7, 1, 9, 8]),\n",
       "   (1, [5, 9, 1, 7, 6, 8, 4, 3, 2]),\n",
       "   (1, [5, 7, 8, 4, 3, 9, 6, 1, 2]),\n",
       "   (1, [7, 4, 5, 8, 9]),\n",
       "   (1, [4, 7, 5, 3, 6, 8]),\n",
       "   (1, [4, 5, 1, 6, 7, 8, 9, 2, 3]),\n",
       "   (1, [2, 8, 7, 5, 3, 4, 9, 6, 1]),\n",
       "   (1, [5, 8, 9, 3, 2, 6, 7, 4, 1]),\n",
       "   (1, [4, 5, 9, 1, 3, 8, 7, 6, 2]),\n",
       "   (1, [5, 4, 7, 2]),\n",
       "   (1, [3, 8, 5, 4, 1, 7]),\n",
       "   (1, [5, 7, 3, 8]),\n",
       "   (1, [5, 3, 1, 9, 6, 4, 7, 2, 8]),\n",
       "   (1, [7, 6, 5, 4, 9, 1]),\n",
       "   (1, [3, 9, 7, 8, 4, 5]),\n",
       "   (1, [4, 7, 1, 3, 9]),\n",
       "   (1, [3, 5, 1, 2, 9, 4, 6, 7, 8]),\n",
       "   (1, [8, 1, 5, 6, 3]),\n",
       "   (1, [9, 4, 1, 6, 7, 5]),\n",
       "   (1, [8, 5, 6, 9, 3, 1, 7, 4, 2]),\n",
       "   (1, [7, 9, 8, 4, 1]),\n",
       "   (1, [5, 8, 3, 2, 1]),\n",
       "   (1, [1, 3, 5, 9, 4]),\n",
       "   (1, [2, 5, 9]),\n",
       "   (1, [7, 9, 8, 5]),\n",
       "   (1, [8, 3, 5, 1, 9]),\n",
       "   (1, [7, 8, 5, 4, 1, 3, 9, 2, 6]),\n",
       "   (1, [1, 9, 5, 4, 7, 3, 8, 6, 2]),\n",
       "   (1, [5, 4, 3, 8]),\n",
       "   (1, [4, 7, 9, 3, 1, 5, 8, 6, 2]),\n",
       "   (1, [8, 5, 4, 3, 9, 7, 2, 6, 1]),\n",
       "   (1, [9, 6, 7, 1, 4, 2, 5, 8, 3]),\n",
       "   (1, [1, 3, 8, 7, 4, 9, 5, 6]),\n",
       "   (1, [9, 7, 1, 4]),\n",
       "   (1, [5, 7, 4, 9, 3, 8, 1, 6, 2]),\n",
       "   (1, [5, 3, 2, 4, 7, 8]),\n",
       "   (1, [6, 1, 3]),\n",
       "   (1, [9, 8, 3, 6]),\n",
       "   (1, [5, 4, 2, 1, 7, 9]),\n",
       "   (1, [5, 1, 8, 3, 9, 4, 7, 6]),\n",
       "   (1, [6, 5, 1, 2]),\n",
       "   (1, [6, 5, 2, 7, 4, 9, 1, 3, 8]),\n",
       "   (1, [6, 9, 1, 5, 4]),\n",
       "   (1, [9, 7, 3, 1, 4, 8, 5, 6, 2]),\n",
       "   (1, [4, 8, 7, 9, 6, 3, 1, 2, 5]),\n",
       "   (1, [5, 8, 3, 7, 4, 6, 1, 9, 2]),\n",
       "   (1, [5, 4, 1, 7, 9, 8, 6, 3, 2]),\n",
       "   (1, [5, 1, 6, 8, 9, 4]),\n",
       "   (1, [4, 7, 3, 5, 8, 9, 6, 2, 1]),\n",
       "   (1, [1, 9, 5, 8]),\n",
       "   (1, [8, 6, 4, 3, 1, 2, 5, 9]),\n",
       "   (1, [1, 9, 5, 7]),\n",
       "   (1, [7, 4, 5, 1, 6, 9]),\n",
       "   (1, [4, 7, 8, 6, 2, 1, 3, 9, 5]),\n",
       "   (1, [9, 4, 7, 1, 3]),\n",
       "   (1, [2, 1, 7, 4, 3, 6, 8, 9, 5]),\n",
       "   (1, [9, 4, 7, 1, 5]),\n",
       "   (1, [5, 6, 7, 9, 8, 4, 1, 3, 2]),\n",
       "   (1, [6, 7, 2, 9, 4]),\n",
       "   (1, [2, 3, 8]),\n",
       "   (1, [7, 5, 6, 8]),\n",
       "   ...]),\n",
       " (12744,\n",
       "  defaultdict(int,\n",
       "              {1: 0.19695543000627747,\n",
       "               2: 0.2405053358443189,\n",
       "               3: 0.24144695543000627,\n",
       "               4: 0.1746704331450094,\n",
       "               5: 0.04770872567482737,\n",
       "               9: 0.0709353421217828,\n",
       "               6: 0.016242937853107344,\n",
       "               8: 0.0045511613308223476,\n",
       "               7: 0.006983678593848085,\n",
       "               0: 0}),\n",
       "  [(702, [7]),\n",
       "   (586, [3, 4]),\n",
       "   (443, [9]),\n",
       "   (442, [4, 3]),\n",
       "   (404, [6]),\n",
       "   (275, [3]),\n",
       "   (262, [4]),\n",
       "   (260, [2]),\n",
       "   (231, [7, 2]),\n",
       "   (192, [6, 2]),\n",
       "   (135, [9, 6]),\n",
       "   (133, [2, 6]),\n",
       "   (125, [6, 9]),\n",
       "   (112, [7, 6]),\n",
       "   (107, [9, 1]),\n",
       "   (96, [3, 4, 6]),\n",
       "   (92, [3, 4, 2]),\n",
       "   (85, [7, 6, 2]),\n",
       "   (85, [7, 2, 6]),\n",
       "   (76, [7, 8]),\n",
       "   (73, [6, 7]),\n",
       "   (73, [4, 3, 6]),\n",
       "   (72, [8]),\n",
       "   (71, [2, 7]),\n",
       "   (62, [3, 4, 6, 2]),\n",
       "   (61, [3, 4, 7]),\n",
       "   (60, [4, 3, 2]),\n",
       "   (58, [6, 2, 7]),\n",
       "   (57, [6, 7, 2]),\n",
       "   (57, [6, 4, 3]),\n",
       "   (56, [1]),\n",
       "   (54, [7, 9]),\n",
       "   (51, [6, 4]),\n",
       "   (51, [6, 9, 2]),\n",
       "   (50, [7, 3, 4]),\n",
       "   (50, [2, 6, 7]),\n",
       "   (46, [4, 3, 7]),\n",
       "   (45, [9, 6, 2]),\n",
       "   (43, [7, 8, 2]),\n",
       "   (41, [2, 5]),\n",
       "   (40, [2, 3, 4]),\n",
       "   (39, [6, 3, 4]),\n",
       "   (39, [6, 2, 3, 4]),\n",
       "   (38, [6, 3]),\n",
       "   (38, [2, 7, 6]),\n",
       "   (37, [6, 2, 9]),\n",
       "   (37, [7, 2, 8]),\n",
       "   (36, [5]),\n",
       "   (36, [9, 1, 6]),\n",
       "   (35, [9, 7]),\n",
       "   (33, [9, 2]),\n",
       "   (32, [3, 6]),\n",
       "   (32, [7, 4]),\n",
       "   (31, [4, 3, 6, 2]),\n",
       "   (31, [4, 3, 2, 6]),\n",
       "   (29, [3, 4, 2, 6]),\n",
       "   (29, [2, 6, 3, 4]),\n",
       "   (28, [3, 2]),\n",
       "   (28, [9, 6, 1]),\n",
       "   (28, [1, 9]),\n",
       "   (28, [6, 3, 4, 2]),\n",
       "   (27, [7, 4, 3]),\n",
       "   (27, [7, 8, 5]),\n",
       "   (27, [6, 9, 7]),\n",
       "   (26, [2, 3]),\n",
       "   (26, [3, 4, 8]),\n",
       "   (26, [7, 2, 5]),\n",
       "   (25, [7, 5, 2]),\n",
       "   (25, [4, 3, 9]),\n",
       "   (24, [6, 7, 9]),\n",
       "   (24, [6, 2, 3]),\n",
       "   (24, [4, 2]),\n",
       "   (23, [2, 6, 9]),\n",
       "   (23, [5, 2]),\n",
       "   (22, [7, 9, 6]),\n",
       "   (22, [9, 2, 6]),\n",
       "   (22, [4, 7]),\n",
       "   (21, [7, 3]),\n",
       "   (21, [2, 3, 4, 6]),\n",
       "   (21, [6, 4, 2]),\n",
       "   (21, [4, 6]),\n",
       "   (20, [9, 1, 2]),\n",
       "   (20, [8, 7]),\n",
       "   (20, [6, 4, 3, 2]),\n",
       "   (19, [4, 6, 2]),\n",
       "   (19, [2, 8]),\n",
       "   (19, [2, 9]),\n",
       "   (19, [7, 8, 5, 2]),\n",
       "   (19, [3, 2, 4]),\n",
       "   (18, [7, 5]),\n",
       "   (18, [6, 2, 4]),\n",
       "   (18, [7, 5, 8]),\n",
       "   (18, [7, 6, 9]),\n",
       "   (18, [3, 4, 9]),\n",
       "   (18, [3, 4, 5]),\n",
       "   (17, [9, 6, 7]),\n",
       "   (17, [4, 2, 6]),\n",
       "   (17, [2, 8, 5]),\n",
       "   (16, [3, 6, 2]),\n",
       "   (16, [9, 1, 2, 6]),\n",
       "   (16, [6, 9, 1]),\n",
       "   (16, [2, 5, 8]),\n",
       "   (16, [9, 7, 6]),\n",
       "   (16, [2, 6, 3]),\n",
       "   (16, [7, 6, 8]),\n",
       "   (15, [4, 6, 3]),\n",
       "   (15, [7, 2, 5, 8]),\n",
       "   (15, [7, 4, 6]),\n",
       "   (15, [7, 2, 6, 8]),\n",
       "   (15, [7, 9, 2]),\n",
       "   (15, [2, 5, 7]),\n",
       "   (15, [4, 8]),\n",
       "   (15, [3, 4, 6, 7]),\n",
       "   (14, [5, 2, 7]),\n",
       "   (14, [6, 7, 8]),\n",
       "   (14, [3, 4, 7, 6]),\n",
       "   (14, [3, 7]),\n",
       "   (14, [7, 3, 2]),\n",
       "   (14, [9, 3, 4]),\n",
       "   (14, [8, 2]),\n",
       "   (14, [3, 4, 8, 2]),\n",
       "   (14, [6, 1, 9]),\n",
       "   (14, [2, 4]),\n",
       "   (14, [4, 3, 8]),\n",
       "   (13, [2, 6, 4, 3]),\n",
       "   (13, [8, 5, 2]),\n",
       "   (13, [2, 4, 3]),\n",
       "   (13, [7, 4, 3, 6]),\n",
       "   (13, [6, 9, 1, 2]),\n",
       "   (13, [6, 2, 4, 3]),\n",
       "   (13, [6, 2, 1]),\n",
       "   (13, [7, 6, 2, 8]),\n",
       "   (13, [8, 7, 2]),\n",
       "   (13, [2, 7, 5]),\n",
       "   (13, [6, 3, 2]),\n",
       "   (12, [2, 7, 3, 4]),\n",
       "   (12, [3, 4, 7, 8]),\n",
       "   (12, [4, 3, 7, 2]),\n",
       "   (12, [4, 7, 3]),\n",
       "   (12, [3, 7, 4]),\n",
       "   (12, [9, 6, 2, 1]),\n",
       "   (12, [7, 8, 6]),\n",
       "   (12, [7, 4, 2]),\n",
       "   (12, [2, 6, 4]),\n",
       "   (12, [6, 4, 3, 9]),\n",
       "   (12, [6, 4, 9]),\n",
       "   (12, [2, 7, 8]),\n",
       "   (12, [6, 9, 4]),\n",
       "   (11, [2, 6, 5]),\n",
       "   (11, [6, 9, 2, 1]),\n",
       "   (11, [6, 9, 4, 3]),\n",
       "   (11, [4, 3, 1]),\n",
       "   (10, [9, 4, 3]),\n",
       "   (10, [4, 6, 7]),\n",
       "   (10, [7, 4, 3, 2]),\n",
       "   (10, [2, 3, 6]),\n",
       "   (10, [2, 4, 3, 6]),\n",
       "   (10, [3, 4, 6, 9, 2]),\n",
       "   (10, [7, 1]),\n",
       "   (10, [3, 7, 6]),\n",
       "   (10, [7, 2, 9]),\n",
       "   (10, [6, 2, 5]),\n",
       "   (10, [3, 4, 9, 6]),\n",
       "   (10, [7, 8, 2, 5]),\n",
       "   (10, [7, 6, 3]),\n",
       "   (10, [2, 5, 6]),\n",
       "   (10, [3, 4, 6, 9]),\n",
       "   (10, [7, 2, 5, 6]),\n",
       "   (10, [8, 3, 4]),\n",
       "   (10, [6, 1]),\n",
       "   (10, [6, 3, 4, 9]),\n",
       "   (9, [7, 8, 6, 2]),\n",
       "   (9, [6, 7, 4]),\n",
       "   (9, [9, 1, 7]),\n",
       "   (9, [3, 4, 7, 2]),\n",
       "   (9, [3, 4, 7, 5]),\n",
       "   (9, [3, 8]),\n",
       "   (9, [2, 7, 6, 5]),\n",
       "   (9, [7, 3, 6]),\n",
       "   (9, [1, 6]),\n",
       "   (9, [2, 7, 8, 5]),\n",
       "   (9, [7, 2, 1]),\n",
       "   (9, [7, 5, 8, 2]),\n",
       "   (9, [7, 6, 2, 4]),\n",
       "   (9, [4, 2, 3]),\n",
       "   (9, [7, 3, 4, 2]),\n",
       "   (9, [6, 3, 2, 4]),\n",
       "   (9, [9, 4]),\n",
       "   (9, [6, 2, 7, 9]),\n",
       "   (8, [3, 4, 2, 7]),\n",
       "   (8, [3, 4, 2, 8]),\n",
       "   (8, [3, 6, 4]),\n",
       "   (8, [7, 2, 3, 4]),\n",
       "   (8, [4, 6, 9]),\n",
       "   (8, [6, 7, 3]),\n",
       "   (8, [7, 6, 4, 3]),\n",
       "   (8, [8, 5, 7]),\n",
       "   (8, [5, 8, 2]),\n",
       "   (8, [9, 6, 1, 2]),\n",
       "   (8, [9, 6, 3]),\n",
       "   (8, [9, 6, 4]),\n",
       "   (8, [3, 6, 9]),\n",
       "   (8, [3, 4, 5, 8]),\n",
       "   (8, [3, 7, 2]),\n",
       "   (8, [6, 2, 8]),\n",
       "   (8, [7, 6, 9, 2]),\n",
       "   (8, [1, 9, 2]),\n",
       "   (8, [2, 4, 6]),\n",
       "   (8, [3, 4, 8, 5]),\n",
       "   (8, [4, 3, 6, 9]),\n",
       "   (8, [4, 3, 6, 7]),\n",
       "   (8, [7, 8, 2, 6]),\n",
       "   (8, [2, 3, 4, 7]),\n",
       "   (8, [3, 4, 1]),\n",
       "   (8, [4, 9]),\n",
       "   (8, [9, 6, 3, 4]),\n",
       "   (8, [7, 3, 4, 6]),\n",
       "   (8, [6, 7, 4, 3]),\n",
       "   (8, [6, 8]),\n",
       "   (8, [7, 2, 3]),\n",
       "   (8, [7, 2, 4]),\n",
       "   (8, [2, 7, 4]),\n",
       "   (8, [7, 2, 6, 5]),\n",
       "   (7, [3, 4, 2, 5]),\n",
       "   (7, [7, 8, 4, 3]),\n",
       "   (7, [7, 9, 1]),\n",
       "   (7, [2, 7, 6, 8]),\n",
       "   (7, [1, 2]),\n",
       "   (7, [7, 8, 4]),\n",
       "   (7, [7, 8, 9]),\n",
       "   (7, [7, 6, 2, 9]),\n",
       "   (7, [9, 4, 3, 6]),\n",
       "   (7, [8, 2, 5]),\n",
       "   (7, [2, 5, 6, 7]),\n",
       "   (7, [3, 4, 6, 8]),\n",
       "   (7, [5, 7, 2]),\n",
       "   (7, [6, 3, 9]),\n",
       "   (7, [7, 6, 4]),\n",
       "   (7, [7, 9, 6, 2]),\n",
       "   (7, [6, 4, 7]),\n",
       "   (6, [3, 4, 2, 1]),\n",
       "   (6, [3, 6, 7]),\n",
       "   (6, [6, 9, 3, 4]),\n",
       "   (6, [9, 6, 7, 2]),\n",
       "   (6, [2, 6, 1]),\n",
       "   (6, [2, 3, 7]),\n",
       "   (6, [6, 2, 9, 1]),\n",
       "   (6, [7, 2, 8, 5]),\n",
       "   (6, [4, 3, 7, 6]),\n",
       "   (6, [4, 7, 2]),\n",
       "   (6, [9, 7, 6, 2]),\n",
       "   (6, [4, 2, 7]),\n",
       "   (6, [2, 5, 3, 4]),\n",
       "   (6, [9, 7, 2]),\n",
       "   (6, [1, 6, 9]),\n",
       "   (6, [8, 5]),\n",
       "   (6, [3, 4, 1, 2]),\n",
       "   (6, [7, 6, 3, 2]),\n",
       "   (6, [7, 6, 3, 4]),\n",
       "   (6, [9, 6, 4, 3]),\n",
       "   (6, [2, 8, 7]),\n",
       "   (6, [1, 9, 6]),\n",
       "   (6, [1, 9, 7]),\n",
       "   (6, [2, 6, 8]),\n",
       "   (6, [6, 7, 9, 2]),\n",
       "   (6, [2, 5, 8, 7]),\n",
       "   (6, [4, 3, 6, 5]),\n",
       "   (6, [4, 6, 3, 2]),\n",
       "   (6, [2, 3, 4, 8]),\n",
       "   (6, [7, 6, 2, 5]),\n",
       "   (6, [6, 4, 3, 7]),\n",
       "   (6, [8, 2, 7]),\n",
       "   (6, [1, 3, 4]),\n",
       "   (6, [5, 2, 7, 6]),\n",
       "   (6, [5, 7, 8]),\n",
       "   (6, [7, 1, 9]),\n",
       "   (6, [9, 7, 1]),\n",
       "   (6, [6, 9, 3]),\n",
       "   (6, [6, 7, 2, 8]),\n",
       "   (6, [6, 1, 2]),\n",
       "   (6, [2, 7, 3]),\n",
       "   (6, [8, 7, 5, 2]),\n",
       "   (6, [8, 4, 3, 2]),\n",
       "   (6, [9, 3]),\n",
       "   (6, [6, 2, 7, 3]),\n",
       "   (6, [7, 2, 6, 9]),\n",
       "   (5, [5, 2, 6]),\n",
       "   (5, [5, 2, 8]),\n",
       "   (5, [7, 6, 8, 2]),\n",
       "   (5, [7, 4, 3, 8]),\n",
       "   (5, [4, 3, 5, 2]),\n",
       "   (5, [4, 2, 6, 7]),\n",
       "   (5, [4, 7, 6]),\n",
       "   (5, [7, 9, 2, 6]),\n",
       "   (5, [9, 7, 6, 1]),\n",
       "   (5, [7, 5, 2, 8]),\n",
       "   (5, [9, 1, 3, 4]),\n",
       "   (5, [3, 5]),\n",
       "   (5, [1, 7]),\n",
       "   (5, [8, 7, 6, 2]),\n",
       "   (5, [4, 3, 9, 6]),\n",
       "   (5, [2, 9, 6]),\n",
       "   (5, [3, 7, 8]),\n",
       "   (5, [7, 8, 1]),\n",
       "   (5, [3, 6, 2, 4]),\n",
       "   (5, [9, 4, 6]),\n",
       "   (5, [2, 8, 5, 6]),\n",
       "   (5, [2, 4, 8]),\n",
       "   (5, [2, 5, 8, 6]),\n",
       "   (5, [4, 3, 8, 2]),\n",
       "   (5, [6, 2, 4, 7]),\n",
       "   (5, [9, 3, 4, 6]),\n",
       "   (5, [4, 6, 2, 3]),\n",
       "   (5, [8, 7, 2, 5]),\n",
       "   (5, [2, 3, 4, 5]),\n",
       "   (5, [6, 9, 2, 7]),\n",
       "   (5, [3, 4, 6, 1]),\n",
       "   (5, [6, 8, 2]),\n",
       "   (5, [6, 8, 4]),\n",
       "   (5, [6, 4, 3, 2, 7]),\n",
       "   (5, [7, 4, 9]),\n",
       "   (5, [7, 4, 2, 6]),\n",
       "   (5, [4, 5]),\n",
       "   (5, [5, 8]),\n",
       "   (5, [3, 4, 6, 2, 7]),\n",
       "   (5, [6, 5]),\n",
       "   (5, [6, 5, 2]),\n",
       "   (5, [8, 7, 6]),\n",
       "   (5, [1, 2, 7]),\n",
       "   (5, [1, 2, 6]),\n",
       "   (5, [7, 6, 2, 5, 8]),\n",
       "   (5, [6, 7, 4, 2]),\n",
       "   (5, [2, 7, 9]),\n",
       "   (5, [6, 3, 4, 7]),\n",
       "   (5, [4, 3, 5]),\n",
       "   (5, [6, 2, 3, 7]),\n",
       "   (5, [6, 2, 7, 8]),\n",
       "   (5, [3, 2, 6]),\n",
       "   (5, [4, 7, 2, 6]),\n",
       "   (5, [3, 7, 4, 6]),\n",
       "   (4, [3, 6, 8]),\n",
       "   (4, [7, 2, 3, 6]),\n",
       "   (4, [7, 3, 2, 6]),\n",
       "   (4, [5, 2, 3]),\n",
       "   (4, [2, 6, 8, 5]),\n",
       "   (4, [4, 3, 7, 8]),\n",
       "   (4, [4, 7, 9]),\n",
       "   (4, [3, 7, 2, 4]),\n",
       "   (4, [3, 2, 6, 4]),\n",
       "   (4, [9, 2, 6, 1]),\n",
       "   (4, [5, 8, 2, 7]),\n",
       "   (4, [9, 2, 1]),\n",
       "   (4, [9, 2, 3]),\n",
       "   (4, [7, 3, 5]),\n",
       "   (4, [1, 4]),\n",
       "   (4, [9, 3, 6]),\n",
       "   (4, [5, 8, 7, 2]),\n",
       "   (4, [2, 6, 7, 3]),\n",
       "   (4, [2, 6, 7, 5]),\n",
       "   (4, [4, 9, 6]),\n",
       "   (4, [2, 7, 8, 6]),\n",
       "   (4, [7, 8, 3]),\n",
       "   (4, [7, 8, 5, 2, 6]),\n",
       "   (4, [2, 8, 5, 7]),\n",
       "   (4, [7, 5, 6]),\n",
       "   (4, [6, 7, 9, 1]),\n",
       "   (4, [4, 3, 6, 8]),\n",
       "   (4, [3, 4, 9, 1]),\n",
       "   (4, [7, 6, 2, 3]),\n",
       "   (4, [2, 7, 3, 4, 6]),\n",
       "   (4, [7, 6, 2, 3, 4]),\n",
       "   (4, [3, 2, 4, 6]),\n",
       "   (4, [8, 2, 6]),\n",
       "   (4, [7, 4, 5]),\n",
       "   (4, [3, 2, 4, 8]),\n",
       "   (4, [6, 4, 9, 2]),\n",
       "   (4, [7, 4, 2, 3]),\n",
