Project euler 90

project_euler/problem_090/sol1.py

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+"""
+Project Euler Problem 90: https://projecteuler.net/problem=90
+Cube digit pairs:
+
+Each of the six faces on a cube has a different digit (0 to 9) written on it;
+the same is done to a second cube.
+In fact, by carefully choosing the digits on both cubes it is possible to display
+all of the square numbers below one-hundred: 01, 04, 09, 16, 25, 36, 49, 64, and 81.
+For example, one way this can be achieved is by placing {0, 5, 6, 7, 8, 9} on one cube
+and {1, 2, 3, 4, 8, 9} on the other cube.
+
+However, for this problem we shall allow the 6 or 9 to be turned upside-down so that
+an arrangement like {0, 5, 6, 7, 8, 9} and {1, 2, 3, 4, 6, 7} allows for all
+nine square numbers to be displayed; otherwise it would be impossible to obtain 09.
+In determining a distinct arrangement we are interested in the digits on each cube,
+not the order.
+{1, 2, 3, 4, 5, 6} is equivalent to {3, 6, 4, 1, 2, 5}
+{1, 2, 3, 4, 5, 6} is distinct from {1, 2, 3, 4, 5, 9}
+
+But because we are allowing 6 and 9 to be reversed, the two distinct sets
+in the last example both represent the extended set {1, 2, 3, 4, 5, 6, 9}
+for the purpose of forming 2-digit numbers.
+How many distinct arrangements of the two cubes allow for all of the
+square numbers to be displayed?
+"""
+
+from itertools import permutations, combinations
+
+
+def set_to_bit(t: tuple) -> int:
+    """
+    returns bit representation of a given iterable, preferably tuple.
+    @param t - tuple representing the 1 bits
+    >>> set_to_bit((0,1))
+    3
+    >>> set_to_bit((1,3))
+    10
+    """
+    res = 0
+    for i in t:
+        res |= 1 << i
+    return res
+
+
+def is_bit_set(n: int, c: int) -> bool:
+    """
+    checks if a given bit is 1 in the given number (work around for 6 and 9)
+    @param n - the number/set to search in
+    @param c - the index to look for
+    >>> is_bit_set(10, 1)
+    True
+    >>> is_bit_set(64, 9)
+    True
+    """
+    if c == 6 or c == 9:
+        return bool((1 << 6 & n) or (1 << 9 & n))
+    return bool(1 << c & n)
+
+
+def validate_cubes(cubes: tuple