       "   (4, [9, 2, 7]),\n",
       "   (4, [6, 2, 3, 4, 9]),\n",
       "   (4, [5, 7]),\n",
       "   (4, [5, 6]),\n",
       "   (4, [2, 6, 7, 3, 4]),\n",
       "   (4, [3, 8, 4]),\n",
       "   (4, [4, 3, 2, 9]),\n",
       "   (4, [4, 8, 3]),\n",
       "   (4, [3, 4, 6, 2, 5]),\n",
       "   (4, [7, 2, 6, 1]),\n",
       "   (4, [7, 3, 4, 8]),\n",
       "   (4, [4, 3, 7, 6, 2]),\n",
       "   (4, [6, 3, 7]),\n",
       "   (4, [8, 5, 2, 7]),\n",
       "   (4, [8, 5, 2, 6]),\n",
       "   (4, [8, 3, 5]),\n",
       "   (4, [9, 7, 4]),\n",
       "   (4, [3, 6, 4, 2]),\n",
       "   (4, [6, 7, 8, 2]),\n",
       "   (4, [6, 7, 2, 9]),\n",
       "   (4, [6, 7, 2, 3]),\n",
       "   (4, [2, 4, 6, 3]),\n",
       "   (4, [9, 1, 4]),\n",
       "   (4, [6, 4, 2, 3]),\n",
       "   (4, [7, 6, 5, 2]),\n",
       "   (4, [4, 2, 3, 6]),\n",
       "   (4, [7, 4, 6, 3]),\n",
       "   (4, [6, 2, 7, 4]),\n",
       "   (4, [6, 7, 3, 4]),\n",
       "   (4, [2, 8, 7, 6]),\n",
       "   (4, [1, 7, 9]),\n",
       "   (3, [3, 4, 2, 9]),\n",
       "   (3, [2, 3, 4, 6, 7]),\n",
       "   (3, [6, 9, 3, 2]),\n",
       "   (3, [6, 7, 5]),\n",
       "   (3, [9, 6, 7, 1]),\n",
       "   (3, [9, 6, 7, 4]),\n",
       "   (3, [3, 6, 7, 8]),\n",
       "   (3, [6, 1, 9, 2]),\n",
       "   (3, [5, 7, 6, 2]),\n",
       "   (3, [9, 7, 3, 4]),\n",
       "   (3, [2, 3, 5]),\n",
       "   (3, [6, 8, 7]),\n",
       "   (3, [6, 2, 9, 3]),\n",
       "   (3, [2, 6, 7, 8, 5, 4, 3, 9, 1]),\n",
       "   (3, [4, 3, 7, 9]),\n",
       "   (3, [7, 2, 4, 3]),\n",
       "   (3, [4, 6, 3, 9]),\n",
       "   (3, [4, 6, 2, 3, 7]),\n",
       "   (3, [2, 8, 3, 4]),\n",
       "   (3, [3, 1]),\n",
       "   (3, [6, 9, 1, 7]),\n",
       "   (3, [3, 4, 2, 6, 9]),\n",
       "   (3, [3, 4, 2, 6, 7]),\n",
       "   (3, [7, 5, 8, 2, 6]),\n",
       "   (3, [7, 2, 5, 6, 8]),\n",
       "   (3, [9, 1, 6, 2]),\n",
       "   (3, [7, 9, 4]),\n",
       "   (3, [7, 5, 2, 6]),\n",
       "   (3, [2, 6, 3, 4, 5]),\n",
       "   (3, [2, 6, 7, 4, 3]),\n",
       "   (3, [9, 7, 2, 6]),\n",
       "   (3, [4, 7, 8]),\n",
       "   (3, [9, 6, 8]),\n",
       "   (3, [8, 2, 5, 7]),\n",
       "   (3, [3, 7, 2, 6]),\n",
       "   (3, [7, 3, 6, 4]),\n",
       "   (3, [2, 7, 6, 5, 8]),\n",
       "   (3, [7, 9, 1, 2]),\n",
       "   (3, [2, 7, 6, 3, 4]),\n",
       "   (3, [2, 5, 7, 6]),\n",
       "   (3, [5, 2, 4, 3]),\n",
       "   (3, [9, 3, 6, 4]),\n",
       "   (3, [6, 3, 9, 7]),\n",
       "   (3, [7, 2, 6, 3, 4]),\n",
       "   (3, [9, 7, 8]),\n",
       "   (3, [2, 8, 6, 5]),\n",
       "   (3, [4, 3, 9, 7]),\n",
       "   (3, [4, 3, 9, 1]),\n",
       "   (3, [1, 2, 3, 4]),\n",
       "   (3, [7, 5, 2, 6, 8]),\n",
       "   (3, [3, 4, 5, 2]),\n",
       "   (3, [3, 4, 5, 6]),\n",
       "   (3, [8, 5, 7, 2]),\n",
       "   (3, [4, 3, 1, 9]),\n",
       "   (3, [6, 2, 1, 9]),\n",
       "   (3, [2, 3, 6, 7]),\n",
       "   (3, [2, 3, 6, 4]),\n",
       "   (3, [4, 9, 3]),\n",
       "   (3, [4, 9, 2]),\n",
       "   (3, [4, 3, 2, 7]),\n",
       "   (3, [3, 7, 9]),\n",
       "   (3, [9, 1, 7, 6]),\n",
       "   (3, [9, 1, 7, 2]),\n",
       "   (3, [4, 6, 7, 9]),\n",
       "   (3, [3, 7, 6, 2]),\n",
       "   (3, [8, 4, 3]),\n",
       "   (3, [7, 2, 9, 8]),\n",
       "   (3, [3, 6, 4, 5]),\n",
       "   (3, [6, 5, 2, 7, 8]),\n",
       "   (3, [6, 2, 3, 4, 7]),\n",
       "   (3, [6, 3, 4, 9, 2]),\n",
       "   (3, [3, 6, 2, 9]),\n",
       "   (3, [2, 5, 4, 3]),\n",
       "   (3, [2, 7, 6, 4, 3]),\n",
       "   (3, [1, 9, 2, 3]),\n",
       "   (3, [1, 9, 4]),\n",
       "   (3, [3, 4, 8, 7]),\n",
       "   (3, [4, 3, 6, 1]),\n",
       "   (3, [6, 2, 8, 7]),\n",
       "   (3, [4, 3, 8, 5]),\n",
       "   (3, [6, 3, 7, 4]),\n",
       "   (3, [2, 4, 8, 7]),\n",
       "   (3, [8, 5, 6, 2]),\n",
       "   (3, [5, 2, 6, 7]),\n",
       "   (3, [3, 4, 9, 2]),\n",
       "   (3, [2, 6, 7, 5, 8]),\n",
       "   (3, [9, 1, 2, 6, 7]),\n",
       "   (3, [9, 8]),\n",
       "   (3, [9, 6, 2, 4]),\n",
       "   (3, [2, 1]),\n",
       "   (3, [2, 6, 9, 1]),\n",
       "   (3, [6, 7, 2, 3, 4]),\n",
       "   (3, [2, 6, 5, 7]),\n",
       "   (3, [2, 6, 5, 8]),\n",
       "   (3, [7, 6, 5]),\n",
       "   (3, [7, 6, 1]),\n",
       "   (3, [8, 6, 2]),\n",
       "   (3, [6, 4, 3, 2, 5]),\n",
       "   (3, [3, 2, 4, 7]),\n",
       "   (3, [2, 5, 3]),\n",
       "   (3, [2, 5, 4]),\n",
       "   (3, [4, 2, 8]),\n",
       "   (3, [7, 6, 8, 2, 5]),\n",
       "   (3, [6, 3, 2, 7]),\n",
       "   (3, [5, 2, 7, 8]),\n",
       "   (3, [7, 6, 2, 3, 4, 9]),\n",
       "   (3, [7, 3, 1]),\n",
       "   (3, [2, 7, 5, 4]),\n",
       "   (3, [6, 9, 7, 8]),\n",
       "   (3, [3, 4, 9, 2, 6]),\n",
       "   (3, [6, 2, 3, 4, 7, 9, 1, 5, 8]),\n",
       "   (3, [8, 5, 4, 3]),\n",
       "   (3, [2, 7, 9, 6]),\n",
       "   (3, [2, 8, 4]),\n",
       "   (3, [2, 8, 6]),\n",
       "   (3, [4, 8, 7]),\n",
       "   (3, [4, 8, 5]),\n",
       "   (3, [5, 7, 3]),\n",
       "   (3, [3, 4, 6, 2, 9]),\n",
       "   (3, [6, 9, 2, 3]),\n",
       "   (3, [8, 7, 5]),\n",
       "   (3, [8, 7, 4]),\n",
       "   (3, [9, 4, 7, 3]),\n",
       "   (3, [2, 7, 5, 3]),\n",
       "   (3, [2, 7, 5, 6]),\n",
       "   (3, [6, 7, 8, 9]),\n",
       "   (3, [2, 1, 9]),\n",
       "   (3, [3, 4, 6, 7, 2]),\n",
       "   (3, [2, 4, 7, 8]),\n",
       "   (3, [4, 3, 2, 6, 7]),\n",
       "   (3, [5, 7, 8, 2]),\n",
       "   (3, [7, 2, 6, 4]),\n",
       "   (3, [3, 2, 7]),\n",
       "   (3, [3, 2, 5]),\n",
       "   (3, [1, 7, 6]),\n",
       "   (3, [3, 7, 4, 8]),\n",
       "   (2, [2, 3, 4, 6, 9]),\n",
       "   (2, [2, 3, 4, 6, 8]),\n",
       "   (2, [4, 2, 3, 1]),\n",
       "   (2, [4, 3, 1, 2]),\n",
       "   (2, [7, 2, 3, 8]),\n",
       "   (2, [9, 4, 7]),\n",
       "   (2, [2, 6, 8, 7]),\n",
       "   (2, [6, 4, 2, 7]),\n",
       "   (2, [9, 1, 2, 7]),\n",
       "   (2, [2, 6, 4, 8]),\n",
       "   (2, [6, 7, 1]),\n",
       "   (2, [7, 6, 4, 9]),\n",
       "   (2, [7, 6, 4, 2]),\n",
       "   (2, [3, 4, 9, 1, 6, 7]),\n",
       "   (2, [3, 6, 9, 2]),\n",
       "   (2, [9, 6, 2, 3, 4]),\n",
       "   (2, [2, 3, 4, 7, 6]),\n",
       "   (2, [7, 6, 8, 3]),\n",
       "   (2, [5, 3, 4]),\n",
       "   (2, [3, 9, 6, 4]),\n",
       "   (2, [6, 1, 9, 4]),\n",
       "   (2, [7, 4, 3, 5]),\n",
       "   (2, [5, 2, 8, 7]),\n",
       "   (2, [9, 1, 8, 6]),\n",
       "   (2, [9, 1, 2, 6, 3, 4]),\n",
       "   (2, [8, 3, 4, 7]),\n",
       "   (2, [7, 2, 8, 4, 3, 5, 6, 1, 9]),\n",
       "   (2, [4, 3, 6, 9, 2]),\n",
       "   (2, [2, 3, 9]),\n",
       "   (2, [4, 2, 6, 8]),\n",
       "   (2, [7, 9, 8, 2]),\n",
       "   (2, [4, 3, 5, 8]),\n",
       "   (2, [4, 3, 5, 6]),\n",
       "   (2, [9, 8, 6]),\n",
       "   (2, [9, 2, 1, 6]),\n",
       "   (2, [7, 2, 8, 6, 5]),\n",
       "   (2, [6, 2, 9, 4]),\n",
       "   (2, [2, 9, 7, 3]),\n",
       "   (2, [4, 2, 6, 3]),\n",
       "   (2, [4, 1, 9]),\n",
       "   (2, [4, 1, 3]),\n",
       "   (2, [4, 1, 6]),\n",
       "   (2, [3, 4, 7, 8, 2]),\n",
       "   (2, [4, 6, 3, 7]),\n",
       "   (2, [9, 2, 3, 4, 6]),\n",
       "   (2, [2, 3, 6, 4, 7]),\n",
       "   (2, [7, 2, 6, 8, 5, 9, 1, 3, 4]),\n",
       "   (2, [3, 4, 2, 5, 7, 6, 1, 9, 8]),\n",
       "   (2, [5, 4, 8]),\n",
       "   (2, [5, 4, 3]),\n",
       "   (2, [7, 4, 2, 1]),\n",
       "   (2, [2, 7, 6, 9]),\n",
       "   (2, [2, 7, 6, 3]),\n",
       "   (2, [2, 7, 6, 4]),\n",
       "   (2, [4, 3, 9, 6, 2]),\n",
       "   (2, [3, 7, 8, 2]),\n",
       "   (2, [1, 2, 6, 9]),\n",
       "   (2, [8, 2, 4, 5]),\n",
       "   (2, [5, 8, 7]),\n",
       "   (2, [6, 8, 2, 7]),\n",
       "   (2, [9, 7, 6, 8]),\n",
       "   (2, [9, 7, 6, 3]),\n",
       "   (2, [9, 7, 6, 4]),\n",
       "   (2, [9, 1, 2, 3, 4, 6, 5, 7, 8]),\n",
       "   (2, [1, 2, 7, 9]),\n",
       "   (2, [8, 4]),\n",
       "   (2, [5, 8, 2, 4]),\n",
       "   (2, [1, 2, 3, 4, 9]),\n",
       "   (2, [4, 3, 7, 6, 9, 8, 5, 2, 1]),\n",
       "   (2, [2, 6, 3, 4, 7]),\n",
       "   (2, [9, 7, 2, 1]),\n",
       "   (2, [7, 8, 2, 6, 9, 1, 3, 4]),\n",
       "   (2, [9, 2, 5]),\n",
       "   (2, [3, 2, 7, 6]),\n",
       "   (2, [7, 3, 8]),\n",
       "   (2, [2, 6, 7, 3, 4, 5, 8, 1, 9]),\n",
       "   (2, [6, 4, 3, 5, 2]),\n",
       "   (2, [8, 2, 5, 1]),\n",
       "   (2, [7, 6, 8, 5, 2]),\n",
       "   (2, [3, 7, 2, 8]),\n",
       "   (2, [7, 3, 6, 2]),\n",
       "   (2, [7, 3, 6, 5]),\n",
       "   (2, [4, 2, 3, 7]),\n",
       "   (2, [2, 7, 8, 5, 6, 3, 4, 9, 1]),\n",
       "   (2, [2, 4, 6, 8]),\n",
       "   (2, [8, 2, 5, 4]),\n",
       "   (2, [2, 5, 7, 8]),\n",
       "   (2, [9, 4, 3, 6, 1]),\n",
       "   (2, [6, 3, 7, 2]),\n",
       "   (2, [5, 8, 3, 4]),\n",
       "   (2, [5, 8, 3, 2]),\n",
       "   (2, [7, 2, 8, 6, 5, 4, 3]),\n",
       "   (2, [6, 3, 9, 4]),\n",
       "   (2, [7, 2, 6, 5, 8]),\n",
       "   (2, [7, 2, 6, 8, 9]),\n",
       "   (2, [1, 4, 9]),\n",
       "   (2, [7, 8, 5, 2, 6, 9, 4, 3, 1]),\n",
       "   (2, [1, 7, 9, 6]),\n",
       "   (2, [8, 9]),\n",
       "   (2, [8, 6]),\n",
       "   (2, [2, 9, 3, 4]),\n",
       "   (2, [3, 4, 1, 9]),\n",
       "   (2, [3, 4, 6, 9, 1]),\n",
       "   (2, [7, 8, 2, 1]),\n",
       "   (2, [7, 6, 2, 5, 8, 4, 3, 9, 1]),\n",
       "   (2, [8, 5, 7, 6]),\n",
       "   (2, [4, 3, 1, 5]),\n",
       "   (2, [5, 8, 7, 6, 2]),\n",
       "   (2, [2, 6, 7, 8, 5]),\n",
       "   (2, [2, 6, 7, 8]),\n",
       "   (2, [2, 6, 7, 4]),\n",
       "   (2, [8, 5, 6, 7]),\n",
       "   (2, [2, 7, 8, 3]),\n",
       "   (2, [3, 4, 6, 5, 2]),\n",
       "   (2, [3, 6, 7, 2, 4]),\n",
       "   (2, [9, 6, 2, 3]),\n",
       "   (2, [6, 4, 1, 3]),\n",
       "   (2, [7, 6, 5, 8]),\n",
       "   (2, [2, 7, 4, 3]),\n",
       "   (2, [7, 6, 1, 9]),\n",
       "   (2, [2, 7, 8, 6, 5]),\n",
       "   (2, [4, 6, 7, 2]),\n",
       "   (2, [6, 5, 4, 3]),\n",
       "   (2, [4, 5, 8, 3]),\n",
       "   (2, [9, 6, 4, 2]),\n",
       "   (2, [8, 4, 7]),\n",
       "   (2, [6, 3, 8]),\n",
       "   (2, [6, 2, 3, 4, 7, 5, 1, 9, 8]),\n",
       "   (2, [3, 6, 8, 2]),\n",
       "   (2, [3, 2, 6, 5]),\n",
       "   (2, [7, 2, 5, 3, 4]),\n",
       "   (2, [4, 8, 2, 3]),\n",
       "   (2, [6, 2, 4, 3, 7]),\n",
       "   (2, [2, 6, 7, 9, 1]),\n",
       "   (2, [3, 4, 2, 6, 8]),\n",
       "   (2, [2, 3, 7, 6]),\n",
       "   (2, [2, 3, 7, 4]),\n",
       "   (2, [6, 7, 8, 2, 5]),\n",
       "   (2, [7, 6, 9, 8]),\n",
       "   (2, [5, 7, 2, 3, 4]),\n",
       "   (2, [4, 3, 6, 2, 9]),\n",
       "   (2, [4, 3, 6, 2, 8]),\n",
       "   (2, [9, 4, 2]),\n",
       "   (2, [9, 4, 1]),\n",
       "   (2, [7, 5, 1]),\n",
       "   (2, [3, 4, 2, 6, 9, 1, 5, 8, 7]),\n",
       "   (2, [1, 9, 2, 6]),\n",
       "   (2, [7, 5, 9]),\n",
       "   (2, [8, 2, 7, 5]),\n",
       "   (2, [2, 4, 7]),\n",
       "   (2, [4, 3, 2, 6, 9, 1, 8, 5, 7]),\n",
       "   (2, [4, 9, 2, 3]),\n",
       "   (2, [9, 1, 3, 4, 6]),\n",
       "   (2, [9, 1, 3, 4, 2]),\n",
       "   (2, [5, 6, 2, 7]),\n",
       "   (2, [3, 4, 9, 1, 6, 2, 7, 5, 8]),\n",
       "   (2, [6, 4, 3, 9, 2]),\n",
       "   (2, [6, 2, 4, 9]),\n",
       "   (2, [7, 2, 9, 6]),\n",
       "   (2, [9, 3, 4, 7]),\n",
       "   (2, [3, 7, 8, 4, 6, 9, 5, 2, 1]),\n",
       "   (2, [8, 2, 7, 5, 6]),\n",
       "   (2, [2, 6, 3, 7]),\n",
       "   (2, [4, 6, 2, 7]),\n",
       "   (2, [8, 7, 2, 6]),\n",
       "   (2, [4, 3, 6, 7, 8]),\n",
       "   (2, [4, 3, 6, 7, 2]),\n",
       "   (2, [2, 8, 4, 6]),\n",
       "   (2, [4, 7, 3, 9]),\n",
       "   (2, [7, 8, 2, 1, 5]),\n",
       "   (2, [2, 3, 4, 9]),\n",
       "   (2, [2, 7, 6, 3, 4, 5, 8, 9, 1]),\n",
       "   (2, [7, 6, 2, 1]),\n",
       "   (2, [7, 3, 4, 6, 9, 2]),\n",
       "   (2, [3, 4, 6, 9, 2, 1, 7, 5, 8]),\n",
       "   (2, [9, 4, 3, 2]),\n",
       "   (2, [2, 6, 9, 3, 4]),\n",
       "   (2, [9, 1, 6, 7, 2]),\n",
       "   (2, [6, 9, 7, 2]),\n",
       "   (2, [6, 9, 2, 4]),\n",
       "   (2, [8, 7, 5, 3]),\n",
       "   (2, [4, 3, 8, 6, 2]),\n",
       "   (2, [3, 4, 6, 5]),\n",
       "   (2, [4, 1, 9, 6]),\n",
       "   (2, [6, 4, 3, 8]),\n",
       "   (2, [6, 4, 1]),\n",
       "   (2, [2, 3, 4, 5, 6, 8]),\n",
       "   (2, [3, 2, 8, 4]),\n",
       "   (2, [6, 8, 3]),\n",
       "   (2, [4, 3, 2, 6, 9]),\n",
       "   (2, [4, 3, 2, 6, 8]),\n",
       "   (2, [7, 4, 8]),\n",
       "   (2, [7, 2, 8, 5, 6, 9, 1, 3, 4]),\n",
       "   (2, [3, 2, 4, 5]),\n",
       "   (2, [4, 7, 8, 6, 2, 3, 5, 9, 1]),\n",
       "   (2, [6, 9, 4, 3, 2]),\n",
       "   (2, [1, 3, 9]),\n",
       "   (2, [5, 6, 3, 2]),\n",
       "   (2, [7, 4, 2, 8]),\n",
       "   (2, [7, 1, 9, 6]),\n",
       "   (2, [7, 1, 9, 4]),\n",
       "   (2, [4, 1]),\n",
       "   (2, [9, 6, 4, 3, 2, 5, 1, 8, 7]),\n",
       "   (2, [6, 3, 2, 9]),\n",
       "   (2, [7, 8, 6, 5, 2]),\n",
       "   (2, [7, 4, 8, 3]),\n",
       "   (2, [7, 4, 8, 2]),\n",
       "   (2, [7, 5, 4, 3]),\n",
       "   (2, [4, 3, 2, 1]),\n",
       "   (2, [3, 8, 5]),\n",
       "   (2, [3, 8, 7]),\n",
       "   (2, [4, 5, 3, 2]),\n",
       "   (2, [2, 9, 6, 1]),\n",
       "   (2, [6, 9, 7, 1]),\n",
       "   (2, [4, 1, 6, 3, 9]),\n",
       "   (2, [4, 3, 2, 7, 6]),\n",
       "   (2, [6, 5, 7, 8, 2, 3, 4]),\n",
       "   (2, [4, 3, 2, 8]),\n",
       "   (2, [6, 4, 3, 7, 2]),\n",
       "   (2, [4, 3, 2, 5]),\n",
       "   (2, [6, 4, 3, 7, 8]),\n",
       "   (2, [2, 8, 3]),\n",
       "   (2, [2, 9, 5]),\n",
       "   (2, [7, 6, 5, 2, 8]),\n",
       "   (2, [7, 6, 2, 4, 3]),\n",
       "   (2, [4, 8, 2]),\n",
       "   (2, [7, 2, 1, 8]),\n",
       "   (2, [2, 7, 9, 5]),\n",
       "   (2, [5, 7, 6]),\n",
       "   (2, [7, 3, 4, 9]),\n",
       "   (2, [7, 3, 4, 1]),\n",
       "   (2, [7, 3, 4, 5]),\n",
       "   (2, [3, 8, 1, 9]),\n",
       "   (2, [7, 2, 8, 5, 6, 1, 3, 4, 9]),\n",
       "   (2, [4, 3, 7, 2, 6]),\n",
       "   (2, [2, 3, 4, 5, 6]),\n",
       "   (2, [3, 4, 6, 8, 7, 5, 2, 1, 9]),\n",
       "   (2, [8, 5, 2, 3]),\n",
       "   (2, [7, 9, 3, 6]),\n",
       "   (2, [6, 5, 9]),\n",
       "   (2, [3, 9, 4]),\n",
       "   (2, [3, 9, 6]),\n",
       "   (2, [3, 9, 1]),\n",
       "   (2, [7, 6, 5, 4, 3, 2, 1, 8, 9]),\n",
       "   (2, [3, 5, 8, 4]),\n",
       "   (2, [6, 9, 5]),\n",
       "   (2, [3, 6, 9, 4]),\n",
       "   (2, [8, 7, 3]),\n",
       "   (2, [7, 4, 3, 9, 6, 2, 5, 1, 8]),\n",
       "   (2, [2, 7, 5, 1]),\n",
       "   (2, [2, 7, 5, 8]),\n",
       "   (2, [3, 4, 2, 5, 8]),\n",
       "   (2, [3, 5, 6]),\n",
       "   (2, [3, 5, 7]),\n",
       "   (2, [9, 1, 5]),\n",
       "   (2, [6, 8, 5, 2]),\n",
       "   (2, [8, 2, 5, 7, 6]),\n",
       "   (2, [6, 4, 7, 3]),\n",
       "   (2, [6, 4, 7, 2]),\n",
       "   (2, [6, 4, 7, 8]),\n",
       "   (2, [2, 6, 7, 3, 4, 1, 5, 8, 9]),\n",
       "   (2, [6, 7, 4, 1]),\n",
       "   (2, [6, 1, 3]),\n",
       "   (2, [6, 7, 4, 8]),\n",
       "   (2, [6, 7, 4, 2, 3]),\n",
       "   (2, [9, 3, 4, 6, 2]),\n",
       "   (2, [7, 4, 3, 2, 1, 9, 5, 6, 8]),\n",
       "   (2, [1, 6, 7]),\n",
       "   (2, [3, 5, 2, 4]),\n",
       "   (2, [6, 7, 2, 5]),\n",
       "   (2, [9, 5, 6]),\n",
       "   (2, [9, 1, 2, 6, 3, 4, 7, 8, 5]),\n",
       "   (2, [2, 1, 4]),\n",
       "   (2, [2, 1, 7]),\n",
       "   (2, [3, 4, 7, 6, 9]),\n",
       "   (2, [7, 9, 6, 4]),\n",
       "   (2, [7, 8, 5, 6]),\n",
       "   (2, [8, 3, 7, 4]),\n",
       "   (2, [6, 7, 4, 3, 2]),\n",
       "   (2, [5, 2, 8, 7, 6]),\n",
       "   (2, [3, 4, 6, 2, 7, 9, 1, 5, 8]),\n",
       "   (2, [6, 3, 4, 7, 8, 9, 2, 1]),\n",
       "   (2, [8, 7, 9, 6]),\n",
       "   (2, [3, 4, 8, 2, 5, 6, 9, 7, 1]),\n",
       "   (2, [8, 2, 7, 6]),\n",
       "   (2, [6, 2, 7, 5]),\n",
       "   (2, [9, 7, 6, 1, 2]),\n",
       "   (2, [3, 2, 9]),\n",
       "   (2, [6, 7, 3, 2]),\n",
       "   (2, [7, 8, 3, 4]),\n",
       "   (1, [4, 2, 3, 5]),\n",
       "   (1, [2, 6, 4, 7, 3, 8]),\n",
       "   (1, [2, 5, 6, 7, 8, 3, 4, 1, 9]),\n",
       "   (1, [3, 6, 5]),\n",
       "   (1, [2, 3, 7, 6, 8]),\n",
       "   (1, [3, 4, 6, 1, 2, 9, 8, 5, 7]),\n",
       "   (1, [2, 3, 4, 6, 5]),\n",
       "   (1, [6, 2, 4, 8, 7, 9]),\n",
       "   (1, [2, 3, 4, 9, 1, 6, 7, 8, 5]),\n",
       "   (1, [6, 2, 1, 7, 3, 4, 5, 8, 9]),\n",
       "   (1, [4, 5, 2, 6, 3]),\n",
       "   (1, [2, 7, 3, 5, 4, 6, 8, 9, 1]),\n",
       "   (1, [1, 3, 5, 6, 2, 4, 7, 8, 9]),\n",
       "   (1, [3, 4, 6, 9, 7, 8, 5, 2, 1]),\n",
       "   (1, [2, 7, 4, 3, 6]),\n",
       "   (1, [6, 8, 2, 7, 4, 3, 9, 1]),\n",
       "   (1, [7, 4, 3, 8, 9, 5, 6, 2, 1]),\n",
       "   (1, [2, 7, 6, 8, 9, 5, 1, 3, 4]),\n",
       "   (1, [3, 4, 8, 6, 5, 2]),\n",
       "   (1, [4, 3, 1, 6, 9, 2, 7, 5, 8]),\n",
       "   (1, [2, 3, 4, 7, 6, 8, 5, 1]),\n",
       "   (1, [2, 7, 9, 8, 6]),\n",
       "   (1, [4, 3, 8, 9, 6, 5, 2, 1, 7]),\n",
       "   (1, [4, 7, 8, 2, 3, 9, 5, 6, 1]),\n",
       "   (1, [3, 4, 2, 1, 6, 7, 5, 9, 8]),\n",
       "   (1, [3, 8, 4, 7]),\n",
       "   (1, [3, 8, 4, 6]),\n",
       "   (1, [3, 8, 4, 5]),\n",
       "   (1, [1, 8, 5, 3, 9, 4, 2, 6, 7]),\n",
       "   (1, [2, 4, 7, 8, 5, 3, 1, 6]),\n",
       "   (1, [3, 4, 5, 2, 1, 6, 8, 9, 7]),\n",
       "   (1, [1, 2, 6, 3, 4, 7, 8, 5, 9]),\n",
       "   (1, [1, 6, 2, 9]),\n",
       "   (1, [4, 1, 2, 9]),\n",
       "   (1, [3, 4, 6, 2, 7, 5, 1, 8, 9]),\n",
       "   (1, [7, 4, 3, 8, 6, 2, 1, 9, 5]),\n",
       "   (1, [3, 2, 4, 5, 8, 6]),\n",
       "   (1, [4, 1, 6, 9]),\n",
       "   (1, [2, 3, 1, 4]),\n",
       "   (1, [7, 3, 4, 5, 6, 2, 1, 8, 9]),\n",
       "   (1, [4, 1, 6, 5]),\n",
       "   (1, [3, 4, 2, 6, 5, 7, 1, 9, 8]),\n",
       "   (1, [2, 3, 4, 6, 7, 9, 1, 5, 8]),\n",
       "   (1, [2, 3, 6, 4, 7, 9, 5, 1, 8]),\n",
       "   (1, [4, 9, 1, 3]),\n",
       "   (1, [9, 3, 4, 1, 7]),\n",
       "   (1, [7, 6, 3, 2, 4, 8, 5, 1, 9]),\n",
       "   (1, [7, 3, 2, 8]),\n",
       "   (1, [9, 3, 4, 1, 2]),\n",
       "   (1, [1, 7, 9, 2]),\n",
       "   (1, [7, 8, 4, 3, 2]),\n",
       "   (1, [3, 7, 2, 1, 5, 8, 9, 4, 6]),\n",
       "   (1, [2, 3, 7, 8, 9, 4]),\n",
       "   (1, [1, 9, 6, 2, 4, 3, 5, 8, 7]),\n",
       "   (1, [4, 3, 5, 7, 8, 6, 2, 9, 1]),\n",
       "   (1, [3, 4, 8, 9, 7, 6, 2, 1]),\n",
       "   (1, [4, 6, 1, 9, 3]),\n",
       "   (1, [3, 4, 2, 6, 8, 5]),\n",
       "   (1, [3, 1, 7]),\n",
       "   (1, [2, 5, 8, 7, 6, 4, 3, 1, 9]),\n",
       "   (1, [6, 9, 3, 8]),\n",
       "   (1, [4, 6, 5, 9, 3]),\n",
       "   (1, [3, 7, 8, 6, 5, 1, 2]),\n",
       "   (1, [7, 9, 6, 8, 5]),\n",
       "   (1, [7, 2, 9, 8, 6, 3, 4, 5, 1]),\n",
       "   (1, [4, 3, 8, 7, 6]),\n",
       "   (1, [5, 7, 4, 3, 8]),\n",
       "   (1, [2, 4, 7, 8, 6]),\n",
       "   (1, [7, 6, 3, 2, 9, 5, 1, 4, 8]),\n",
       "   (1, [8, 2, 4, 3, 6]),\n",
       "   (1, [2, 3, 4, 5, 7, 8, 6, 1, 9]),\n",
       "   (1, [9, 7, 6, 4, 3, 1, 2, 5, 8]),\n",
       "   (1, [4, 6, 8]),\n",
       "   (1, [3, 7, 9, 4]),\n",
       "   (1, [4, 6, 5]),\n",
       "   (1, [6, 4, 2, 1]),\n",
       "   (1, [7, 8, 2, 5, 6, 1, 3, 4, 9]),\n",
       "   (1, [6, 4, 2, 5]),\n",
       "   (1, [2, 9, 3, 4, 6, 1, 5, 7, 8]),\n",
       "   (1, [2, 4, 3, 6, 5, 7, 9, 8, 1]),\n",
       "   (1, [2, 3, 5, 8, 9, 6, 7, 1]),\n",
       "   (1, [9, 1, 2, 3]),\n",
       "   (1, [6, 7, 2, 4, 1, 3, 9, 5, 8]),\n",
       "   (1, [9, 7, 2, 6, 1, 3, 4, 5, 8]),\n",
       "   (1, [9, 6, 7, 2, 3, 4]),\n",
       "   (1, [9, 8, 6, 2, 1, 3, 7, 4, 5]),\n",
       "   (1, [7, 8, 3, 4, 6]),\n",
       "   (1, [9, 6, 7, 2, 3, 4, 1, 8, 5]),\n",
       "   (1, [7, 4, 3, 1, 6, 9, 2, 8, 5]),\n",
       "   (1, [7, 9, 6, 2, 1]),\n",
       "   (1, [9, 7, 6, 4, 3, 8]),\n",
       "   (1, [9, 1, 4, 6]),\n",
       "   (1, [8, 2, 5, 7, 4, 3, 6, 1, 9]),\n",
       "   (1, [5, 3, 2, 8]),\n",
       "   (1, [9, 1, 4, 2]),\n",
       "   (1, [3, 2, 7, 4, 8]),\n",
       "   (1, [3, 4, 2, 6, 5, 1, 8, 9, 7]),\n",
       "   (1, [3, 4, 2, 7, 5]),\n",
       "   (1, [3, 4, 2, 7, 8]),\n",
       "   (1, [8, 4, 6, 2, 9, 7, 3, 5, 1]),\n",
       "   (1, [6, 1, 8, 9]),\n",
       "   (1, [6, 8, 2, 4, 3, 5, 1, 9, 7]),\n",
       "   (1, [6, 9, 2, 8, 3]),\n",
       "   (1, [2, 8, 5, 4, 3, 6]),\n",
       "   (1, [3, 2, 8, 7, 6, 5, 4, 1, 9]),\n",
       "   (1, [7, 6, 4, 5]),\n",
       "   (1, [4, 3, 2, 5, 6, 8, 9, 1, 7]),\n",
       "   (1, [9, 6, 7, 3]),\n",
       "   (1, [2, 6, 3, 4, 7, 9, 1, 5, 8]),\n",
       "   (1, [1, 9, 2, 5, 8, 7, 6, 3, 4]),\n",
       "   (1, [6, 7, 9, 1, 2]),\n",
       "   (1, [3, 4, 6, 9, 1, 2, 7, 8, 5]),\n",
       "   (1, [6, 9, 1, 2, 8, 4]),\n",
       "   (1, [4, 6, 8, 2]),\n",
       "   (1, [2, 6, 3, 4, 7, 5, 8, 9, 1]),\n",
       "   (1, [3, 4, 7, 2, 6, 9, 1, 5, 8]),\n",
       "   (1, [7, 6, 3, 4, 8, 5, 2, 9, 1]),\n",
       "   (1, [3, 4, 9, 1, 6, 2]),\n",
       "   (1, [7, 2, 6, 1, 5, 8, 9]),\n",
       "   (1, [3, 6, 4, 2, 9, 1, 5, 8, 7]),\n",
       "   (1, [7, 3, 4, 5, 8]),\n",
       "   (1, [7, 3, 4, 5, 6]),\n",
       "   (1, [4, 3, 6, 2, 9, 7, 5, 1, 8]),\n",
       "   (1, [1, 9, 6, 5, 7, 8, 4, 2, 3]),\n",
       "   (1, [3, 2, 6, 5, 4]),\n",
       "   (1, [3, 6, 7, 2]),\n",
       "   (1, [3, 4, 6, 2, 9, 1, 5, 8, 7]),\n",
       "   (1, [6, 5, 8, 4, 7]),\n",
       "   (1, [3, 4, 8, 5, 2, 6, 7, 9, 1]),\n",
       "   (1, [7, 2, 5, 8, 6]),\n",
       "   (1, [4, 3, 1, 2, 6]),\n",
       "   (1, [3, 4, 6, 2, 7, 9]),\n",
       "   (1, [2, 3, 4, 5, 6, 7, 8, 9, 1]),\n",
       "   (1, [3, 5, 4, 7]),\n",
       "   (1, [3, 2, 5, 8]),\n",
       "   (1, [3, 2, 5, 4]),\n",
       "   (1, [3, 2, 5, 6]),\n",
       "   (1, [3, 4, 8, 7, 2, 5, 6, 1, 9]),\n",
       "   (1, [8, 5, 3]),\n",
       "   (1, [5, 8, 7, 2, 4, 3]),\n",
       "   (1, [6, 4, 8, 3, 2]),\n",
       "   (1, [4, 5, 1, 2]),\n",
       "   (1, [1, 6, 2, 3, 4, 9, 7, 5, 8]),\n",
       "   (1, [3, 1, 7, 6]),\n",
       "   (1, [2, 7, 5, 3, 4]),\n",
       "   (1, [3, 5, 8, 4, 7, 1, 2, 6, 9]),\n",
       "   (1, [7, 1, 4]),\n",
       "   (1, [2, 8, 7, 5, 3, 4]),\n",
       "   (1, [2, 7, 3, 5]),\n",
       "   (1, [3, 2, 6, 9, 4]),\n",
       "   (1, [2, 5, 8, 6, 7]),\n",
       "   (1, [3, 9, 4, 6]),\n",
       "   (1, [2, 5, 7, 8, 1]),\n",
       "   (1, [3, 9, 4, 2]),\n",
       "   (1, [3, 9, 4, 1]),\n",
       "   (1, [4, 6, 8, 3, 2]),\n",
       "   (1, [6, 7, 9, 2, 1, 5, 8]),\n",
       "   (1, [2, 7, 3, 4, 5, 8, 6, 9, 1]),\n",
       "   (1, [4, 3, 7, 6, 8, 5, 2]),\n",
       "   (1, [5, 3, 8]),\n",
       "   (1, [9, 6, 2, 1, 7]),\n",
       "   (1, [7, 3, 4, 2, 6, 8, 5, 9, 1]),\n",
       "   (1, [9, 6, 2, 1, 3]),\n",
       "   (1, [4, 3, 1, 8, 7, 9, 5, 2, 6]),\n",
       "   (1, [9, 6, 3, 4, 1]),\n",
       "   (1, [5, 2, 6, 7, 8]),\n",
       "   ...]),\n",
       " (9567,\n",
       "  defaultdict(int,\n",
       "              {1: 0.30615657991010764,\n",
       "               2: 0.18386118950559213,\n",
       "               3: 0.23047977422389465,\n",
       "               4: 0.14205079962370648,\n",
       "               5: 0.038883662590153655,\n",
       "               9: 0.07410891606564231,\n",
       "               8: 0.006062506532873419,\n",
       "               6: 0.013379324762203408,\n",
       "               7: 0.005017246785826278,\n",
       "               0: 0}),\n",
       "  [(1112, [6]),\n",
       "   (584, [3]),\n",
       "   (499, [4]),\n",
       "   (190, [8]),\n",
       "   (188, [7]),\n",
       "   (160, [5]),\n",
       "   (108, [4, 2]),\n",
       "   (105, [4, 5]),\n",
       "   (93, [6, 3]),\n",
       "   (86, [6, 7]),\n",
       "   (86, [6, 5]),\n",
       "   (85, [2]),\n",
       "   (84, [3, 7]),\n",
       "   (82, [7, 5]),\n",
       "   (73, [9]),\n",
       "   (73, [3, 6]),\n",
       "   (72, [5, 7]),\n",
       "   (72, [8, 3]),\n",
       "   (59, [4, 2, 5]),\n",
       "   (52, [4, 7]),\n",
       "   (50, [3, 5]),\n",
       "   (49, [6, 5, 7]),\n",
       "   (47, [6, 8]),\n",
       "   (46, [3, 9]),\n",
       "   (44, [7, 3]),\n",
       "   (42, [6, 8, 3]),\n",
       "   (42, [5, 6]),\n",
       "   (41, [4, 5, 7]),\n",
       "   (38, [2, 4]),\n",
       "   (38, [1]),\n",
       "   (37, [7, 6]),\n",
       "   (36, [7, 5, 6]),\n",
       "   (36, [6, 7, 5]),\n",
       "   (36, [3, 8]),\n",
       "   (36, [6, 4]),\n",
       "   (36, [3, 5, 7]),\n",
       "   (35, [5, 7, 6]),\n",
       "   (33, [3, 7, 5]),\n",
       "   (33, [6, 3, 8]),\n",
       "   (31, [4, 5, 2]),\n",
       "   (29, [4, 6]),\n",
       "   (28, [6, 9]),\n",
       "   (28, [6, 4, 5]),\n",
       "   (27, [3, 7, 9]),\n",
       "   (27, [3, 4]),\n",
       "   (27, [8, 3, 6]),\n",
       "   (26, [8, 6, 3]),\n",
       "   (26, [6, 5, 3]),\n",
       "   (25, [4, 3]),\n",
       "   (25, [5, 4]),\n",
       "   (25, [5, 6, 7]),\n",
       "   (24, [6, 5, 4]),\n",
       "   (23, [5, 7, 2]),\n",
       "   (22, [3, 6, 7]),\n",
       "   (22, [3, 6, 5]),\n",
       "   (22, [9, 3]),\n",
       "   (21, [2, 4, 5]),\n",
       "   (21, [5, 4, 7]),\n",
       "   (19, [4, 6, 5]),\n",
       "   (19, [4, 2, 7]),\n",
       "   (19, [9, 7]),\n",
       "   (19, [6, 3, 7]),\n",
       "   (19, [3, 7, 1]),\n",
       "   (19, [7, 5, 3]),\n",
       "   (18, [5, 3, 7]),\n",
       "   (18, [2, 5]),\n",
       "   (18, [4, 7, 5]),\n",
       "   (18, [7, 3, 5]),\n",
       "   (18, [6, 3, 5]),\n",
       "   (18, [8, 3, 9]),\n",
       "   (18, [5, 2]),\n",
       "   (17, [7, 6, 5]),\n",
       "   (17, [3, 2]),\n",
       "   (17, [8, 6]),\n",
       "   (17, [6, 2]),\n",
       "   (17, [4, 3, 5]),\n",
       "   (16, [3, 9, 1]),\n",
       "   (16, [3, 9, 7]),\n",
       "   (16, [3, 4, 5]),\n",
       "   (16, [8, 6, 3, 1]),\n",
       "   (16, [7, 9]),\n",
       "   (16, [7, 2]),\n",
       "   (16, [4, 5, 6]),\n",
       "   (15, [2, 7]),\n",
       "   (15, [5, 3]),\n",
       "   (15, [5, 7, 4]),\n",
       "   (15, [3, 1]),\n",
       "   (15, [3, 7, 2]),\n",
       "   (14, [3, 6, 1]),\n",
       "   (14, [3, 6, 8]),\n",
       "   (14, [7, 5, 4]),\n",
       "   (14, [4, 2, 5, 7]),\n",
       "   (14, [6, 3, 9]),\n",
       "   (14, [8, 3, 1]),\n",
       "   (14, [7, 1]),\n",
       "   (14, [3, 6, 8, 9]),\n",
       "   (13, [4, 7, 2]),\n",
       "   (13, [3, 4, 7]),\n",
       "   (13, [6, 3, 4]),\n",
       "   (12, [2, 5, 7]),\n",
       "   (12, [7, 3, 9]),\n",
       "   (12, [7, 3, 6]),\n",
       "   (12, [3, 4, 6]),\n",
       "   (12, [2, 4, 5, 7]),\n",
       "   (12, [6, 9, 7]),\n",
       "   (12, [7, 4]),\n",
       "   (12, [4, 5, 3]),\n",
       "   (12, [6, 3, 8, 1]),\n",
       "   (11, [3, 8, 1, 6]),\n",
       "   (11, [3, 6, 9]),\n",
       "   (11, [7, 5, 2]),\n",
       "   (11, [6, 4, 5, 7]),\n",
       "   (11, [3, 8, 9]),\n",
       "   (11, [4, 5, 7, 2]),\n",
       "   (11, [9, 1]),\n",
       "   (11, [6, 3, 1]),\n",
       "   (11, [7, 2, 5]),\n",
       "   (11, [4, 3, 7]),\n",
       "   (10, [7, 5, 1]),\n",
       "   (10, [6, 7, 9]),\n",
       "   (10, [5, 3, 6]),\n",
       "   (10, [9, 7, 1]),\n",
       "   (10, [5, 4, 2]),\n",
       "   (10, [7, 5, 6, 3]),\n",
       "   (9, [6, 8, 1]),\n",
       "   (9, [2, 4, 7]),\n",
       "   (9, [8, 3, 6, 1]),\n",
       "   (9, [7, 3, 1]),\n",
       "   (9, [4, 2, 8]),\n",
       "   (9, [3, 8, 6]),\n",
       "   (9, [3, 8, 1]),\n",
       "   (9, [5, 7, 3]),\n",
       "   (9, [3, 5, 6]),\n",
       "   (9, [6, 1]),\n",
       "   (9, [4, 5, 2, 7]),\n",
       "   (8, [6, 8, 2]),\n",
       "   (8, [5, 2, 7]),\n",
       "   (8, [4, 6, 7]),\n",
       "   (8, [7, 9, 3]),\n",
       "   (8, [2, 3]),\n",
       "   (8, [7, 6, 3]),\n",
       "   (8, [2, 5, 4]),\n",
       "   (8, [4, 3, 2]),\n",
       "   (8, [3, 8, 2]),\n",
       "   (8, [6, 7, 3]),\n",
       "   (8, [3, 7, 9, 1]),\n",
       "   (8, [6, 9, 3]),\n",
       "   (8, [3, 5, 4]),\n",
       "   (8, [7, 6, 5, 3]),\n",
       "   (8, [6, 7, 8]),\n",
       "   (8, [5, 6, 9]),\n",
       "   (8, [3, 9, 5]),\n",
       "   (7, [3, 6, 7, 5]),\n",
       "   (7, [3, 1, 7, 9]),\n",
       "   (7, [6, 1, 3]),\n",
       "   (7, [9, 6, 3]),\n",
       "   (7, [3, 1, 6]),\n",
       "   (7, [3, 1, 7]),\n",
       "   (7, [3, 1, 9]),\n",
       "   (7, [6, 4, 3]),\n",
       "   (7, [6, 8, 9]),\n",
       "   (7, [7, 4, 5]),\n",
       "   (7, [5, 4, 6]),\n",
       "   (7, [8, 6, 3, 5]),\n",
       "   (7, [6, 7, 2]),\n",
       "   (7, [4, 7, 5, 2]),\n",
       "   (7, [1, 6]),\n",
       "   (7, [3, 5, 7, 9]),\n",
       "   (7, [3, 4, 5, 6]),\n",
       "   (7, [7, 3, 1, 9]),\n",
       "   (7, [3, 7, 8]),\n",
       "   (7, [5, 6, 3]),\n",
       "   (6, [3, 7, 5, 9]),\n",
       "   (6, [5, 2, 4]),\n",
       "   (6, [4, 6, 3]),\n",
       "   (6, [4, 2, 5, 3]),\n",
       "   (6, [3, 8, 9, 1]),\n",
       "   (6, [3, 1, 5]),\n",
       "   (6, [4, 2, 7, 5]),\n",
       "   (6, [7, 9, 1]),\n",
       "   (6, [6, 8, 5]),\n",
       "   (6, [7, 6, 9]),\n",
       "   (6, [7, 4, 2]),\n",
       "   (6, [5, 6, 7, 9]),\n",
       "   (6, [3, 9, 1, 7]),\n",
       "   (6, [4, 3, 9]),\n",
       "   (6, [9, 3, 7]),\n",
       "   (6, [5, 7, 8]),\n",
       "   (6, [5, 7, 9]),\n",
       "   (6, [6, 7, 4]),\n",
       "   (6, [8, 1, 3]),\n",
       "   (6, [3, 7, 5, 6]),\n",
       "   (6, [6, 9, 1]),\n",
       "   (6, [5, 2, 4, 7]),\n",
       "   (6, [2, 7, 5, 4]),\n",
       "   (6, [3, 5, 8]),\n",
       "   (6, [5, 9, 3]),\n",
       "   (6, [7, 2, 6]),\n",
       "   (6, [2, 7, 5]),\n",
       "   (6, [2, 4, 7, 5]),\n",
       "   (6, [4, 3, 6]),\n",
       "   (6, [6, 3, 7, 9]),\n",
       "   (6, [9, 6]),\n",
       "   (6, [3, 2, 5]),\n",
       "   (5, [6, 9, 5]),\n",
       "   (5, [6, 5, 7, 9]),\n",
       "   (5, [5, 6, 4, 7]),\n",
       "   (5, [2, 8]),\n",
       "   (5, [7, 5, 6, 9]),\n",
       "   (5, [7, 3, 5, 6]),\n",
       "   (5, [9, 3, 8]),\n",
       "   (5, [9, 1, 3, 7]),\n",
       "   (5, [7, 5, 4, 2]),\n",
       "   (5, [3, 8, 5]),\n",
       "   (5, [3, 8, 7]),\n",
       "   (5, [5, 3, 6, 7]),\n",
       "   (5, [3, 7, 2, 5]),\n",
       "   (5, [4, 7, 5, 6]),\n",
       "   (5, [6, 3, 2]),\n",
       "   (5, [3, 9, 6]),\n",
       "   (5, [2, 5, 7, 4]),\n",
       "   (5, [1, 7]),\n",
       "   (5, [1, 3]),\n",
       "   (5, [6, 3, 5, 7]),\n",
       "   (5, [9, 3, 6]),\n",
       "   (5, [3, 5, 2]),\n",
       "   (5, [6, 4, 7, 5]),\n",
       "   (5, [8, 2]),\n",
       "   (5, [9, 7, 3]),\n",
       "   (5, [6, 7, 4, 5]),\n",
       "   (5, [3, 4, 5, 2]),\n",
       "   (5, [3, 4, 5, 7]),\n",
       "   (5, [3, 5, 6, 7]),\n",
       "   (5, [2, 7, 4]),\n",
       "   (5, [3, 7, 4]),\n",
       "   (5, [7, 6, 3, 5]),\n",
       "   (5, [3, 6, 8, 1]),\n",
       "   (5, [4, 5, 6, 7]),\n",
       "   (5, [9, 3, 7, 1]),\n",
       "   (5, [3, 8, 6, 1]),\n",
       "   (5, [7, 9, 1, 3]),\n",
       "   (4, [7, 1, 5]),\n",
       "   (4, [7, 1, 9]),\n",
       "   (4, [5, 4, 2, 7]),\n",
       "   (4, [6, 3, 7, 2]),\n",
       "   (4, [6, 3, 7, 5]),\n",
       "   (4, [3, 5, 4, 7]),\n",
       "   (4, [5, 3, 9]),\n",
       "   (4, [5, 3, 4]),\n",
       "   (4, [3, 9, 6, 8]),\n",
       "   (4, [8, 9, 6]),\n",
       "   (4, [8, 9, 3]),\n",
       "   (4, [3, 7, 1, 9]),\n",
       "   (4, [6, 1, 7]),\n",
       "   (4, [6, 8, 9, 3]),\n",
       "   (4, [6, 3, 1, 8]),\n",
       "   (4, [5, 4, 3, 7]),\n",
       "   (4, [2, 9]),\n",
       "   (4, [4, 6, 3, 5]),\n",
       "   (4, [2, 3, 7]),\n",
       "   (4, [6, 1, 3, 8]),\n",
       "   (4, [6, 3, 8, 5]),\n",
       "   (4, [4, 3, 5, 7]),\n",
       "   (4, [4, 5, 2, 8]),\n",
       "   (4, [7, 5, 3, 6]),\n",
       "   (4, [8, 6, 1]),\n",
       "   (4, [8, 6, 4]),\n",
       "   (4, [2, 5, 6]),\n",
       "   (4, [4, 2, 3]),\n",
       "   (4, [5, 6, 7, 3]),\n",
       "   (4, [4, 3, 6, 7]),\n",
       "   (4, [4, 7, 6]),\n",
       "   (4, [5, 4, 3]),\n",
       "   (4, [5, 2, 7, 4]),\n",
       "   (4, [2, 6]),\n",
       "   (4, [5, 7, 2, 4]),\n",
       "   (4, [5, 9]),\n",
       "   (4, [7, 8, 5]),\n",
       "   (4, [7, 5, 6, 1]),\n",
       "   (4, [3, 4, 1]),\n",
       "   (4, [3, 4, 2]),\n",
       "   (4, [9, 3, 1]),\n",
       "   (4, [7, 4, 5, 3]),\n",
       "   (4, [6, 7, 5, 3]),\n",
       "   (4, [8, 6, 1, 3]),\n",
       "   (4, [3, 9, 4]),\n",
       "   (4, [6, 3, 5, 8]),\n",
       "   (4, [1, 9]),\n",
       "   (4, [3, 7, 5, 4]),\n",
       "   (4, [3, 1, 9, 7]),\n",
       "   (4, [9, 7, 4]),\n",
       "   (4, [6, 8, 3, 5]),\n",
       "   (4, [6, 8, 3, 1]),\n",
       "   (4, [6, 5, 2, 7]),\n",
       "   (4, [3, 7, 6]),\n",
       "   (4, [9, 1, 7]),\n",
       "   (4, [9, 1, 3]),\n",
       "   (4, [6, 4, 7]),\n",
       "   (4, [6, 2, 7]),\n",
       "   (4, [6, 2, 5]),\n",
       "   (4, [7, 5, 1, 9]),\n",
       "   (4, [9, 1, 7, 6]),\n",
       "   (4, [9, 3, 1, 5]),\n",
       "   (4, [4, 5, 6, 2]),\n",
       "   (4, [9, 2]),\n",
       "   (4, [3, 2, 7]),\n",
       "   (4, [1, 7, 9]),\n",
       "   (4, [7, 9, 5]),\n",
       "   (4, [7, 2, 8]),\n",
       "   (4, [3, 5, 7, 2]),\n",
       "   (4, [3, 5, 7, 4]),\n",
       "   (4, [3, 5, 7, 6]),\n",
       "   (3, [4, 2, 3, 5]),\n",
       "   (3, [3, 6, 2]),\n",
       "   (3, [3, 6, 4]),\n",
       "   (3, [2, 4, 3]),\n",
       "   (3, [2, 4, 6]),\n",
       "   (3, [7, 5, 8]),\n",
       "   (3, [7, 5, 9]),\n",
       "   (3, [2, 6, 4]),\n",
       "   (3, [9, 7, 6]),\n",
       "   (3, [8, 6, 9, 3, 1]),\n",
       "   (3, [3, 2, 9, 7]),\n",
       "   (3, [6, 5, 7, 4]),\n",
       "   (3, [3, 5, 2, 7]),\n",
       "   (3, [4, 2, 5, 8]),\n",
       "   (3, [3, 5, 6, 4]),\n",
       "   (3, [1, 3, 7]),\n",
       "   (3, [5, 3, 2]),\n",
       "   (3, [6, 8, 7, 5]),\n",
       "   (3, [8, 5]),\n",
       "   (3, [7, 4, 3, 5]),\n",
       "   (3, [3, 6, 8, 1, 4, 9, 7, 5, 2]),\n",
       "   (3, [6, 1, 9]),\n",
       "   (3, [2, 6, 5]),\n",
       "   (3, [7, 5, 8, 2]),\n",
       "   (3, [7, 6, 5, 9]),\n",
       "   (3, [2, 3, 9]),\n",
       "   (3, [3, 1, 4]),\n",
       "   (3, [6, 3, 9, 5]),\n",
       "   (3, [6, 5, 4, 8]),\n",
       "   (3, [5, 7, 1, 9]),\n",
       "   (3, [5, 7, 1, 6]),\n",
       "   (3, [6, 9, 3, 7]),\n",
       "   (3, [8, 3, 6, 9]),\n",
       "   (3, [6, 4, 2]),\n",
       "   (3, [7, 5, 3, 4]),\n",
       "   (3, [9, 1, 7, 3]),\n",
       "   (3, [1, 2, 3, 4, 5, 6, 7, 8, 9]),\n",
       "   (3, [3, 6, 1, 2, 4, 7, 5, 8, 9]),\n",
       "   (3, [7, 6, 1]),\n",
       "   (3, [8, 6, 5]),\n",
       "   (3, [1, 9, 3, 7]),\n",
       "   (3, [3, 9, 7, 5]),\n",
       "   (3, [3, 9, 7, 1]),\n",
       "   (3, [6, 9, 1, 5]),\n",
       "   (3, [1, 3, 6]),\n",
       "   (3, [5, 6, 3, 4]),\n",
       "   (3, [6, 7, 9, 5]),\n",
       "   (3, [4, 7, 3]),\n",
       "   (3, [5, 7, 4, 2]),\n",
       "   (3, [6, 5, 3, 7]),\n",
       "   (3, [7, 3, 4]),\n",
       "   (3, [3, 9, 5, 6]),\n",
       "   (3, [8, 6, 3, 9]),\n",
       "   (3, [9, 6, 1]),\n",
       "   (3, [6, 5, 9, 7]),\n",
       "   (3, [6, 7, 5, 4]),\n",
       "   (3, [6, 7, 2, 8]),\n",
       "   (3, [7, 9, 2]),\n",
       "   (3, [5, 3, 6, 4]),\n",
       "   (3, [4, 5, 7, 6]),\n",
       "   (3, [7, 3, 8]),\n",
       "   (3, [5, 7, 1]),\n",
       "   (3, [6, 3, 7, 5, 4]),\n",
       "   (3, [4, 7, 5, 3]),\n",
       "   (3, [6, 5, 9]),\n",
       "   (3, [3, 8, 1, 9]),\n",
       "   (3, [3, 9, 8]),\n",
       "   (3, [3, 9, 2]),\n",
       "   (3, [6, 3, 5, 4]),\n",
       "   (3, [8, 6, 9, 3]),\n",
       "   (3, [9, 5]),\n",
       "   (3, [7, 6, 5, 2]),\n",
       "   (3, [9, 7, 5]),\n",
       "   (3, [8, 9, 3, 1]),\n",
       "   (3, [3, 9, 2, 1]),\n",
       "   (3, [3, 5, 1]),\n",
       "   (3, [3, 5, 9]),\n",
       "   (3, [2, 9, 5, 7]),\n",
       "   (3, [9, 3, 2]),\n",
       "   (3, [6, 8, 5, 7]),\n",
       "   (3, [3, 6, 7, 9]),\n",
       "   (3, [5, 3, 7, 6]),\n",
       "   (3, [1, 6, 3]),\n",
       "   (3, [7, 2, 3]),\n",
       "   (3, [7, 2, 4]),\n",
       "   (3, [2, 7, 3]),\n",
       "   (3, [7, 3, 1, 5]),\n",
       "   (3, [7, 1, 3]),\n",
       "   (3, [9, 7, 5, 4]),\n",
       "   (3, [4, 5, 8]),\n",
       "   (3, [4, 6, 7, 5]),\n",
       "   (3, [6, 5, 4, 7]),\n",
       "   (3, [7, 8, 3]),\n",
       "   (3, [7, 8, 2]),\n",
       "   (3, [6, 5, 4, 3]),\n",
       "   (3, [3, 7, 6, 5]),\n",
       "   (3, [2, 9, 7]),\n",
       "   (3, [3, 7, 2, 8]),\n",
       "   (3, [3, 2, 4]),\n",
       "   (3, [4, 6, 5, 2]),\n",
       "   (3, [3, 2, 8]),\n",
       "   (3, [1, 7, 5]),\n",
       "   (3, [5, 6, 4]),\n",
       "   (3, [5, 6, 8]),\n",
       "   (2, [6, 7, 1, 5]),\n",
       "   (2, [6, 7, 3, 1, 2]),\n",
       "   (2, [4, 8, 6, 2]),\n",
       "   (2, [5, 3, 4, 6]),\n",
       "   (2, [8, 1, 6]),\n",
       "   (2, [8, 1, 3, 9]),\n",
       "   (2, [3, 1, 5, 4, 6]),\n",
       "   (2, [7, 8, 2, 5]),\n",
       "   (2, [1, 9, 7]),\n",
       "   (2, [1, 9, 3]),\n",
       "   (2, [6, 5, 1, 2]),\n",
       "   (2, [5, 7, 6, 9]),\n",
       "   (2, [3, 9, 8, 1, 6, 5, 7, 2, 4]),\n",
       "   (2, [2, 6, 7]),\n",
       "   (2, [5, 2, 3]),\n",
       "   (2, [6, 9, 3, 1]),\n",
       "   (2, [8, 6, 3, 1, 5]),\n",
       "   (2, [8, 1, 3, 6]),\n",
       "   (2, [5, 4, 6, 8]),\n",
       "   (2, [5, 4, 6, 7]),\n",
       "   (2, [5, 4, 6, 2]),\n",
       "   (2, [5, 6, 2, 8]),\n",
       "   (2, [4, 6, 8]),\n",
       "   (2, [6, 9, 8]),\n",
       "   (2, [4, 6, 2]),\n",
       "   (2, [6, 3, 9, 8]),\n",
       "   (2, [9, 6, 7, 5]),\n",
       "   (2, [5, 9, 7, 3]),\n",
       "   (2, [4, 5, 3, 2, 6, 7, 8, 9, 1]),\n",
       "   (2, [8, 6, 9]),\n",
       "   (2, [2, 4, 8, 7]),\n",
       "   (2, [8, 3, 6, 2]),\n",
       "   (2, [3, 6, 9, 7]),\n",
       "   (2, [3, 5, 4, 2]),\n",
       "   (2, [6, 9, 7, 5, 2, 3, 4, 8]),\n",
       "   (2, [3, 2, 5, 7]),\n",
       "   (2, [7, 2, 9, 5]),\n",
       "   (2, [7, 2, 9, 1]),\n",
       "   (2, [9, 3, 7, 5]),\n",
       "   (2, [4, 2, 8, 5]),\n",
       "   (2, [3, 9, 6, 1]),\n",
       "   (2, [8, 5, 3]),\n",
       "   (2, [6, 7, 5, 4, 2, 8, 3, 1, 9]),\n",
       "   (2, [3, 5, 2, 8]),\n",
       "   (2, [8, 3, 1, 6, 9]),\n",
       "   (2, [6, 7, 5, 1]),\n",
       "   (2, [3, 6, 1, 8]),\n",
       "   (2, [3, 6, 1, 5]),\n",
       "   (2, [6, 5, 1]),\n",
       "   (2, [7, 1, 5, 3, 9]),\n",
       "   (2, [3, 6, 5, 9]),\n",
       "   (2, [6, 3, 1, 7]),\n",
       "   (2, [3, 6, 5, 4]),\n",
       "   (2, [2, 3, 4, 6]),\n",
       "   (2, [6, 1, 5]),\n",
       "   (2, [6, 5, 2]),\n",
       "   (2, [3, 7, 5, 4, 2]),\n",
       "   (2, [2, 4, 5, 7, 6]),\n",
       "   (2, [5, 4, 7, 2]),\n",
       "   (2, [5, 4, 7, 3]),\n",
       "   (2, [5, 7, 3, 6]),\n",
       "   (2, [2, 8, 4, 5, 3, 7]),\n",
       "   (2, [6, 8, 9, 1]),\n",
       "   (2, [3, 8, 9, 6]),\n",
       "   (2, [6, 7, 3, 2]),\n",
       "   (2, [5, 4, 3, 6]),\n",
       "   (2, [8, 3, 1, 7]),\n",
       "   (2, [2, 3, 5]),\n",
       "   (2, [6, 9, 7, 5]),\n",
       "   (2, [6, 1, 3, 9]),\n",
       "   (2, [3, 1, 2]),\n",
       "   (2, [6, 1, 3, 4]),\n",
       "   (2, [3, 1, 8]),\n",
       "   (2, [4, 3, 5, 6]),\n",
       "   (2, [9, 7, 1, 3]),\n",
       "   (2, [6, 8, 3, 1, 9]),\n",
       "   (2, [6, 9, 1, 2, 5, 7]),\n",
       "   (2, [2, 6, 5, 7]),\n",
       "   (2, [9, 1, 6]),\n",
       "   (2, [6, 1, 7, 9]),\n",
       "   (2, [6, 4, 1]),\n",
       "   (2, [7, 2, 8, 9]),\n",
       "   (2, [8, 3, 6, 1, 2, 4, 5, 7, 9]),\n",
       "   (2, [3, 4, 7, 5]),\n",
       "   (2, [8, 3, 9, 1]),\n",
       "   (2, [8, 3, 9, 6]),\n",
       "   (2, [6, 9, 8, 3]),\n",
       "   (2, [7, 4, 3]),\n",
       "   (2, [7, 2, 4, 5]),\n",
       "   (2, [7, 9, 2, 1]),\n",
       "   (2, [1, 3, 5]),\n",
       "   (2, [7, 5, 9, 2]),\n",
       "   (2, [7, 5, 9, 1]),\n",
       "   (2, [5, 7, 3, 4]),\n",
       "   (2, [8, 7, 5, 3, 1, 9, 2]),\n",
       "   (2, [6, 7, 9, 3]),\n",
       "   (2, [4, 9, 5, 1, 2, 7, 8, 6, 3]),\n",
       "   (2, [5, 6, 3, 9]),\n",
       "   (2, [5, 6, 3, 7]),\n",
       "   (2, [7, 4, 2, 5]),\n",
       "   (2, [3, 7, 5, 2]),\n",
       "   (2, [3, 7, 6, 9]),\n",
       "   (2, [2, 4, 5, 3]),\n",
       "   (2, [2, 4, 5, 6, 7]),\n",
       "   (2, [4, 8]),\n",
       "   (2, [3, 9, 1, 5]),\n",
       "   (2, [8, 6, 1, 9, 5, 2, 7, 4, 3]),\n",
       "   (2, [6, 5, 1, 7, 9, 8, 2, 3, 4]),\n",
       "   (2, [5, 6, 4, 7, 2]),\n",
       "   (2, [3, 9, 1, 8, 2, 7, 5, 4, 6]),\n",
       "   (2, [3, 7, 8, 9]),\n",
       "   (2, [3, 8, 1, 9, 6]),\n",
       "   (2, [6, 9, 1, 3]),\n",
       "   (2, [5, 8, 2]),\n",
       "   (2, [6, 8, 2, 5]),\n",
       "   (2, [9, 3, 5, 2]),\n",
       "   (2, [3, 9, 5, 7]),\n",
       "   (2, [6, 9, 5, 7]),\n",
       "   (2, [1, 7, 2, 5]),\n",
       "   (2, [7, 1, 9, 6]),\n",
       "   (2, [3, 5, 9, 7]),\n",
       "   (2, [3, 9, 1, 2, 7, 8, 5, 4, 6]),\n",
       "   (2, [8, 9]),\n",
       "   (2, [6, 7, 5, 2]),\n",
       "   (2, [6, 7, 5, 8]),\n",
       "   (2, [6, 7, 5, 9]),\n",
       "   (2, [8, 1]),\n",
       "   (2, [7, 9, 6]),\n",
       "   (2, [7, 5, 2, 8]),\n",
       "   (2, [6, 1, 4, 2]),\n",
       "   (2, [9, 3, 6, 5]),\n",
       "   (2, [3, 4, 9]),\n",
       "   (2, [3, 4, 8]),\n",
       "   (2, [9, 2, 8]),\n",
       "   (2, [9, 2, 7]),\n",
       "   (2, [3, 2, 7, 5]),\n",
       "   (2, [6, 8, 1, 9, 7]),\n",
       "   (2, [4, 3, 2, 7]),\n",
       "   (2, [9, 6, 5]),\n",
       "   (2, [4, 5, 3, 2]),\n",
       "   (2, [4, 5, 3, 6]),\n",
       "   (2, [4, 5, 3, 7]),\n",
       "   (2, [6, 7, 4, 9]),\n",
       "   (2, [3, 8, 9, 6, 1, 2, 5, 4, 7]),\n",
       "   (2, [7, 2, 5, 6]),\n",
       "   (2, [3, 4, 7, 5, 2]),\n",
       "   (2, [7, 3, 6, 5]),\n",
       "   (2, [3, 4, 7, 5, 6, 2, 8, 1, 9]),\n",
       "   (2, [7, 9, 5, 1]),\n",
       "   (2, [2, 4, 6, 5]),\n",
       "   (2, [2, 7, 6, 5]),\n",
       "   (2, [3, 2, 6]),\n",
       "   (2, [9, 6, 7, 1]),\n",
       "   (2, [4, 8, 2, 6]),\n",
       "   (2, [3, 9, 1, 2]),\n",
       "   (2, [2, 5, 7, 9]),\n",
       "   (2, [2, 5, 7, 3]),\n",
       "   (2, [6, 3, 5, 8, 7, 9, 4, 1, 2]),\n",
       "   (2, [1, 4]),\n",
       "   (2, [3, 1, 5, 6, 7]),\n",
       "   (2, [3, 1, 9, 5]),\n",
       "   (2, [3, 6, 9, 5]),\n",
       "   (2, [8, 3, 5]),\n",
       "   (2, [8, 3, 7]),\n",
       "   (2, [8, 7, 3]),\n",
       "   (2, [8, 7, 5]),\n",
       "   (2, [3, 1, 5, 7]),\n",
       "   (2, [3, 1, 5, 6]),\n",
       "   (2, [6, 1, 5, 3]),\n",
       "   (2, [6, 3, 9, 7]),\n",
       "   (2, [6, 3, 1, 2, 4, 5, 7, 8, 9]),\n",
       "   (2, [3, 4, 2, 5, 7]),\n",
       "   (2, [8, 3, 1, 7, 9]),\n",
       "   (2, [1, 2]),\n",
       "   (2, [1, 3, 9, 2]),\n",
       "   (2, [3, 6, 1, 7, 5, 4, 2, 8, 9]),\n",
       "   (2, [3, 5, 7, 4, 6]),\n",
       "   (2, [6, 7, 8, 3]),\n",
       "   (2, [6, 7, 8, 5]),\n",
       "   (2, [1, 2, 3, 4]),\n",
       "   (2, [5, 8, 1, 3]),\n",
       "   (2, [5, 9, 6]),\n",
       "   (2, [1, 5, 7, 9]),\n",
       "   (2, [1, 7, 5, 3]),\n",
       "   (2, [8, 9, 7]),\n",
       "   (2, [1, 7, 5, 6]),\n",
       "   (2, [6, 5, 1, 9, 7]),\n",
       "   (2, [4, 7, 5, 9, 2]),\n",
       "   (2, [6, 4, 7, 5, 2]),\n",
       "   (2, [6, 7, 2, 5]),\n",
       "   (2, [5, 3, 7, 1]),\n",
       "   (2, [5, 3, 7, 2]),\n",
       "   (2, [5, 6, 9, 7]),\n",
       "   (2, [5, 6, 9, 4]),\n",
       "   (2, [9, 6, 3, 1, 2, 4, 5, 7, 8]),\n",
       "   (2, [7, 6, 4]),\n",
       "   (2, [1, 9, 7, 5]),\n",
       "   (2, [7, 8]),\n",
       "   (2, [9, 1, 7, 6, 5]),\n",
       "   (2, [3, 5, 7, 1, 8, 9, 2, 4, 6]),\n",
       "   (2, [6, 3, 5, 1]),\n",
       "   (2, [6, 2, 9]),\n",
       "   (2, [7, 6, 5, 4]),\n",
       "   (2, [6, 2, 4]),\n",
       "   (2, [6, 3, 5, 2]),\n",
       "   (2, [7, 5, 1, 2, 9, 6, 3]),\n",
       "   (2, [3, 1, 9, 8, 2, 5, 4, 7, 6]),\n",
       "   (2, [2, 7, 3, 4]),\n",
       "   (2, [4, 5, 8, 2]),\n",
       "   (2, [9, 3, 1, 5, 7]),\n",
       "   (2, [3, 1, 9, 5, 7]),\n",
       "   (2, [3, 9, 7, 2, 8]),\n",
       "   (2, [3, 6, 4, 5]),\n",
       "   (2, [4, 5, 2, 6]),\n",
       "   (2, [6, 2, 7, 3]),\n",
       "   (2, [4, 6, 5, 7]),\n",
       "   (2, [3, 2, 9]),\n",
       "   (2, [4, 9, 7]),\n",
       "   (2, [1, 7, 3]),\n",
       "   (2, [4, 7, 2, 5]),\n",
       "   (2, [3, 5, 1, 7]),\n",
       "   (2, [6, 4, 7, 2, 1, 9, 8, 5]),\n",
       "   (2, [3, 5, 7, 1]),\n",
       "   (2, [4, 6, 2, 8]),\n",
       "   (2, [3, 5, 1, 9]),\n",
       "   (2, [4, 5, 6, 3]),\n",
       "   (2, [3, 8, 6, 5]),\n",
       "   (2, [5, 7, 8, 2]),\n",
       "   (2, [5, 6, 2]),\n",
       "   (2, [4, 2, 6, 5]),\n",
       "   (1, [7, 2, 4, 6, 5]),\n",
       "   (1, [3, 4, 2, 7]),\n",
       "   (1, [3, 4, 2, 6]),\n",
       "   (1, [7, 2, 8, 6, 5, 1, 9, 4, 3]),\n",
       "   (1, [4, 2, 3, 8]),\n",
       "   (1, [3, 9, 1, 7, 6, 5]),\n",
       "   (1, [2, 5, 7, 6]),\n",
       "   (1, [3, 9, 5, 6, 7]),\n",
       "   (1, [5, 3, 4, 2]),\n",
       "   (1, [5, 7, 1, 2, 9]),\n",
       "   (1, [2, 6, 4, 5, 7, 8, 9, 1, 3]),\n",
       "   (1, [5, 7, 2, 6]),\n",
       "   (1, [9, 4, 5]),\n",
       "   (1, [6, 7, 2, 8, 3, 9, 5, 4, 1]),\n",
       "   (1, [7, 6, 5, 3, 2, 8, 4, 1, 9]),\n",
       "   (1, [8, 3, 5, 7, 2, 6, 4, 1, 9]),\n",
       "   (1, [4, 5, 7, 8, 2, 6, 1, 3, 9]),\n",
       "   (1, [7, 6, 3, 1, 5]),\n",
       "   (1, [5, 8]),\n",
       "   (1, [3, 5, 4, 7, 6]),\n",
       "   (1, [3, 5, 4, 7, 2]),\n",
       "   (1, [8, 1, 2]),\n",
       "   (1, [1, 9, 2, 8]),\n",
       "   (1, [2, 8, 5, 4]),\n",
       "   (1, [7, 3, 6, 5, 4, 8, 9, 1, 2]),\n",
       "   (1, [6, 1, 8, 5]),\n",
       "   (1, [6, 3, 1, 9, 8, 7, 5, 4, 2]),\n",
       "   (1, [6, 1, 8, 3]),\n",
       "   (1, [8, 3, 7, 4, 5, 6, 2, 1, 9]),\n",
       "   (1, [7, 6, 5, 4, 3, 8, 2, 1, 9]),\n",
       "   (1, [6, 1, 7, 5, 9, 8, 2, 4, 3]),\n",
       "   (1, [7, 6, 3, 5, 1]),\n",
       "   (1, [9, 7, 6, 1]),\n",
       "   (1, [6, 7, 4, 5, 2, 8, 1, 9, 3]),\n",
       "   (1, [5, 2, 7, 6, 3]),\n",
       "   (1, [6, 9, 3, 1, 5, 2, 8, 7, 4]),\n",
       "   (1, [6, 8, 3, 1, 7]),\n",
       "   (1, [6, 7, 5, 3, 8]),\n",
       "   (1, [2, 4, 8]),\n",
       "   (1, [2, 4, 9]),\n",
       "   (1, [6, 8, 1, 2, 3, 5, 9, 7, 4]),\n",
       "   (1, [7, 1, 4]),\n",
       "   (1, [7, 1, 6]),\n",
       "   (1, [7, 1, 8]),\n",
       "   (1, [9, 7, 3, 1]),\n",
       "   (1, [7, 5, 3, 2, 9, 8, 6, 4, 1]),\n",
       "   (1, [6, 1, 4, 5, 9, 8, 7, 3, 2]),\n",
       "   (1, [9, 2, 1, 8, 7, 6, 3, 5, 4]),\n",
       "   (1, [3, 8, 4, 2]),\n",
       "   (1, [7, 5, 3, 4, 6]),\n",
       "   (1, [4, 8, 5, 3]),\n",
       "   (1, [3, 8, 7, 2, 1, 5]),\n",
       "   (1, [5, 7, 6, 4]),\n",
       "   (1, [3, 6, 8, 9, 1, 5, 4, 2, 7]),\n",
       "   (1, [9, 4, 3, 7, 6, 1, 5, 2, 8]),\n",
       "   (1, [3, 2, 7, 8, 9, 1, 5]),\n",
       "   (1, [5, 7, 6, 2]),\n",
       "   (1, [3, 7, 5, 8]),\n",
       "   (1, [3, 7, 5, 1, 2, 4, 8, 9, 6]),\n",
       "   (1, [6, 8, 3, 2, 1, 5, 4, 7, 9]),\n",
       "   (1, [3, 7, 2, 5, 8, 1, 9, 4, 6]),\n",
       "   (1, [6, 5, 1, 2, 3, 4, 7, 8, 9]),\n",
       "   (1, [5, 3, 7, 9, 4, 6]),\n",
       "   (1, [3, 1, 7, 2, 4, 6, 5, 8]),\n",
       "   (1, [2, 6, 8]),\n",
       "   (1, [2, 6, 9]),\n",
       "   (1, [6, 5, 4, 8, 2, 7, 1, 9, 3]),\n",
       "   (1, [7, 2, 3, 6]),\n",
       "   (1, [6, 8, 7, 5, 2]),\n",
       "   (1, [3, 1, 9, 6, 7, 2, 5, 8, 4]),\n",
       "   (1, [4, 5, 6, 8, 2]),\n",
       "   (1, [6, 4, 1, 3, 5, 7, 8, 9, 2]),\n",
       "   (1, [3, 9, 8, 5, 4, 6]),\n",
       "   (1, [3, 6, 1, 2, 5, 8, 9, 4, 7]),\n",
       "   (1, [7, 5, 8, 6]),\n",
       "   (1, [9, 8, 3, 2, 5, 7, 1, 6, 4]),\n",
       "   (1, [6, 4, 5, 7, 3]),\n",
       "   (1, [6, 3, 8, 1, 9, 7, 2, 5, 4]),\n",
       "   (1, [4, 2, 6, 5, 7, 8, 3, 1, 9]),\n",
       "   (1, [9, 4, 7]),\n",
       "   (1, [6, 7, 9, 8]),\n",
       "   (1, [4, 7, 3, 8, 9, 2, 1, 5, 6]),\n",
       "   (1, [9, 8, 7]),\n",
       "   (1, [3, 7, 5, 1, 8, 6]),\n",
       "   (1, [6, 1, 3, 2, 4, 9, 5, 7, 8]),\n",
       "   (1, [9, 8, 3]),\n",
       "   (1, [6, 5, 3, 1, 2, 7, 8, 9]),\n",
       "   (1, [7, 5, 4, 8, 2, 6, 3, 9, 1]),\n",
       "   (1, [3, 5, 7, 2, 6, 9]),\n",
       "   (1, [3, 9, 7, 8, 1, 2]),\n",
       "   (1, [1, 3, 5, 7, 2, 8, 4, 6, 9]),\n",
       "   (1, [7, 3, 2, 9]),\n",
       "   (1, [5, 7, 9, 3, 6]),\n",
       "   (1, [4, 3, 8, 5, 2]),\n",
       "   (1, [8, 9, 6, 1, 3, 7, 4, 5, 2]),\n",
       "   (1, [5, 2, 1]),\n",
       "   (1, [6, 5, 4, 2, 7]),\n",
       "   (1, [3, 2, 4, 5, 7, 8, 1, 6, 9]),\n",
       "   (1, [5, 2, 6]),\n",
       "   (1, [5, 2, 8]),\n",
       "   (1, [6, 5, 4, 8, 2, 7]),\n",
       "   (1, [6, 9, 3, 5]),\n",
       "   (1, [6, 5, 7, 4, 2, 8, 1, 9, 3]),\n",
       "   (1, [3, 6, 5, 9, 4, 8, 1, 2, 7]),\n",
       "   (1, [3, 7, 1, 2, 9, 8, 4]),\n",
       "   (1, [6, 9, 3, 2]),\n",
       "   (1, [8, 5, 2, 7]),\n",
       "   (1, [4, 3, 7, 5, 6, 1, 2, 9, 8]),\n",
       "   (1, [6, 9, 3, 8]),\n",
       "   (1, [9, 3, 5, 6, 4, 2, 1, 7, 8]),\n",
       "   (1, [8, 9, 5, 7, 3, 2, 1, 4, 6]),\n",
       "   (1, [3, 5, 7, 6, 9, 1]),\n",
       "   (1, [7, 9, 1, 6, 8, 5, 3]),\n",
       "   (1, [5, 4, 7, 6, 3, 1, 9, 2]),\n",
       "   (1, [1, 8, 9, 2]),\n",
       "   (1, [5, 3, 7, 6, 9, 2, 8, 1, 4]),\n",
       "   (1, [9, 3, 6, 7, 5]),\n",
       "   (1, [4, 6, 5, 7, 3, 2, 1, 8, 9]),\n",
       "   (1, [9, 5, 3, 7, 6]),\n",
       "   (1, [3, 7, 4, 5, 2]),\n",
       "   (1, [3, 2, 9, 8, 4]),\n",
       "   (1, [2, 6, 8, 3]),\n",
       "   (1, [6, 4, 2, 7, 3, 8, 1, 9]),\n",
       "   (1, [6, 3, 1, 7, 9, 8, 2, 5, 4]),\n",
       "   (1, [6, 5, 2, 7, 8, 4, 3, 1, 9]),\n",
       "   (1, [6, 8, 3, 2, 1, 4, 5, 7, 9]),\n",
       "   (1, [3, 9, 8, 2, 5]),\n",
       "   (1, [5, 3, 1, 7]),\n",
       "   (1, [3, 8, 6, 1, 9, 7, 5, 2, 4]),\n",
       "   (1, [2, 4, 6, 5, 3]),\n",
       "   (1, [8, 3, 1, 9, 5, 4, 2, 7, 6]),\n",
       "   (1, [3, 6, 8, 1, 7, 2, 4, 5, 9]),\n",
       "   (1, [5, 4, 6, 3]),\n",
       "   (1, [7, 1, 8, 9]),\n",
       "   (1, [3, 2, 4, 7, 8]),\n",
       "   (1, [7, 5, 3, 6, 4, 2, 1, 8, 9]),\n",
       "   (1, [5, 3, 6, 1, 9, 7, 4, 2, 8]),\n",
       "   (1, [7, 6, 9, 5, 2]),\n",
       "   (1, [5, 6, 2, 1]),\n",
       "   (1, [8, 6, 3, 4, 5, 2, 1, 9, 7]),\n",
       "   (1, [5, 4, 2, 9]),\n",
       "   (1, [5, 4, 2, 8]),\n",
       "   (1, [7, 1, 9, 2, 3, 8, 5, 4, 6]),\n",
       "   (1, [6, 4, 2, 7]),\n",
       "   (1, [3, 5, 6, 7, 4, 9, 8, 1, 2]),\n",
       "   (1, [7, 6, 3, 5, 1, 2, 9, 8, 4]),\n",
       "   (1, [8, 6, 3, 7, 9, 1, 2, 4, 5]),\n",
       "   (1, [4, 5, 3, 6, 2, 8]),\n",
       "   (1, [3, 7, 4, 6, 5, 9, 1, 2, 8]),\n",
       "   (1, [4, 6, 7, 9, 8, 5, 2]),\n",
       "   (1, [6, 4, 2, 5, 7, 8]),\n",
       "   (1, [4, 5, 3, 6, 2, 7]),\n",
       "   (1, [6, 4, 5, 7, 1, 9, 8, 2, 3]),\n",
       "   (1, [6, 3, 4, 5, 1, 7, 8, 9, 2]),\n",
       "   (1, [9, 3, 5, 7, 6, 4, 8, 2, 1]),\n",
       "   (1, [1, 9, 8, 7, 6, 5, 3, 2, 4]),\n",
       "   (1, [3, 5, 6, 7, 2, 9, 4]),\n",
       "   (1, [3, 5, 7, 4, 6, 9, 1, 2, 8]),\n",
       "   (1, [3, 2, 9, 4]),\n",
       "   (1, [3, 2, 9, 6]),\n",
       "   (1, [8, 6, 3, 7]),\n",
       "   (1, [4, 6, 5, 3, 2]),\n",
       "   (1, [4, 3, 2, 9, 8]),\n",
       "   (1, [4, 2, 8, 6, 5]),\n",
       "   (1, [6, 8, 1, 3, 9, 7, 4, 2, 5]),\n",
       "   (1, [7, 4, 3, 5, 6]),\n",
       "   (1, [6, 5, 7, 2, 8, 1, 3, 4, 9]),\n",
       "   (1, [3, 5, 6, 2, 7, 8, 9, 1, 4]),\n",
       "   (1, [7, 6, 5, 4, 3, 1, 9, 8, 2]),\n",
       "   (1, [5, 7, 4, 2, 1]),\n",
       "   (1, [7, 3, 2, 4]),\n",
       "   (1, [3, 4, 2, 7, 9]),\n",
       "   (1, [1, 7, 9, 8, 2, 3, 6, 5, 4]),\n",
       "   (1, [3, 9, 1, 2, 8, 7, 6, 5, 4]),\n",
       "   (1, [6, 5, 7, 9, 1, 4, 8, 2, 3]),\n",
       "   (1, [5, 4, 2, 6, 8, 7]),\n",
       "   (1, [9, 3, 1, 7, 5, 2, 8, 4, 6]),\n",
       "   (1, [9, 7, 1, 5, 8, 2, 3, 6, 4]),\n",
       "   (1, [6, 5, 7, 8]),\n",
       "   (1, [9, 3, 2, 8, 1, 7, 4, 5, 6]),\n",
       "   (1, [6, 5, 3, 7, 8, 2, 1, 4, 9]),\n",
       "   (1, [2, 7, 9, 1, 8, 5, 3, 4, 6]),\n",
       "   (1, [3, 7, 5, 6, 1, 4, 2, 9, 8]),\n",
       "   (1, [6, 5, 7, 2]),\n",
       "   (1, [6, 5, 7, 3]),\n",
       "   (1, [4, 7, 3, 9, 5, 2, 6, 8, 1]),\n",
       "   (1, [8, 6, 3, 1, 2, 4, 5, 9, 7]),\n",
       "   (1, [6, 5, 7, 4, 2]),\n",
       "   (1, [6, 3, 9, 7, 8]),\n",
       "   (1, [4, 5, 7, 8, 6]),\n",
       "   (1, [5, 2, 4, 7, 1, 6, 3]),\n",
       "   (1, [6, 5, 3, 9]),\n",
       "   (1, [8, 3, 9, 7, 5, 6, 4, 2, 1]),\n",
       "   (1, [7, 3, 5, 9, 8, 2]),\n",
       "   (1, [7, 5, 2, 3, 6]),\n",
       "   (1, [6, 5, 3, 1]),\n",
       "   (1, [4, 6, 7, 2, 8, 5, 3, 1, 9]),\n",
       "   (1, [8, 6, 2]),\n",
       "   (1, [6, 7, 9, 1, 5]),\n",
       "   (1, [4, 3, 5, 7, 2]),\n",
       "   (1, [5, 9, 7, 2]),\n",
       "   (1, [5, 9, 7, 1]),\n",
       "   (1, [4, 2, 7, 6, 5, 8, 3, 9, 1]),\n",
       "   (1, [3, 8, 1, 9, 2, 6, 7]),\n",
       "   (1, [6, 4, 7, 5, 2, 3, 1, 8, 9]),\n",
       "   (1, [4, 6, 8, 3]),\n",
       "   (1, [1, 3, 6, 7, 9, 8, 5, 2, 4]),\n",
       "   (1, [9, 7, 8, 1, 2, 3, 5, 6, 4]),\n",
       "   (1, [6, 7, 5, 2, 3, 8, 4]),\n",
       "   (1, [6, 7, 9, 1, 3, 5, 8, 4]),\n",
       "   (1, [6, 3, 7, 4]),\n",
       "   (1, [6, 3, 7, 8]),\n",
       "   (1, [8, 3, 1, 4, 9, 6, 7, 5, 2]),\n",
       "   (1, [4, 2, 5, 9]),\n",
       "   (1, [4, 8, 2, 5]),\n",
       "   (1, [3, 2, 5, 8, 9, 7]),\n",
       "   (1, [4, 2, 5, 1]),\n",
       "   (1, [9, 6, 3, 5, 7, 1]),\n",
       "   (1, [4, 2, 5, 6]),\n",
       "   (1, [6, 5, 7, 3, 1, 9, 8, 2, 4]),\n",
       "   (1, [8, 6, 5, 3]),\n",
       "   (1, [7, 1, 3, 9, 6, 8, 2, 5, 4]),\n",
       "   (1, [9, 3, 5, 7, 6, 4, 2, 8, 1]),\n",
       "   (1, [6, 3, 4, 5, 2, 1, 7, 8, 9]),\n",
       "   (1, [8, 6, 3, 9, 1, 2, 4, 5, 7]),\n",
       "   (1, [3, 9, 8, 6, 1, 7]),\n",
       "   (1, [3, 6, 7, 2]),\n",
       "   (1, [6, 5, 8, 7]),\n",
       "   (1, [3, 6, 7, 1]),\n",
       "   (1, [8, 2, 7]),\n",
       "   (1, [3, 6, 7, 8, 2, 1]),\n",
       "   (1, [2, 5, 8, 4, 6, 3, 1, 7, 9]),\n",
       "   (1, [3, 5, 6, 8]),\n",
       "   (1, [3, 5, 6, 9]),\n",
       "   (1, [7, 2, 9, 6]),\n",
       "   (1, [4, 2, 8, 5, 7, 3, 6, 1, 9]),\n",
       "   (1, [6, 7, 5, 4, 3]),\n",
       "   (1, [5, 7, 8, 2, 3, 4, 6]),\n",
       "   (1, [7, 5, 2, 9, 8, 6, 1]),\n",
       "   (1, [5, 4, 7, 8, 9, 6, 3, 2, 1]),\n",
       "   (1, [3, 2, 5, 9]),\n",
       "   (1, [5, 6, 7, 3, 2, 8, 1]),\n",
       "   (1, [6, 9, 8, 7, 2]),\n",
       "   (1, [8, 5, 2]),\n",
       "   (1, [6, 4, 3, 8, 5, 2]),\n",
       "   (1, [3, 4, 9, 8, 7, 1, 2, 5, 6]),\n",
       "   (1, [8, 5, 6]),\n",
       "   (1, [8, 3, 4]),\n",
       "   (1, [6, 7, 8, 2, 5, 4, 1, 9, 3]),\n",
       "   (1, [3, 4, 5, 7, 2, 6, 8, 9]),\n",
       "   (1, [3, 2, 1, 8]),\n",
       "   (1, [7, 5, 8, 2, 3, 4, 1, 6, 9]),\n",
       "   (1, [3, 6, 8, 9, 1, 2, 5, 4, 7]),\n",
       "   (1, [4, 5, 1, 6]),\n",
       "   (1, [7, 1, 3, 5, 9, 2, 8, 4]),\n",
       "   (1, [6, 5, 4, 7, 2, 8, 9, 1, 3]),\n",
       "   (1, [6, 5, 3, 1, 8, 2, 7, 9, 4]),\n",
       "   (1, [6, 7, 4, 9, 8, 1, 2, 5, 3]),\n",
       "   (1, [6, 2, 4, 5]),\n",
       "   (1, [7, 4, 9, 2]),\n",
       "   (1, [6, 2, 4, 9]),\n",
       "   (1, [8, 3, 2]),\n",
       "   (1, [5, 6, 8, 7]),\n",
       "   (1, [3, 4, 5, 2, 9, 7, 1, 8, 6]),\n",
       "   (1, [6, 4, 5, 7, 8, 2, 1, 3, 9]),\n",
       "   (1, [1, 7, 3, 9, 8]),\n",
       "   (1, [7, 9, 6, 5, 4, 8, 1]),\n",
       "   (1, [4, 2, 8, 6]),\n",
       "   (1, [2, 6, 8, 7, 9, 5, 1, 3, 4]),\n",
       "   (1, [2, 7, 1, 9]),\n",
       "   (1, [3, 4, 6, 7, 5, 8]),\n",
       "   (1, [8, 3, 9, 5, 6, 7, 4, 1, 2]),\n",
       "   (1, [8, 3, 1, 5, 4, 7]),\n",
       "   (1, [6, 5, 2, 8, 9]),\n",
       "   (1, [7, 4, 5, 2, 3, 6]),\n",
       "   (1, [9, 6, 3, 1, 8]),\n",
       "   (1, [9, 1, 3, 5, 7]),\n",
       "   (1, [3, 9, 6, 8, 7, 5, 2, 4, 1]),\n",
       "   (1, [8, 2, 3]),\n",
       "   (1, [8, 9, 5, 3, 2, 1, 7, 6, 4]),\n",
       "   (1, [8, 3, 1, 9, 2, 4, 5, 7, 6]),\n",
       "   (1, [8, 6, 1, 3, 2, 4, 5, 7, 9]),\n",
       "   (1, [6, 9, 8, 5]),\n",
       "   (1, [4, 2, 5, 7, 3]),\n",
       "   (1, [5, 6, 3, 1, 9]),\n",
       "   (1, [2, 6, 3, 7, 4, 5, 8, 9, 1]),\n",
       "   (1, [7, 6, 8, 5]),\n",
       "   (1, [4, 6, 2, 5]),\n",
       "   (1, [3, 4, 9, 8]),\n",
       "   (1, [5, 1, 2]),\n",
       "   (1, [7, 3, 5, 1, 2, 4, 6, 8, 9]),\n",
       "   (1, [8, 7, 2, 5]),\n",
       "   (1, [6, 5, 9, 1, 7, 4, 3, 2, 8]),\n",
       "   (1, [2, 1, 3, 4, 6, 5, 7, 9, 8]),\n",
       "   (1, [7, 6, 5, 1, 9]),\n",
       "   (1, [9, 6, 7, 2, 1, 5]),\n",
       "   (1, [3, 8, 2, 5, 7, 4, 6, 9, 1]),\n",
       "   (1, [8, 2, 3, 9]),\n",
       "   (1, [6, 9, 1, 3, 7]),\n",
       "   (1, [3, 9, 6, 5]),\n",
       "   (1, [3, 9, 6, 7]),\n",
       "   (1, [9, 8, 3, 1, 4, 2]),\n",
       "   (1, [7, 4, 5, 6, 8, 9, 2, 1]),\n",
       "   (1, [3, 5, 8, 1, 7, 9, 6, 4, 2]),\n",
       "   (1, [7, 1, 9, 8, 2]),\n",
       "   (1, [3, 7, 9, 1, 2, 8, 5, 6, 4]),\n",
       "   (1, [5, 6, 4, 3]),\n",
       "   (1, [7, 5, 3, 2]),\n",
       "   (1, [1, 8, 9, 7, 2, 4, 3, 5, 6]),\n",
       "   (1, [6, 1, 9, 8]),\n",
       "   (1, [6, 1, 5, 8, 3]),\n",
       "   (1, [7, 9, 2, 1, 3]),\n",
       "   (1, [6, 4, 7, 2]),\n",
       "   (1, [8, 5, 1, 3]),\n",
       "   (1, [8, 9, 2]),\n",
       "   (1, [3, 6, 1, 8, 2, 4, 5, 7, 9]),\n",
       "   (1, [6, 7, 5, 8, 9, 2, 1, 4, 3]),\n",
       "   (1, [3, 8, 5, 7]),\n",
       "   (1, [7, 6, 2, 8, 9, 4, 5, 3, 1]),\n",
       "   (1, [3, 5, 6, 4, 7, 9, 1, 2, 8]),\n",
       "   (1, [3, 1, 4, 5, 7, 8, 9, 2]),\n",
       "   (1, [1, 3, 7, 5]),\n",
       "   (1, [3, 6, 7, 5, 4]),\n",
       "   (1, [7, 9, 8, 3, 1]),\n",
       "   (1, [6, 8, 3, 1, 4, 7, 9, 5, 2]),\n",
       "   (1, [3, 5, 2, 4]),\n",
       "   (1, [3, 2, 4, 9]),\n",
       "   (1, [3, 5, 2, 6]),\n",
       "   (1, [2, 8, 4, 5]),\n",
       "   (1, [1, 3, 7, 9]),\n",
       "   (1, [3, 9, 1, 2, 4, 7, 5, 8, 6]),\n",
       "   (1, [6, 8, 7, 2, 9, 5, 4, 3, 1]),\n",
       "   (1, [7, 9, 8, 2, 1, 4]),\n",
       "   (1, [4, 7, 3, 2]),\n",
       "   (1, [5, 3, 4, 9, 2, 1, 7, 6, 8]),\n",
       "   (1, [7, 4, 3, 2]),\n",
       "   (1, [3, 1, 6, 7, 9, 4, 5, 2, 8]),\n",
       "   (1, [7, 9, 5, 4, 3, 2, 8, 1, 6]),\n",
       "   (1, [6, 3, 9, 8, 1, 2, 4, 5, 7]),\n",
       "   (1, [9, 8, 6, 1]),\n",
       "   (1, [3, 5, 2, 8, 1, 9]),\n",
       "   (1, [6, 5, 7, 8, 3, 1]),\n",
       "   (1, [3, 4, 7, 5, 6, 8, 9, 2, 1]),\n",
       "   (1, [5, 4, 2, 9, 7]),\n",
       "   (1, [6, 4, 5, 2]),\n",
       "   (1, [3, 6, 1, 9, 7, 8, 5, 4, 2]),\n",
       "   (1, [7, 3, 1, 2, 5, 9, 8, 4, 6]),\n",
       "   (1, [5, 2, 8, 7]),\n",
       "   (1, [4, 3, 2, 5, 8, 7, 1, 9, 6]),\n",
       "   (1, [3, 9, 5, 6, 2, 1, 7, 8, 4]),\n",
       "   (1, [3, 1, 2, 5, 6, 7, 8, 9]),\n",
       "   (1, [6, 3, 1, 4]),\n",
       "   (1, [3, 1, 5, 4, 8, 2, 9]),\n",
       "   (1, [1, 3, 5, 9]),\n",
       "   ...]),\n",
       " (8738,\n",
       "  defaultdict(int,\n",
       "              {1: 0.25028610666056306,\n",
       "               3: 0.28633554589150834,\n",
       "               2: 0.17360952162966353,\n",
       "               4: 0.14602883955138476,\n",
       "               9: 0.07862211032272831,\n",
       "               5: 0.04005493247882811,\n",
       "               6: 0.015106431677729458,\n",
       "               8: 0.004463263904783703,\n",
       "               7: 0.005493247882810712,\n",
       "               0: 0}),\n",
       "  [(741, [2]),\n",
       "   (393, [7]),\n",
       "   (392, [9]),\n",
       "   (346, [7, 8, 9]),\n",
       "   (246, [2, 3]),\n",
       "   (234, [7, 9, 8]),\n",
       "   (204, [9, 7, 8]),\n",
       "   (171, [4]),\n",
       "   (168, [9, 8, 7]),\n",
       "   (128, [1]),\n",
       "   (127, [8, 7, 9]),\n",
       "   (115, [5]),\n",
       "   (109, [9, 7]),\n",
       "   (99, [8]),\n",
       "   (87, [7, 9]),\n",
       "   (79, [2, 1]),\n",
       "   (79, [8, 2]),\n",
       "   (77, [6]),\n",
       "   (71, [3]),\n",
       "   (70, [1, 5]),\n",
       "   (68, [2, 6]),\n",
       "   (66, [2, 8]),\n",
       "   (49, [2, 5]),\n",
       "   (46, [5, 1]),\n",
       "   (44, [1, 2]),\n",
       "   (41, [2, 7]),\n",
       "   (40, [2, 5, 1]),\n",
       "   (34, [2, 3, 1]),\n",
       "   (32, [2, 6, 1]),\n",
       "   (32, [2, 3, 6]),\n",
       "   (32, [2, 1, 5]),\n",
       "   (31, [4, 5]),\n",
       "   (30, [1, 3]),\n",
       "   (28, [7, 8, 9, 5]),\n",
       "   (25, [7, 2]),\n",
       "   (23, [2, 6, 3]),\n",
       "   (23, [4, 1]),\n",
       "   (23, [1, 2, 5]),\n",
       "   (23, [1, 5, 2]),\n",
       "   (22, [8, 9, 7]),\n",
       "   (22, [7, 8, 9, 1]),\n",
       "   (22, [2, 1, 3]),\n",
       "   (21, [9, 8]),\n",
       "   (20, [5, 1, 2]),\n",
       "   (19, [2, 9]),\n",
       "   (19, [7, 8, 9, 2]),\n",
       "   (19, [7, 9, 5]),\n",
       "   (19, [7, 8]),\n",
       "   (19, [2, 1, 6]),\n",
       "   (18, [5, 2]),\n",
       "   (18, [7, 6]),\n",
       "   (17, [6, 2]),\n",
       "   (16, [2, 1, 3, 6]),\n",
       "   (16, [6, 3]),\n",
       "   (15, [7, 9, 8, 2]),\n",
       "   (15, [7, 8, 9, 3]),\n",
       "   (15, [3, 1]),\n",
       "   (15, [7, 5]),\n",
       "   (15, [2, 3, 6, 1]),\n",
       "   (15, [4, 5, 1]),\n",
       "   (15, [5, 2, 1]),\n",
       "   (14, [7, 9, 8, 1]),\n",
       "   (14, [6, 7]),\n",
       "   (14, [1, 6]),\n",
       "   (14, [2, 7, 8, 9]),\n",
       "   (14, [6, 2, 3]),\n",
       "   (13, [8, 2, 7]),\n",
       "   (13, [4, 2]),\n",
       "   (13, [4, 7]),\n",
       "   (13, [7, 9, 8, 5]),\n",
       "   (12, [2, 4]),\n",
       "   (12, [7, 8, 9, 6]),\n",
       "   (12, [5, 4, 1]),\n",
       "   (12, [5, 4]),\n",
       "   (12, [2, 8, 5]),\n",
       "   (11, [8, 2, 7, 9]),\n",
       "   (11, [6, 3, 1, 2]),\n",
       "   (11, [1, 2, 6]),\n",
       "   (11, [2, 5, 3]),\n",
       "   (11, [8, 9]),\n",
       "   (11, [6, 3, 2]),\n",
       "   (11, [7, 1]),\n",
       "   (11, [2, 7, 9]),\n",
       "   (11, [2, 6, 3, 1]),\n",
       "   (11, [6, 2, 1]),\n",
       "   (11, [9, 6]),\n",
       "   (10, [2, 4, 5]),\n",
       "   (10, [1, 4]),\n",
       "   (10, [1, 7]),\n",
       "   (10, [9, 7, 3]),\n",
       "   (10, [4, 1, 5]),\n",
       "   (10, [9, 7, 8, 2]),\n",
       "   (10, [9, 7, 8, 1]),\n",
       "   (10, [7, 9, 1]),\n",
       "   (10, [2, 1, 5, 3]),\n",
       "   (10, [9, 7, 5]),\n",
       "   (10, [1, 6, 3]),\n",
       "   (10, [7, 9, 3]),\n",
       "   (10, [7, 3]),\n",
       "   (9, [7, 9, 8, 6]),\n",
       "   (9, [3, 1, 2]),\n",
       "   (9, [2, 1, 6, 3]),\n",
       "   (9, [2, 5, 4]),\n",
       "   (9, [8, 7, 2]),\n",
       "   (9, [5, 7]),\n",
       "   (9, [8, 7, 9, 2]),\n",
       "   (9, [7, 9, 2]),\n",
       "   (9, [2, 8, 1]),\n",
       "   (9, [6, 3, 1]),\n",
       "   (9, [6, 1]),\n",
       "   (9, [1, 5, 7]),\n",
       "   (9, [9, 7, 2]),\n",
       "   (9, [9, 2]),\n",
       "   (9, [5, 1, 4]),\n",
       "   (9, [2, 5, 4, 1]),\n",
       "   (8, [7, 5, 1]),\n",
       "   (8, [2, 6, 7]),\n",
       "   (8, [2, 6, 5]),\n",
       "   (8, [6, 7, 9]),\n",
       "   (8, [9, 8, 7, 5]),\n",
       "   (8, [9, 8, 7, 1]),\n",
       "   (8, [2, 3, 5]),\n",
       "   (8, [7, 9, 8, 3]),\n",
       "   (8, [2, 7, 8]),\n",
       "   (8, [1, 3, 5]),\n",
       "   (8, [4, 2, 1]),\n",
       "   (8, [9, 5]),\n",
       "   (8, [5, 7, 1]),\n",
       "   (8, [1, 4, 5]),\n",
       "   (8, [9, 7, 6]),\n",
       "   (8, [9, 8, 7, 2]),\n",
       "   (7, [2, 3, 1, 6]),\n",
       "   (7, [5, 1, 7]),\n",
       "   (7, [8, 7]),\n",
       "   (7, [6, 1, 2]),\n",
       "   (7, [7, 6, 9]),\n",
       "   (7, [1, 3, 6]),\n",
       "   (7, [1, 3, 2]),\n",
       "   (7, [5, 4, 2]),\n",
       "   (7, [3, 6]),\n",
       "   (7, [1, 2, 6, 5]),\n",
       "   (7, [2, 8, 7]),\n",
       "   (7, [7, 9, 6]),\n",
       "   (7, [9, 1]),\n",
       "   (7, [2, 6, 1, 3]),\n",
       "   (7, [9, 8, 7, 6]),\n",
       "   (7, [1, 5, 3]),\n",
       "   (7, [2, 1, 5, 6]),\n",
       "   (6, [2, 3, 1, 5]),\n",
       "   (6, [7, 1, 8, 9]),\n",
       "   (6, [3, 2, 1, 6]),\n",
       "   (6, [2, 3, 8]),\n",
       "   (6, [3, 1, 5]),\n",
       "   (6, [9, 1, 7]),\n",
       "   (6, [2, 9, 7, 8]),\n",
       "   (6, [2, 5, 7]),\n",
       "   (6, [2, 5, 8]),\n",
       "   (6, [3, 7]),\n",
       "   (6, [3, 5]),\n",
       "   (6, [3, 2]),\n",
       "   (6, [5, 9]),\n",
       "   (6, [1, 5, 2, 3]),\n",
       "   (6, [9, 3]),\n",
       "   (6, [2, 8, 6]),\n",
       "   (6, [9, 7, 8, 5]),\n",
       "   (6, [2, 9, 7]),\n",
       "   (6, [8, 2, 9]),\n",
       "   (6, [8, 2, 9, 7]),\n",
       "   (6, [7, 9, 5, 1]),\n",
       "   (6, [1, 2, 7]),\n",
       "   (6, [1, 7, 9]),\n",
       "   (6, [6, 7, 8, 9]),\n",
       "   (6, [7, 2, 3]),\n",
       "   (6, [2, 7, 1]),\n",
       "   (6, [2, 1, 7]),\n",
       "   (6, [4, 5, 7]),\n",
       "   (6, [6, 2, 1, 3]),\n",
       "   (6, [1, 7, 8, 9]),\n",
       "   (6, [3, 2, 1]),\n",
       "   (6, [2, 8, 7, 9]),\n",
       "   (6, [1, 5, 6]),\n",
       "   (6, [8, 7, 9, 5]),\n",
       "   (5, [2, 6, 8]),\n",
       "   (5, [5, 2, 4]),\n",
       "   (5, [5, 1, 3]),\n",
       "   (5, [2, 1, 3, 4, 5, 6, 7, 8, 9]),\n",
       "   (5, [3, 6, 1, 5]),\n",
       "   (5, [6, 1, 3]),\n",
       "   (5, [9, 8, 7, 3]),\n",
       "   (5, [2, 1, 6, 5]),\n",
       "   (5, [2, 6, 5, 1]),\n",
       "   (5, [2, 6, 5, 3]),\n",
       "   (5, [8, 2, 5]),\n",
       "   (5, [2, 5, 9]),\n",
       "   (5, [2, 4, 5, 1]),\n",
       "   (5, [9, 7, 8, 3]),\n",
       "   (5, [7, 9, 8, 5, 1]),\n",
       "   (5, [4, 2, 5]),\n",
       "   (5, [9, 2, 5]),\n",
       "   (5, [9, 2, 1]),\n",
       "   (5, [9, 6, 3]),\n",
       "   (5, [2, 8, 9]),\n",
       "   (5, [6, 4]),\n",
       "   (5, [6, 3, 7]),\n",
       "   (5, [6, 3, 5]),\n",
       "   (5, [1, 2, 3]),\n",
       "   (5, [7, 4]),\n",
       "   (5, [7, 2, 5]),\n",
       "   (5, [2, 7, 5]),\n",
       "   (5, [1, 6, 3, 5]),\n",
       "   (5, [7, 8, 3]),\n",
       "   (5, [7, 2, 9]),\n",
       "   (5, [1, 3, 6, 5]),\n",
       "   (5, [5, 7, 8, 9]),\n",
       "   (5, [5, 6, 1]),\n",
       "   (4, [3, 6, 2]),\n",
       "   (4, [3, 6, 1]),\n",
       "   (4, [7, 1, 5]),\n",
       "   (4, [2, 4, 1]),\n",
       "   (4, [2, 6, 9]),\n",
       "   (4, [4, 1, 2, 5]),\n",
       "   (4, [7, 5, 8, 9]),\n",
       "   (4, [2, 1, 3, 5]),\n",
       "   (4, [1, 6, 2]),\n",
       "   (4, [8, 2, 6]),\n",
       "   (4, [2, 4, 6]),\n",
       "   (4, [5, 1, 6]),\n",
       "   (4, [7, 3, 9]),\n",
       "   (4, [1, 3, 5, 6]),\n",
       "   (4, [6, 1, 5]),\n",
       "   (4, [6, 1, 7]),\n",
       "   (4, [2, 3, 9]),\n",
       "   (4, [3, 1, 6]),\n",
       "   (4, [5, 7, 8, 9, 1]),\n",
       "   (4, [7, 6, 8, 9]),\n",
       "   (4, [1, 5, 4]),\n",
       "   (4, [4, 1, 2]),\n",
       "   (4, [8, 6, 9]),\n",
       "   (4, [7, 1, 9, 8]),\n",
       "   (4, [2, 8, 3, 4]),\n",
       "   (4, [4, 7, 9]),\n",
       "   (4, [5, 3]),\n",
       "   (4, [7, 6, 9, 3]),\n",
       "   (4, [7, 9, 1, 5]),\n",
       "   (4, [7, 9, 1, 3]),\n",
       "   (4, [2, 5, 7, 8]),\n",
       "   (4, [2, 5, 7, 9]),\n",
       "   (4, [1, 2, 5, 7]),\n",
       "   (4, [1, 2, 5, 3]),\n",
       "   (4, [1, 7, 9, 8]),\n",
       "   (4, [2, 6, 1, 5]),\n",
       "   (4, [5, 9, 7]),\n",
       "   (4, [9, 5, 2]),\n",
       "   (4, [2, 7, 3]),\n",
       "   (4, [7, 5, 4]),\n",
       "   (4, [1, 6, 3, 2]),\n",
       "   (4, [2, 3, 6, 7]),\n",
       "   (4, [2, 1, 4]),\n",
       "   (4, [4, 7, 8, 9, 5]),\n",
       "   (4, [9, 5, 1]),\n",
       "   (4, [2, 6, 3, 5]),\n",
       "   (4, [6, 2, 8]),\n",
       "   (4, [5, 2, 1, 3]),\n",
       "   (4, [6, 2, 3, 1]),\n",
       "   (4, [2, 9, 1]),\n",
       "   (4, [2, 9, 3]),\n",
       "   (4, [3, 2, 6]),\n",
       "   (4, [1, 7, 5]),\n",
       "   (4, [2, 3, 5, 6]),\n",
       "   (4, [7, 5, 9, 8]),\n",
       "   (4, [2, 8, 1, 3]),\n",
       "   (4, [2, 8, 1, 5]),\n",
       "   (4, [3, 6, 2, 1]),\n",
       "   (4, [2, 1, 5, 4]),\n",
       "   (3, [5, 7, 2, 1]),\n",
       "   (3, [1, 2, 6, 3, 5]),\n",
       "   (3, [9, 7, 5, 8]),\n",
       "   (3, [5, 7, 9, 8]),\n",
       "   (3, [1, 9, 8]),\n",
       "   (3, [7, 1, 2]),\n",
       "   (3, [1, 6, 2, 5]),\n",
       "   (3, [8, 2, 3]),\n",
       "   (3, [1, 2, 6, 5, 3]),\n",
       "   (3, [2, 9, 8, 7]),\n",
       "   (3, [5, 4, 2, 1]),\n",
       "   (3, [2, 5, 1, 4]),\n",
       "   (3, [7, 5, 9]),\n",
       "   (3, [2, 3, 1, 6, 7, 8, 9, 5, 4]),\n",
       "   (3, [7, 8, 9, 6, 2]),\n",
       "   (3, [7, 8, 9, 2, 1]),\n",
       "   (3, [7, 1, 9]),\n",
       "   (3, [7, 8, 9, 4]),\n",
       "   (3, [7, 3, 5]),\n",
       "   (3, [7, 3, 8, 9]),\n",
       "   (3, [3, 6, 1, 2]),\n",
       "   (3, [2, 3, 8, 5]),\n",
       "   (3, [1, 5, 7, 8, 9]),\n",
       "   (3, [6, 3, 1, 5]),\n",
       "   (3, [1, 3, 5, 2]),\n",
       "   (3, [1, 2, 5, 3, 6]),\n",
       "   (3, [3, 6, 7, 8, 9]),\n",
       "   (3, [1, 6, 5, 3]),\n",
       "   (3, [3, 9, 7]),\n",
       "   (3, [2, 3, 7]),\n",
       "   (3, [7, 9, 8, 2, 5]),\n",
       "   (3, [5, 1, 7, 9, 8]),\n",
       "   (3, [5, 2, 7]),\n",
       "   (3, [7, 6, 3]),\n",
       "   (3, [7, 6, 2]),\n",
       "   (3, [7, 8, 1]),\n",
       "   (3, [2, 3, 4]),\n",
       "   (3, [9, 3, 5]),\n",
       "   (3, [9, 3, 6]),\n",
       "   (3, [6, 7, 9, 3]),\n",
       "   (3, [2, 4, 1, 5]),\n",
       "   (3, [1, 3, 2, 6]),\n",
       "   (3, [2, 8, 3, 1]),\n",
       "   (3, [2, 8, 3, 6]),\n",
       "   (3, [6, 3, 1, 2, 5]),\n",
       "   (3, [3, 2, 9]),\n",
       "   (3, [7, 3, 6]),\n",
       "   (3, [5, 3, 6]),\n",
       "   (3, [8, 7, 9, 3]),\n",
       "   (3, [9, 2, 7]),\n",
       "   (3, [4, 5, 7, 2]),\n",
       "   (3, [9, 3, 7]),\n",
       "   (3, [9, 6, 7]),\n",
       "   (3, [2, 8, 3]),\n",
       "   (3, [9, 4]),\n",
       "   (3, [5, 7, 3]),\n",
       "   (3, [7, 9, 3, 5]),\n",
       "   (3, [6, 9]),\n",
       "   (3, [5, 1, 7, 8, 9]),\n",
       "   (3, [6, 5, 3]),\n",
       "   (3, [1, 9]),\n",
       "   (3, [9, 7, 4, 1]),\n",
       "   (3, [9, 7, 4]),\n",
       "   (3, [3, 5, 1]),\n",
       "   (3, [1, 5, 3, 6]),\n",
       "   (3, [3, 6, 7, 9]),\n",
       "   (3, [1, 6, 5]),\n",
       "   (3, [5, 9, 1]),\n",
       "   (3, [2, 3, 6, 9]),\n",
       "   (3, [2, 3, 6, 4]),\n",
       "   (3, [1, 3, 2, 5, 6]),\n",
       "   (3, [4, 5, 2]),\n",
       "   (3, [9, 7, 8, 6]),\n",
       "   (3, [9, 7, 1]),\n",
       "   (3, [7, 8, 9, 5, 1]),\n",
       "   (3, [8, 7, 9, 6]),\n",
       "   (3, [1, 5, 2, 6]),\n",
       "   (3, [9, 7, 3, 1]),\n",
       "   (3, [1, 5, 2, 8]),\n",
       "   (3, [3, 2, 5]),\n",
       "   (3, [7, 8, 9, 5, 6, 4, 3, 2, 1]),\n",
       "   (3, [2, 1, 8]),\n",
       "   (3, [6, 3, 2, 1, 5]),\n",
       "   (3, [2, 1, 5, 7]),\n",
       "   (3, [3, 7, 9]),\n",
       "   (2, [5, 1, 2, 7]),\n",
       "   (2, [5, 1, 2, 4]),\n",
       "   (2, [5, 1, 2, 3]),\n",
       "   (2, [7, 8, 9, 6, 5, 4, 3, 2, 1]),\n",
       "   (2, [1, 3, 5, 2, 6]),\n",
       "   (2, [3, 6, 8]),\n",
       "   (2, [2, 6, 1, 3, 5, 4]),\n",
       "   (2, [7, 5, 2]),\n",
       "   (2, [7, 5, 8]),\n",
       "   (2, [1, 9, 6]),\n",
       "   (2, [1, 5, 8, 2]),\n",
       "   (2, [2, 6, 1, 5, 3]),\n",
       "   (2, [2, 6, 4]),\n",
       "   (2, [9, 8, 3]),\n",
       "   (2, [2, 7, 8, 9, 1]),\n",
       "   (2, [2, 1, 3, 7]),\n",
       "   (2, [1, 8, 9, 7]),\n",
       "   (2, [8, 9, 5]),\n",
       "   (2, [2, 6, 8, 3]),\n",
       "   (2, [9, 2, 8, 7]),\n",
       "   (2, [5, 6, 2, 1]),\n",
       "   (2, [4, 6, 2]),\n",
       "   (2, [8, 2, 7, 3, 9]),\n",
       "   (2, [2, 6, 4, 5]),\n",
       "   (2, [7, 1, 6]),\n",
       "   (2, [6, 7, 8]),\n",
       "   (2, [7, 9, 8, 6, 3, 1, 5, 2, 4]),\n",
       "   (2, [6, 3, 2, 1, 5, 4, 7, 8, 9]),\n",
       "   (2, [8, 6, 3]),\n",
       "   (2, [2, 6, 1, 5, 3, 4, 7, 8, 9]),\n",
       "   (2, [7, 8, 2, 9]),\n",
       "   (2, [2, 3, 4, 5, 6, 7, 8, 9, 1]),\n",
       "   (2, [8, 5, 1]),\n",
       "   (2, [4, 5, 1, 2]),\n",
       "   (2, [8, 2, 3, 6]),\n",
       "   (2, [8, 2, 1, 5]),\n",
       "   (2, [7, 2, 9, 3]),\n",
       "   (2, [6, 9, 8, 7]),\n",
       "   (2, [2, 6, 3, 7]),\n",
       "   (2, [8, 7, 2, 9]),\n",
       "   (2, [7, 4, 5]),\n",
       "   (2, [2, 8, 5, 7, 9, 1, 3, 4, 6]),\n",
       "   (2, [3, 2, 1, 6, 5]),\n",
       "   (2, [2, 8, 4, 1]),\n",
       "   (2, [1, 3, 6, 5, 2]),\n",
       "   (2, [1, 5, 2, 6, 3]),\n",
       "   (2, [1, 3, 5, 8]),\n",
       "   (2, [6, 1, 4]),\n",
       "   (2, [6, 1, 9]),\n",
       "   (2, [2, 1, 6, 5, 3]),\n",
       "   (2, [8, 7, 9, 5, 1]),\n",
       "   (2, [3, 6, 2, 1, 5]),\n",
       "   (2, [7, 8, 9, 6, 3, 1, 5, 4, 2]),\n",
       "   (2, [6, 3, 9, 2, 5, 1, 7, 8]),\n",
       "   (2, [3, 1, 6, 7]),\n",
       "   (2, [7, 6, 8, 2]),\n",
       "   (2, [9, 3, 7, 6]),\n",
       "   (2, [7, 8, 9, 1, 5, 4, 3, 6, 2]),\n",
       "   (2, [9, 7, 1, 5]),\n",
       "   (2, [5, 7, 1, 9]),\n",
       "   (2, [5, 7, 1, 2]),\n",
       "   (2, [5, 2, 1, 7, 8, 9]),\n",
       "   (2, [9, 8, 5]),\n",
       "   (2, [9, 2, 1, 7]),\n",
       "   (2, [7, 9, 4]),\n",
       "   (2, [6, 4, 5]),\n",
       "   (2, [9, 8, 7, 5, 2, 1, 3, 6, 4]),\n",
       "   (2, [2, 3, 8, 7]),\n",
       "   (2, [6, 8, 2]),\n",
       "   (2, [6, 8, 9]),\n",
       "   (2, [1, 2, 7, 8, 9, 6, 5, 3, 4]),\n",
       "   (2, [9, 8, 2]),\n",
       "   (2, [6, 3, 8, 7, 1, 5, 9]),\n",
       "   (2, [7, 6, 8]),\n",
       "   (2, [7, 9, 6, 3]),\n",
       "   (2, [6, 2, 5, 1]),\n",
       "   (2, [6, 2, 5, 3]),\n",
       "   (2, [2, 1, 8, 5]),\n",
       "   (2, [9, 8, 2, 7]),\n",
       "   (2, [2, 5, 6]),\n",
       "   (2, [8, 9, 7, 6]),\n",
       "   (2, [8, 9, 7, 3]),\n",
       "   (2, [7, 3, 9, 2]),\n",
       "   (2, [7, 9, 2, 3]),\n",
       "   (2, [1, 2, 5, 6, 3]),\n",
       "   (2, [4, 7, 5]),\n",
       "   (2, [4, 7, 1]),\n",
       "   (2, [9, 1, 3, 7]),\n",
       "   (2, [5, 3, 1]),\n",
       "   (2, [6, 3, 2, 9]),\n",
       "   (2, [6, 3, 2, 1]),\n",
       "   (2, [5, 2, 3, 1]),\n",
       "   (2, [3, 7, 8, 9]),\n",
       "   (2, [9, 7, 6, 3]),\n",
       "   (2, [7, 8, 9, 1, 5]),\n",
       "   (2, [4, 6, 5]),\n",
       "   (2, [5, 8, 3]),\n",
       "   (2, [1, 2, 3, 6, 5]),\n",
       "   (2, [5, 8]),\n",
       "   (2, [5, 2, 7, 8, 9]),\n",
       "   (2, [1, 5, 3, 6, 2, 7, 8, 9, 4]),\n",
       "   (2, [9, 7, 8, 5, 2, 1, 4, 3, 6]),\n",
       "   (2, [2, 9, 6, 3]),\n",
       "   (2, [1, 2, 4, 7]),\n",
       "   (2, [1, 2, 4, 5]),\n",
       "   (2, [7, 8, 9, 5, 4, 2, 1, 3, 6]),\n",
       "   (2, [8, 3]),\n",
       "   (2, [7, 5, 2, 4]),\n",
       "   (2, [9, 7, 2, 6]),\n",
       "   (2, [6, 7, 2, 1, 3]),\n",
       "   (2, [7, 2, 1, 8]),\n",
       "   (2, [2, 1, 5, 3, 6]),\n",
       "   (2, [9, 2, 6]),\n",
       "   (2, [8, 7, 9, 6, 1]),\n",
       "   (2, [7, 3, 2]),\n",
       "   (2, [4, 5, 7, 9]),\n",
       "   (2, [9, 6, 1]),\n",
       "   (2, [9, 6, 5]),\n",
       "   (2, [2, 7, 9, 8]),\n",
       "   (2, [2, 7, 9, 5]),\n",
       "   (2, [5, 7, 9]),\n",
       "   (2, [5, 7, 4]),\n",
       "   (2, [5, 7, 6]),\n",
       "   (2, [5, 7, 2]),\n",
       "   (2, [2, 8, 1, 4, 5]),\n",
       "   (2, [7, 2, 5, 1]),\n",
       "   (2, [8, 9, 2]),\n",
       "   (2, [6, 5]),\n",
       "   (2, [2, 5, 6, 3]),\n",
       "   (2, [5, 8, 7, 9, 1, 3, 6]),\n",
       "   (2, [4, 7, 5, 2]),\n",
       "   (2, [7, 4, 5, 6]),\n",
       "   (2, [1, 2, 3, 5, 6]),\n",
       "   (2, [9, 8, 6]),\n",
       "   (2, [9, 6, 3, 2]),\n",
       "   (2, [8, 2, 9, 1]),\n",
       "   (2, [2, 6, 3, 1, 5]),\n",
       "   (2, [2, 5, 3, 1]),\n",
       "   (2, [6, 5, 2]),\n",
       "   (2, [2, 5, 7, 1]),\n",
       "   (2, [3, 9, 1]),\n",
       "   (2, [8, 1, 7]),\n",
       "   (2, [8, 1, 3]),\n",
       "   (2, [6, 1, 2, 3]),\n",
       "   (2, [1, 8]),\n",
       "   (2, [6, 9, 2]),\n",
       "   (2, [8, 2, 7, 9, 1]),\n",
       "   (2, [1, 7, 9, 8, 5]),\n",
       "   (2, [9, 5, 6, 3]),\n",
       "   (2, [2, 3, 1, 6, 5, 4, 7, 8, 9]),\n",
       "   (2, [8, 7, 5]),\n",
       "   (2, [8, 7, 4]),\n",
       "   (2, [1, 2, 5, 6]),\n",
       "   (2, [3, 1, 5, 2]),\n",
       "   (2, [9, 3, 6, 7]),\n",
       "   (2, [2, 7, 5, 1]),\n",
       "   (2, [3, 5, 6]),\n",
       "   (2, [2, 1, 5, 6, 3, 7, 8, 9, 4]),\n",
       "   (2, [2, 1, 3, 5, 6, 7, 8, 9, 4]),\n",
       "   (2, [1, 2, 8]),\n",
       "   (2, [9, 3, 2]),\n",
       "   (2, [7, 5, 4, 8, 9]),\n",
       "   (2, [4, 8, 2]),\n",
       "   (2, [2, 8, 6, 1]),\n",
       "   (2, [7, 6, 3, 9, 8, 2, 5, 1, 4]),\n",
       "   (2, [1, 5, 8]),\n",
       "   (2, [1, 3, 5, 7, 6, 4, 9, 8, 2]),\n",
       "   (2, [2, 3, 1, 5, 6]),\n",
       "   (2, [1, 2, 7, 9]),\n",
       "   (2, [2, 6, 1, 4]),\n",
       "   (2, [9, 7, 8, 5, 4]),\n",
       "   (2, [1, 2, 3, 6]),\n",
       "   (2, [5, 8, 7, 3]),\n",
       "   (2, [1, 4, 5, 7]),\n",
       "   (2, [1, 6, 7]),\n",
       "   (2, [1, 6, 9]),\n",
       "   (2, [7, 9, 1, 8]),\n",
       "   (2, [2, 1, 5, 4, 3]),\n",
       "   (2, [2, 5, 6, 1]),\n",
       "   (2, [7, 5, 1, 8, 9]),\n",
       "   (2, [7, 2, 6]),\n",
       "   (2, [2, 7, 6]),\n",
       "   (2, [2, 1, 6, 3, 5]),\n",
       "   (2, [9, 7, 8, 5, 4, 1]),\n",
       "   (2, [9, 8, 7, 5, 1]),\n",
       "   (2, [2, 6, 7, 9]),\n",
       "   (2, [2, 6, 7, 3]),\n",
       "   (2, [2, 3, 6, 5]),\n",
       "   (2, [3, 7, 5]),\n",
       "   (2, [2, 1, 4, 5]),\n",
       "   (2, [9, 8, 7, 6, 5, 4, 3, 2, 1]),\n",
       "   (2, [7, 9, 3, 6]),\n",
       "   (2, [7, 2, 4]),\n",
       "   (2, [8, 2, 6, 4]),\n",
       "   (2, [7, 2, 8]),\n",
       "   (2, [6, 2, 9]),\n",
       "   (2, [6, 2, 5]),\n",
       "   (2, [7, 5, 1, 2]),\n",
       "   (2, [2, 3, 1, 6, 5]),\n",
       "   (2, [7, 8, 5]),\n",
       "   (2, [5, 2, 1, 9]),\n",
       "   (2, [4, 3, 5]),\n",
       "   (2, [7, 8, 9, 3, 2]),\n",
       "   (2, [7, 8, 9, 3, 1]),\n",
       "   (2, [2, 6, 7, 5]),\n",
       "   (2, [8, 7, 9, 4]),\n",
       "   (2, [8, 7, 9, 1]),\n",
       "   (2, [2, 3, 8, 7, 9]),\n",
       "   (2, [3, 2, 6, 1]),\n",
       "   (2, [6, 2, 3, 4]),\n",
       "   (2, [4, 5, 2, 1]),\n",
       "   (2, [3, 1, 2, 6]),\n",
       "   (2, [6, 1, 2, 5]),\n",
       "   (2, [6, 7, 3, 2]),\n",
       "   (2, [1, 3, 6, 2]),\n",
       "   (2, [1, 3, 6, 9]),\n",
       "   (2, [1, 8, 7, 9]),\n",
       "   (2, [7, 6, 9, 8]),\n",
       "   (2, [2, 3, 5, 4]),\n",
       "   (2, [1, 5, 2, 3, 6]),\n",
       "   (2, [2, 3, 5, 8]),\n",
       "   (2, [2, 3, 7, 8]),\n",
       "   (2, [2, 1, 5, 9]),\n",
       "   (1, [4, 2, 3, 5]),\n",
       "   (1, [6, 7, 1, 3]),\n",
       "   (1, [7, 8, 9, 1, 4, 5, 6, 2, 3]),\n",
       "   (1, [2, 3, 1, 7, 4, 8, 5, 9, 6]),\n",
       "   (1, [7, 8, 9, 2, 6, 1, 3, 4, 5]),\n",
       "   (1, [5, 1, 2, 9]),\n",
       "   (1, [5, 1, 7, 8, 9, 3, 2, 6, 4]),\n",
       "   (1, [3, 6, 7]),\n",
       "   (1, [2, 8, 7, 9, 5, 3, 1, 4]),\n",
       "   (1, [8, 1, 5, 7, 9]),\n",
       "   (1, [5, 9, 1, 4]),\n",
       "   (1, [3, 6, 9]),\n",
       "   (1, [9, 8, 7, 6, 2]),\n",
       "   (1, [5, 7, 1, 2, 9]),\n",
       "   (1, [9, 4, 7]),\n",
       "   (1, [9, 4, 5]),\n",
       "   (1, [5, 4, 1, 7, 3, 9, 6, 2, 8]),\n",
       "   (1, [8, 9, 7, 5, 4, 3, 2, 1, 6]),\n",
       "   (1, [9, 5, 7, 8]),\n",
       "   (1, [6, 5, 7, 4, 3, 1, 9, 2, 8]),\n",
       "   (1, [2, 8, 4, 9, 1, 5, 3, 6, 7]),\n",
       "   (1, [2, 8, 5, 7]),\n",
       "   (1, [2, 8, 5, 6]),\n",
       "   (1, [7, 5, 6]),\n",
       "   (1, [7, 5, 3]),\n",
       "   (1, [1, 9, 2, 3]),\n",
       "   (1, [2, 5, 1, 6, 3, 7, 8, 9, 4]),\n",
       "   (1, [6, 1, 8, 2]),\n",
       "   (1, [1, 9, 2, 5]),\n",
       "   (1, [1, 9, 7]),\n",
       "   (1, [1, 9, 4]),\n",
       "   (1, [1, 3, 2, 5, 7, 9, 6, 8, 4]),\n",
       "   (1, [1, 9, 2]),\n",
       "   (1, [7, 9, 5, 2]),\n",
       "   (1, [9, 7, 1, 5, 6, 2]),\n",
       "   (1, [1, 5, 2, 7, 8, 9, 6, 3, 4]),\n",
       "   (1, [2, 3, 7, 6]),\n",
       "   (1, [2, 4, 8]),\n",
       "   (1, [7, 1, 3]),\n",
       "   (1, [7, 1, 8]),\n",
       "   (1, [1, 5, 8, 6]),\n",
       "   (1, [2, 4, 3]),\n",
       "   (1, [1, 5, 8, 3]),\n",
       "   (1, [7, 9, 8, 1, 5, 6, 3, 2, 4]),\n",
       "   (1, [8, 9, 6, 7, 2, 5, 3, 1, 4]),\n",
       "   (1, [5, 7, 9, 3, 2, 6, 1, 8, 4]),\n",
       "   (1, [7, 9, 5, 4]),\n",
       "   (1, [9, 7, 8, 2, 5, 3, 1, 6]),\n",
       "   (1, [1, 6, 4, 5, 3, 2, 9, 8, 7]),\n",
       "   (1, [1, 6, 2, 8]),\n",
       "   (1, [6, 2, 1, 5, 7, 8, 9, 3, 4]),\n",
       "   (1, [9, 7, 8, 3, 2, 6, 1, 5, 4]),\n",
       "   (1, [1, 6, 2, 3]),\n",
       "   (1, [4, 7, 1, 9]),\n",
       "   (1, [2, 1, 9, 7, 5, 8, 6, 3, 4]),\n",
       "   (1, [9, 8, 7, 5, 3, 1, 6, 4, 2]),\n",
       "   (1, [6, 1, 8, 5, 3]),\n",
       "   (1, [6, 1, 8, 5, 7]),\n",
       "   (1, [6, 7, 5, 3, 9, 8, 1, 2, 4]),\n",
       "   (1, [7, 9, 1, 3, 5, 2, 8, 6, 4]),\n",
       "   (1, [2, 4, 3, 1, 5, 6, 7, 8, 9]),\n",
       "   (1, [2, 3, 1, 4]),\n",
       "   (1, [7, 2, 3, 6]),\n",
       "   (1, [4, 2, 1, 5, 7, 8, 9, 6, 3]),\n",
       "   (1, [9, 7, 8, 5, 3, 2]),\n",
       "   (1, [1, 5, 6, 3, 2]),\n",
       "   (1, [9, 1, 7, 8, 4, 5, 6, 3, 2]),\n",
       "   (1, [1, 5, 6, 3, 7]),\n",
       "   (1, [2, 1, 7, 9]),\n",
       "   (1, [1, 9, 8, 7]),\n",
       "   (1, [2, 1, 7, 5]),\n",
       "   (1, [6, 7, 9, 2]),\n",
       "   (1, [4, 1, 5, 2, 7, 8, 9, 6, 3]),\n",
       "   (1, [6, 7, 9, 5]),\n",
       "   (1, [9, 8, 1]),\n",
       "   (1, [7, 9, 8, 2, 6, 5, 1, 3, 4]),\n",
       "   (1, [2, 5, 1, 3, 6, 8, 7, 9, 4]),\n",
       "   (1, [7, 3, 2, 6]),\n",
       "   (1, [9, 8, 7, 6, 3, 4, 5, 1, 2]),\n",
       "   (1, [2, 8, 7, 5, 1]),\n",
       "   (1, [2, 1, 3, 8]),\n",
       "   (1, [7, 8, 6, 9]),\n",
       "   (1, [6, 7, 8, 3, 1, 2, 9, 5, 4]),\n",
       "   (1, [2, 5, 8, 9]),\n",
       "   (1, [1, 2, 4, 3, 5, 6, 7, 8]),\n",
       "   (1, [5, 2, 8]),\n",
       "   (1, [9, 5, 6, 7, 4]),\n",
       "   (1, [2, 1, 3, 4]),\n",
       "   (1, [5, 1, 2, 9, 4]),\n",
       "   (1, [7, 5, 1, 3]),\n",
       "   (1, [7, 9, 8, 6, 2, 3, 1, 5, 4]),\n",
       "   (1, [5, 2, 9, 7, 8]),\n",
       "   (1, [8, 7, 1, 9, 5]),\n",
       "   (1, [9, 4, 2, 6]),\n",
       "   (1, [4, 5, 7, 8, 9, 1, 6, 2, 3]),\n",
       "   (1, [2, 1, 7, 5, 8, 9, 3, 4, 6]),\n",
       "   (1, [9, 4, 2, 1]),\n",
       "   (1, [7, 9, 6, 8, 2]),\n",
       "   (1, [7, 5, 2, 8, 9, 6, 3, 4, 1]),\n",
       "   (1, [7, 9, 5, 8, 1]),\n",
       "   (1, [4, 3, 6, 9]),\n",
       "   (1, [6, 1, 2, 5, 3, 4, 7, 9, 8]),\n",
       "   (1, [2, 6, 8, 9]),\n",
       "   (1, [7, 8, 9, 6, 4, 2, 5, 3, 1]),\n",
       "   (1, [1, 4, 5, 9, 8, 7, 2]),\n",
       "   (1, [9, 7, 8, 6, 3, 5, 1, 2, 4]),\n",
       "   (1, [1, 2, 5, 8, 3, 6]),\n",
       "   (1, [7, 9, 8, 5, 4, 1, 2, 3, 6]),\n",
       "   (1, [6, 1, 5, 3, 2, 4, 7, 8, 9]),\n",
       "   (1, [7, 3, 1, 5, 4]),\n",
       "   (1, [7, 8, 9, 6, 3, 5, 2, 1, 4]),\n",
       "   (1, [9, 2, 5, 7, 8, 4, 3, 1, 6]),\n",
       "   (1, [5, 3, 6, 7, 8, 9, 4, 1, 2]),\n",
       "   (1, [1, 3, 5, 7, 8, 9]),\n",
       "   (1, [2, 9, 8, 5]),\n",
       "   (1, [7, 9, 8, 5, 6]),\n",
       "   (1, [1, 5, 3, 6, 2]),\n",
       "   (1, [2, 5, 9, 7, 8]),\n",
       "   (1, [1, 5, 3, 6, 7]),\n",
       "   (1, [5, 1, 3, 2]),\n",
       "   (1, [2, 7, 6, 3, 4, 5, 1, 8, 9]),\n",
       "   (1, [2, 6, 4, 3, 1, 7, 8, 9]),\n",
       "   (1, [8, 7, 9, 1, 3, 6]),\n",
       "   (1, [7, 9, 8, 1, 5]),\n",
       "   (1, [4, 3, 2, 5, 6, 1, 7, 8, 9]),\n",
       "   (1, [7, 9, 8, 1, 2]),\n",
       "   (1, [7, 8, 9, 2, 5, 1, 3, 6, 4]),\n",
       "   (1, [7, 8, 9, 6, 3, 4, 2, 1, 5]),\n",
       "   (1, [9, 7, 8, 2, 5, 6]),\n",
       "   (1, [2, 8, 1, 9, 6, 7, 3, 4, 5]),\n",
       "   (1, [8, 2, 7, 9, 4, 3]),\n",
       "   (1, [7, 8, 9, 5, 2, 1, 3, 4]),\n",
       "   (1, [2, 6, 4, 1]),\n",
       "   (1, [7, 3, 9, 8, 6, 5, 4, 1, 2]),\n",
       "   (1, [7, 8, 9, 2, 4, 6, 3, 5, 1]),\n",
       "   (1, [2, 8, 7, 9, 5, 1, 3, 4]),\n",
       "   (1, [7, 5, 2, 1, 3]),\n",
       "   (1, [1, 7, 8, 2, 3, 9, 5, 6, 4]),\n",
       "   (1, [1, 3, 2, 5, 6, 7, 8, 9, 4]),\n",
       "   (1, [5, 3, 2, 6]),\n",
       "   (1, [2, 8, 3, 7, 9, 5, 1, 6, 4]),\n",
       "   (1, [5, 3, 2, 1]),\n",
       "   (1, [7, 6, 8, 9, 3]),\n",
       "   (1, [2, 4, 5, 1, 3, 6, 7, 8, 9]),\n",
       "   (1, [8, 6, 3, 2]),\n",
       "   (1, [6, 2, 1, 3, 8, 9, 7, 5, 4]),\n",
       "   (1, [9, 7, 4, 1, 5]),\n",
       "   (1, [9, 7, 8, 6, 5, 3, 1, 4, 2]),\n",
       "   (1, [6, 7, 5]),\n",
       "   (1, [9, 7, 8, 1, 3, 5, 4, 2, 6]),\n",
       "   (1, [6, 7, 2]),\n",
       "   (1, [9, 7, 6, 2, 1, 5, 8, 3, 4]),\n",
       "   (1, [7, 8, 9, 3, 5, 4, 1, 2, 6]),\n",
       "   (1, [5, 1, 9, 7, 8, 4, 6, 3, 2]),\n",
       "   (1, [3, 1, 6, 7, 8, 9]),\n",
       "   (1, [9, 7, 8, 2, 4, 1, 5, 6, 3]),\n",
       "   (1, [2, 5, 1, 6]),\n",
       "   (1, [2, 3, 4, 5, 1, 6]),\n",
       "   (1, [2, 5, 1, 3]),\n",
       "   (1, [2, 5, 9, 7, 6, 4, 1, 8, 3]),\n",
       "   (1, [2, 3, 1, 6, 5, 8, 7, 9, 4]),\n",
       "   (1, [4, 5, 6, 2, 1, 3, 7, 8, 9]),\n",
       "   (1, [6, 1, 5, 3, 2]),\n",
       "   (1, [9, 7, 8, 4, 3, 5, 6, 2, 1]),\n",
       "   (1, [7, 6, 4, 1]),\n",
       "   (1, [5, 2, 3, 7, 8, 1, 9, 6]),\n",
       "   (1, [9, 7, 8, 5, 6, 1, 2, 3, 4]),\n",
       "   (1, [5, 7, 1, 3, 6, 9, 8]),\n",
       "   (1, [9, 6, 7, 2]),\n",
       "   (1, [2, 3, 1, 7, 8, 9, 6]),\n",
       "   (1, [9, 3, 1, 8, 7, 6, 4, 5, 2]),\n",
       "   (1, [9, 6, 7, 8]),\n",
       "   (1, [1, 7, 8, 9, 4, 5]),\n",
       "   (1, [9, 7, 8, 6, 5, 4, 2, 1, 3]),\n",
       "   (1, [5, 9, 7, 3]),\n",
       "   (1, [7, 3, 2, 9, 1, 6, 5, 8, 4]),\n",
       "   (1, [2, 3, 6, 1, 5, 4, 7, 8, 9]),\n",
       "   (1, [4, 2, 1, 9]),\n",
       "   (1, [5, 9, 7, 4]),\n",
       "   (1, [7, 8, 9, 1, 5, 3, 6]),\n",
       "   (1, [4, 2, 1, 5]),\n",
       "   (1, [1, 2, 3, 5, 6, 4]),\n",
       "   (1, [9, 8, 7, 6, 1, 2, 3, 5, 4]),\n",
       "   (1, [8, 7, 9, 2, 3, 4, 5, 1, 6]),\n",
       "   (1, [6, 3, 7, 1]),\n",
       "   (1, [6, 3, 7, 8]),\n",
       "   (1, [6, 3, 7, 9]),\n",
       "   (1, [9, 2, 6, 7]),\n",
       "   (1, [7, 3, 2, 1, 6]),\n",
       "   (1, [2, 3, 5, 6, 1, 8, 7, 9, 4]),\n",
       "   (1, [5, 7, 9, 8, 1, 4, 3, 6, 2]),\n",
       "   (1, [7, 3, 8, 9, 1]),\n",
       "   (1, [7, 9, 8, 2, 5, 4, 1, 6, 3]),\n",
       "   (1, [2, 1, 5, 9, 7, 6, 8, 4, 3]),\n",
       "   (1, [8, 3, 6, 7]),\n",
       "   (1, [2, 3, 6, 1, 4, 5]),\n",
       "   (1, [6, 8, 3, 9, 2, 4, 1, 5, 7]),\n",
       "   (1, [8, 3, 6, 2]),\n",
       "   (1, [3, 6, 7, 2]),\n",
       "   (1, [4, 5, 9, 1]),\n",
       "   (1, [1, 5, 2, 7, 8, 9]),\n",
       "   (1, [9, 8, 7, 3, 5, 1]),\n",
       "   (1, [3, 6, 7, 5]),\n",
       "   (1, [9, 7, 8, 2, 1, 5, 3, 6, 4]),\n",
       "   (1, [8, 3, 9]),\n",
       "   (1, [7, 8, 2, 1]),\n",
       "   (1, [2, 3, 5, 7]),\n",
       "   (1, [7, 8, 2, 3]),\n",
       "   (1, [3, 5, 6, 7]),\n",
       "   (1, [7, 4, 3, 8, 9, 6, 1, 5, 2]),\n",
       "   (1, [1, 5, 3, 7, 8, 9, 6, 4, 2]),\n",
       "   (1, [3, 2, 5, 6]),\n",
       "   (1, [6, 3, 1, 2, 7, 9, 8, 5, 4]),\n",
       "   (1, [8, 3, 5]),\n",
       "   (1, [1, 5, 3, 6, 2, 8, 9, 7, 4]),\n",
       "   (1, [8, 5, 6, 7]),\n",
       "   (1, [9, 7, 8, 5, 1, 2, 4, 3, 6]),\n",
       "   (1, [9, 7, 8, 1, 2, 5, 3, 4, 6]),\n",
       "   (1, [5, 2, 7, 8, 9, 1]),\n",
       "   (1, [5, 6, 1, 3, 7, 8, 9, 2, 4]),\n",
       "   (1, [3, 1, 7, 9]),\n",
       "   (1, [9, 7, 5, 8, 4]),\n",
       "   (1, [4, 5, 1, 6]),\n",
       "   (1, [4, 5, 1, 7]),\n",
       "   (1, [8, 7, 6, 2]),\n",
       "   (1, [1, 6, 3, 5, 2, 7, 8, 9, 4]),\n",
       "   (1, [7, 8, 9, 2, 6]),\n",
       "   (1, [6, 1, 4, 5, 8, 7, 9, 3, 2]),\n",
       "   (1, [7, 9, 8, 1, 2, 3, 5, 6, 4]),\n",
       "   (1, [7, 8, 9, 2, 3]),\n",
       "   (1, [7, 9, 8, 5, 3, 1, 2, 4, 6]),\n",
       "   (1, [1, 5, 7, 8, 9, 2, 3, 4, 6]),\n",
       "   (1, [7, 9, 8, 5, 1, 6, 3, 4, 2]),\n",
       "   (1, [2, 7, 1, 4]),\n",
       "   (1, [2, 7, 1, 5]),\n",
       "   (1, [2, 8, 6, 3, 1, 9, 5, 4, 7]),\n",
       "   (1, [2, 1, 8, 9, 7, 6, 5, 3]),\n",
       "   (1, [9, 8, 6, 7]),\n",
       "   (1, [6, 2, 3, 1, 7]),\n",
       "   (1, [2, 7, 1, 8]),\n",
       "   (1, [2, 7, 1, 9]),\n",
       "   (1, [2, 7, 3, 8]),\n",
       "   (1, [2, 1, 9, 6, 7, 8, 3, 5, 4]),\n",
       "   (1, [2, 6, 5, 3, 1]),\n",
       "   (1, [7, 6, 1, 9, 4]),\n",
       "   (1, [7, 8, 9, 3, 6, 4]),\n",
       "   (1, [2, 6, 3, 1, 4, 5, 7, 8, 9]),\n",
       "   (1, [7, 8, 9, 5, 6, 1]),\n",
       "   (1, [6, 5, 1, 3, 2, 8, 7, 4, 9]),\n",
       "   (1, [2, 6, 3, 1, 5, 4, 8]),\n",
       "   (1, [2, 7, 9, 8, 4, 5]),\n",
       "   (1, [2, 1, 3, 5, 6, 4]),\n",
       "   (1, [5, 3, 8]),\n",
       "   (1, [5, 1, 8]),\n",
       "   (1, [5, 9, 6, 7, 3, 1]),\n",
       "   (1, [9, 1, 4, 7]),\n",
       "   (1, [9, 7, 8, 4, 1, 2, 5, 3, 6]),\n",
       "   (1, [5, 3, 2]),\n",
       "   (1, [5, 7, 8, 9, 1, 4, 6, 3, 2]),\n",
       "   (1, [2, 8, 9, 5, 3, 1, 4]),\n",
       "   (1, [5, 2, 6, 3, 7]),\n",
       "   (1, [8, 7, 2, 5]),\n",
       "   (1, [7, 9, 3, 6, 8]),\n",
       "   (1, [1, 5, 4, 9, 8, 7, 3, 2, 6]),\n",
       "   (1, [2, 5, 1, 7, 4, 9, 6, 8, 3]),\n",
       "   (1, [3, 1, 9, 8, 7, 6, 5, 2, 4]),\n",
       "   (1, [2, 1, 7, 3, 5]),\n",
       "   (1, [7, 9, 8, 5, 3, 2, 1, 6]),\n",
       "   (1, [1, 9, 5, 4, 3]),\n",
       "   (1, [8, 1, 2, 4, 6]),\n",
       "   (1, [7, 1, 8, 9, 5, 2]),\n",
       "   (1, [9, 8, 2, 5]),\n",
       "   (1, [8, 7, 5, 9]),\n",
       "   (1, [2, 4, 5, 3, 6, 7, 8, 1, 9]),\n",
       "   (1, [2, 6, 4, 1, 5, 3, 7, 9, 8]),\n",
       "   (1, [6, 1, 9, 2]),\n",
       "   (1, [5, 6, 4, 1]),\n",
       "   (1, [9, 7, 8, 5, 1, 4, 2]),\n",
       "   (1, [6, 8, 7, 5]),\n",
       "   (1, [7, 1, 5, 3, 6, 2, 8, 9]),\n",
       "   (1, [6, 1, 9, 8]),\n",
       "   (1, [9, 6, 7, 3, 2, 1, 4, 5, 8]),\n",
       "   (1, [7, 8, 9, 1, 6, 3, 2, 5, 4]),\n",
       "   (1, [7, 8, 9, 3, 4, 6, 5, 2, 1]),\n",
       "   (1, [1, 3, 2, 5, 8, 7, 9]),\n",
       "   (1, [7, 3, 8, 2]),\n",
       "   (1, [7, 9, 8, 3, 6]),\n",
       "   (1, [7, 9, 8, 2, 3, 5, 6, 4, 1]),\n",
       "   (1, [8, 9, 6]),\n",
       "   (1, [4, 6, 7]),\n",
       "   (1, [7, 4, 2]),\n",
       "   (1, [5, 6, 7, 9, 8]),\n",
       "   (1, [2, 4, 6, 3, 1, 5]),\n",
       "   (1, [3, 6, 9, 1, 2]),\n",
       "   (1, [7, 9, 8, 5, 6, 1, 2, 3, 4]),\n",
       "   (1, [2, 4, 1, 5, 3, 6, 7, 8, 9]),\n",
       "   (1, [5, 4, 9, 7, 8, 1, 6, 3, 2]),\n",
       "   (1, [3, 5, 2, 8]),\n",
       "   (1, [2, 8, 4, 6]),\n",
       "   (1, [3, 5, 2, 6]),\n",
       "   (1, [7, 8, 9, 6, 1, 5, 3, 2, 4]),\n",
       "   (1, [2, 3, 4, 5, 1, 6, 7, 8, 9]),\n",
       "   (1, [2, 3, 1, 5, 6, 4, 7, 8, 9]),\n",
       "   (1, [4, 1, 3, 6]),\n",
       "   (1, [4, 5, 7, 8, 9, 1, 2, 6, 3]),\n",
       "   (1, [9, 7, 8, 2, 1, 3, 5, 4, 6]),\n",
       "   (1, [1, 6, 3, 2, 5, 7]),\n",
       "   (1, [4, 2, 5, 7, 8, 9, 1, 3, 6]),\n",
       "   (1, [5, 9, 8, 7, 4, 6]),\n",
       "   (1, [7, 4, 3, 2]),\n",
       "   (1, [5, 4, 1, 3, 7, 9, 6, 8, 2]),\n",
       "   (1, [7, 8, 9, 5, 6, 4, 1, 3, 2]),\n",
       "   (1, [5, 7, 6, 1]),\n",
       "   (1, [7, 3, 1, 2, 4, 5, 6, 8, 9]),\n",
       "   (1, [8, 7, 9, 5, 1, 6, 3, 4, 2]),\n",
       "   (1, [3, 6, 1, 9]),\n",
       "   (1, [4, 1, 7, 8]),\n",
       "   (1, [7, 8, 5, 9, 2]),\n",
       "   (1, [9, 8, 7, 1, 3, 6, 5, 2, 4]),\n",
       "   (1, [9, 7, 8, 5, 1, 2, 6, 3, 4]),\n",
       "   (1, [5, 1, 4, 9, 8, 7, 2, 3, 6]),\n",
       "   (1, [1, 5, 2, 6, 4]),\n",
       "   (1, [7, 8, 9, 6, 1, 2, 5, 3, 4]),\n",
       "   (1, [2, 8, 3, 7, 5]),\n",
       "   (1, [3, 2, 4, 1, 7, 5, 6, 9, 8]),\n",
       "   (1, [2, 3, 8, 4]),\n",
       "   (1, [1, 3, 6, 8, 9, 7]),\n",
       "   (1, [2, 8, 7, 1, 9, 6]),\n",
       "   (1, [2, 7, 8, 9, 1, 4, 5, 3, 6]),\n",
       "   (1, [1, 2, 5, 8]),\n",
       "   (1, [1, 3, 5, 7]),\n",
       "   (1, [5, 4, 7, 6, 8]),\n",
       "   (1, [7, 8, 9, 6, 5, 3, 2, 1, 4]),\n",
       "   (1, [3, 6, 5, 4]),\n",
       "   (1, [2, 3, 4, 5]),\n",
       "   (1, [7, 2, 8, 3, 5, 9, 1, 6, 4]),\n",
       "   (1, [2, 3, 4, 6]),\n",
       "   (1, [9, 7, 8, 1, 3, 2, 4, 5, 6]),\n",
       "   (1, [2, 5, 6, 7, 4, 8, 1, 3, 9]),\n",
       "   (1, [3, 5, 1, 6]),\n",
       "   (1, [2, 5, 1, 9, 8, 7, 6, 3, 4]),\n",
       "   (1, [2, 5, 9, 3]),\n",
       "   (1, [2, 5, 9, 1]),\n",
       "   (1, [1, 7, 8, 9, 5]),\n",
       "   (1, [1, 7, 8, 9, 4]),\n",
       "   (1, [2, 9, 7, 8, 5]),\n",
       "   (1, [5, 7, 8, 9, 4]),\n",
       "   (1, [1, 7, 8, 9, 2]),\n",
       "   (1, [7, 3, 2, 6, 8, 9, 5, 1, 4]),\n",
       "   (1, [4, 5, 1, 2, 7, 8, 9, 3, 6]),\n",
       "   (1, [3, 4, 1, 6, 5, 7, 9, 2, 8]),\n",
       "   (1, [3, 2, 5, 7, 8]),\n",
       "   (1, [6, 3, 9, 8, 7, 5, 2]),\n",
       "   (1, [8, 2, 3, 6, 5, 4, 1, 7, 9]),\n",
       "   (1, [9, 8, 7, 2, 5, 1, 3, 4, 6]),\n",
       "   (1, [9, 8, 7, 4, 3, 5, 2, 1, 6]),\n",
       "   (1, [8, 1, 7, 9]),\n",
       "   (1, [2, 1, 6, 3, 5, 4]),\n",
       "   (1, [3, 1, 5, 6]),\n",
       "   (1, [3, 2, 1, 6, 4, 5, 7, 8, 9]),\n",
       "   (1, [2, 6, 1, 5, 4, 3]),\n",
       "   (1, [5, 2, 4, 1]),\n",
       "   (1, [9, 8, 7, 3, 5]),\n",
       "   (1, [1, 2, 9, 5, 3, 8, 7, 6, 4]),\n",
       "   (1, [9, 6, 2, 5]),\n",
       "   (1, [9, 7, 8, 6, 3, 4, 1, 5, 2]),\n",
       "   (1, [2, 4, 5, 7, 8, 9, 6, 3, 1]),\n",
       "   (1, [7, 9, 1, 5, 3, 4, 2, 8, 6]),\n",
       "   (1, [7, 2, 3, 5, 1]),\n",
       "   (1, [9, 6, 5, 3, 8, 7]),\n",
       "   (1, [7, 5, 6, 3, 1, 2, 9, 8, 4]),\n",
       "   (1, [7, 9, 8, 5, 1, 6, 4, 3, 2]),\n",
       "   (1, [9, 7, 2, 1, 3, 4, 5, 6, 8]),\n",
       "   (1, [5, 6, 2, 4, 1, 3, 9, 7, 8]),\n",
       "   (1, [2, 8, 1, 4, 5, 7, 9, 3, 6]),\n",
       "   (1, [6, 1, 5, 3]),\n",
       "   (1, [1, 6, 5, 2]),\n",
       "   (1, [2, 5, 6, 3, 1, 8, 7, 9, 4]),\n",
       "   (1, [6, 8, 9, 4]),\n",
       "   (1, [3, 6, 5, 2]),\n",
       "   (1, [5, 4, 7, 8]),\n",
       "   (1, [2, 5, 8, 9, 1]),\n",
       "   (1, [7, 8, 5, 1]),\n",
       "   (1, [4, 5, 3, 7, 8, 9, 1, 2, 6]),\n",
       "   (1, [8, 7, 9, 1, 2]),\n",
       "   (1, [7, 6, 2, 3, 1, 5]),\n",
       "   (1, [3, 6, 5, 1, 7, 8, 9]),\n",
       "   (1, [1, 3, 5, 6, 2, 7, 8, 4, 9]),\n",
       "   (1, [3, 6, 2, 1, 8]),\n",
       "   (1, [9, 7, 8, 3, 2, 1]),\n",
       "   (1, [6, 3, 1, 9]),\n",
       "   (1, [1, 7, 5, 9, 8]),\n",
       "   (1, [5, 2, 6, 1, 8]),\n",
       "   (1, [2, 1, 3, 6, 5, 4, 9, 8, 7]),\n",
       "   (1, [5, 1, 9]),\n",
       "   (1, [2, 4, 7, 8, 9, 5]),\n",
       "   (1, [7, 8, 9, 5, 2, 1, 4, 3, 6]),\n",
       "   (1, [1, 8, 9, 7, 5, 3, 2, 6, 4]),\n",
       "   (1, [7, 1, 5, 4, 8, 9]),\n",
       "   (1, [2, 3, 7, 5, 1, 4, 9, 8, 6]),\n",
       "   (1, [1, 3, 2, 6, 5]),\n",
       "   (1, [8, 2, 1, 5, 9, 7, 4, 3, 6]),\n",
       "   (1, [1, 7, 3, 5]),\n",
       "   (1, [1, 5, 4, 2, 3]),\n",
       "   (1, [9, 4, 1]),\n",
       "   (1, [9, 7, 8, 6, 1, 3, 5, 4, 2]),\n",
       "   (1, [2, 1, 6, 5, 3, 4, 8, 7, 9]),\n",
       "   (1, [1, 8, 9, 7, 3, 5, 6, 2, 4]),\n",
       "   (1, [4, 1, 5, 2]),\n",
       "   (1, [2, 6, 8, 9, 3, 1, 5, 7, 4]),\n",
       "   (1, [2, 6, 1, 4, 8]),\n",
       "   (1, [3, 2, 8, 9, 7]),\n",
       "   (1, [1, 9, 7, 8, 4, 5, 2, 6, 3]),\n",
       "   (1, [8, 7, 9, 1, 2, 4, 3, 5, 6]),\n",
       "   (1, [9, 8, 7, 5, 1, 4, 3, 6, 2]),\n",
       "   (1, [7, 9, 8, 2, 5, 6, 3, 1, 4]),\n",
       "   (1, [8, 9, 7, 2, 3, 5, 4]),\n",
       "   (1, [2, 4, 7, 8, 6, 9, 5, 3, 1]),\n",
       "   ...])]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_,_,temp = list_of_votes[0]\n",
    "known = [tuple(x) for x in list(zip(*temp))[1]]\n",
    "pos = all_possible_votes(5)\n",
    "len(pos)\n",
    "# [x for x in pos if x not in known]\n",
    "# len(known)\n",
    "list_of_votes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "import collections\n",
    "#collections.OrderedDict(sorted(list_of_votes[0][1]))\n",
    "def default_to_sorted(dictdict):\n",
    "    lens_dict = dictdict\n",
    "    od = collections.OrderedDict(sorted(lens_dict.items()))\n",
    "    if 0 in od:\n",
    "        del(od[0])\n",
    "    return od"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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      "6900\n"
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      "2031\n",
      "8682\n"
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      "2114\n",
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      "2831\n",
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      "2367\n",
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    {
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     "output_type": "stream",
     "text": [
      "2172\n"
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    }
   ],
   "source": [
    "def squaredDif(a, b):\n",
    "#     if a == 0 and b == 0:\n",
    "#         return 0\n",
    "#     if a != 0:\n",
    "#         a = math.log(abs(a))\n",
    "\n",
    "    # a = math.log(a)\n",
    "    return (a - b)**2\n",
    "\n",
    "table = []\n",
    "col_names = ['Number_Votes','Mallow\\'s_Divergence','Plackett-Luce_Divergence']\n",
    "\n",
    "for i in tqdm_notebook(range(len(list_of_votes))):\n",
    "    num_votes, lengths, votes = list_of_votes[i]\n",
    "    print(num_votes)\n",
    "    lengths = default_to_sorted(lengths)\n",
    "    num_alternatives = len(lengths)\n",
    "    sigma, phi = mallows_params[i]\n",
    "    pl_weights = pl_params[i]\n",
    "    \n",
    "    delta_mallows = 0\n",
    "    delta_plackett = 0\n",
    "       \n",
    "    freq_list = list(zip(*votes))[0]\n",
    "    known_votes = [tuple(x) for x in list(zip(*votes))[1]]\n",
    "    # possibles = all_possible_votes(num_alternatives)\n",
    "    # empirical_zero = [x for x in possibles if x not in known_votes]\n",
    "    vote_in = 0\n",
    "    for vote in tqdm_notebook(all_possible_votes(num_alternatives)):\n",
    "        if (vote in known_votes):\n",
    "            vote_in += 1\n",
    "            index = known_votes.index(vote)\n",
    "            num_occurances = freq_list[index]\n",
    "            \n",
    "            empirical = num_occurances / num_votes\n",
    "            mallows = mallowsProb(vote, sigma, phi)\n",
    "            plackett = probPlackett(vote, pl_weights)\n",
    "            delta_mallows += squaredDif(mallows, empirical)\n",
    "            delta_plackett += squaredDif(plackett, empirical)\n",
    "            # print(vote)\n",
    "        else:\n",
    "            empirical = 0\n",
    "            mallows = mallowsProb(vote, sigma, phi)\n",
    "            plackett = probPlackett(vote, pl_weights)\n",
    "            # print(mallows, plackett)\n",
    "            delta_mallows += squaredDif(mallows, empirical)\n",
    "            delta_plackett += squaredDif(plackett, empirical)\n",
    "    print(vote_in)\n",
    "#     for entry in votes:\n",
    "#         num_occurances, vote = entry\n",
    "#         empirical = num_occurances / num_votes\n",
    "#         mallows = mallowsProb(vote, sigma, phi)\n",
    "#         plackett = probPlackett(vote, pl_weights)\n",
    "#         delta_mallows += squaredDif(mallows, empirical)\n",
    "#         delta_plackett += squaredDif(plackett, empirical)\n",
    "    \n",
    "    table.append([num_votes, delta_mallows, delta_plackett])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Number_Votes</th>\n",
       "      <th>Mallow's_Divergence</th>\n",
       "      <th>Plackett-Luce_Divergence</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>6900.0</td>\n",
       "      <td>28241.038542</td>\n",
       "      <td>50.404369</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>8682.0</td>\n",
       "      <td>30140.875207</td>\n",
       "      <td>47.690520</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>9334.0</td>\n",
       "      <td>15228.450599</td>\n",
       "      <td>35.152489</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>9901.0</td>\n",
       "      <td>14887.191457</td>\n",
       "      <td>37.450739</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>8654.0</td>\n",
       "      <td>29305.686858</td>\n",
       "      <td>48.567345</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>12744.0</td>\n",
       "      <td>30721.945313</td>\n",
       "      <td>76.856822</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>9567.0</td>\n",
       "      <td>23401.130750</td>\n",
       "      <td>47.750907</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>8738.0</td>\n",
       "      <td>29799.859947</td>\n",
       "      <td>61.462438</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Number_Votes  Mallow's_Divergence  Plackett-Luce_Divergence\n",
       "1        6900.0         28241.038542                 50.404369\n",
       "2        8682.0         30140.875207                 47.690520\n",
       "3        9334.0         15228.450599                 35.152489\n",
       "4        9901.0         14887.191457                 37.450739\n",
       "5        8654.0         29305.686858                 48.567345\n",
       "6       12744.0         30721.945313                 76.856822\n",
       "7        9567.0         23401.130750                 47.750907\n",
       "8        8738.0         29799.859947                 61.462438"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "npdata = np.array(table)\n",
    "results_df = pd.DataFrame(data=npdata,columns=col_names)\n",
    "results_df.index += 1\n",
    "# pickle.dump(results_df, open('pickle/divergence10k.p','wb'))\n",
    "results_df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "perms = list(itertools.permutations([1,2,3]))\n",
    "print(perms)\n",
    "(1,2,3) in perms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pickle.dump([mallows_params, pl_params, table], open('./pickle/ireland_0001_table_10k.p', 'wb'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mallows_params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pl_params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "122.55 \t -5.62\n",
      "178.23 \t -6.37\n",
      "2893.52 \t -11.94\n",
      "4990.13 \t -13.03\n",
      "207.46 \t -6.67\n",
      "476.65 \t -8.33\n",
      "197.95 \t -6.58\n",
      "101.64 \t -5.24\n"
     ]
    }
   ],
   "source": [
    "mallows_debian = [122.55,178.23,2893.52,4990.13,207.46,476.65,197.95,101.64]\n",
    "plackett_debian = [7.68,12.24,26.22,36.26,50.27,12.20,7.78,35.42]\n",
    "num_alternatives = [4,5,7,8,9,5,4,8]\n",
    "\n",
    "def AIC(likelihood, k):\n",
    "    return (-2 * math.log(likelihood)) + (2*k)\n",
    "\n",
    "for l in mallows_debian:\n",
    "    aic = AIC(l, 2)\n",
    "    print(l,\"\\t\",\"%2.2f\"%aic)\n",
    "\n",
    "# for i in range(len(plackett_debian)):\n",
    "#     l = plackett_debian[i]\n",
    "#     aic = AIC(l, num_alternatives[i])\n",
    "#     print(l,\"\\t\",\"%2.2f\"%aic)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7.68 \t 12.56\n",
      "12.24 \t 23.91\n",
      "26.22 \t 60.14\n",
      "36.26 \t 85.66\n",
      "50.27 \t 116.38\n",
      "12.2 \t 23.92\n",
      "7.78 \t 12.53\n",
      "35.42 \t 85.70\n"
     ]
    }
   ],
   "source": [
    "def BIC(likelihood, k, n):\n",
    "    return k*math.log(n) - 2*math.log(likelihood)\n",
    "\n",
    "# for i in range(len(mallows_debian)):\n",
    "#     l = mallows_debian[i]\n",
    "#     k = 2\n",
    "#     n = len(all_possible_votes(num_alternatives[i]))\n",
    "#     bic = BIC(l, k, n)\n",
    "#     print(l,'\\t','%2.2f'%bic)\n",
    "    \n",
    "for i in range(len(plackett_debian)):\n",
    "    l = plackett_debian[i]\n",
    "    k = num_alternatives[i]\n",
    "    n = len(all_possible_votes(num_alternatives[i]))\n",
    "    bic = BIC(l, k, n)\n",
    "    print(l,'\\t','%2.2f'%bic)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "my_list = []\n",
    "my_list.append(5.3)\n",
    "\n",
    "# del(my_list)\n",
    "my_list.clear()\n",
    "my_list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[20.88708072075542,\n",
       " 24.841886817329524,\n",
       " 23.909707357480325,\n",
       " 22.38608095497758,\n",
       " 21.737695500995276,\n",
       " 21.084308956239717,\n",
       " 23.916254008170263,\n",
       " 24.816013235268084,\n",
       " 21.78457264693931,\n",
       " 23.88857753883989,\n",
       " 23.906442050632556,\n",
       " 23.93106279651868,\n",
       " 23.9476466394649,\n",
       " 23.94431885167551,\n",
       " 23.952648726274077,\n",
       " 23.904811394203044,\n",
       " 23.921178097649815,\n",
       " 23.921178097649815,\n",
       " 23.94431885167551,\n",
       " 23.921178097649815,\n",
       " 23.92446757207557,\n",
       " 23.95097997356984,\n",
       " 21.07914356149404,\n",
       " 21.189682447028417,\n",
       " 21.799876675135337,\n",
       " 21.67311249172223,\n",
       " 21.153113798933376,\n",
       " 20.23515474137529,\n",
       " 21.187167777795207,\n",
       " 20.682352798244008]"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PL_APAUK = [55.48,7.68,12.24,26.22,36.26,50.27,12.2,7.78,35.42,12.37,12.26,12.11,12.01,12.03,11.98,12.27,12.17,12.17,12.03,12.17,12.15,11.99,50.40,47.69,35.15,37.45,48.57,76.86,47.75,61.46]\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "bic_li = []\n",
    "aic_li = []\n",
    "\n",
    "for i in range(len(PL_APAUK)):\n",
    "    l = PL_APAUK[i]\n",
    "    k = 5# num_alternatives[i]\n",
    "    n = len(all_possible_votes(k))\n",
    "    bic = BIC(l, k, n)\n",
    "    bic_li.append(bic)\n",
    "    aic = AIC(l, k)\n",
    "    aic_li.append(aic)\n",
    "    \n",
    "bic_li"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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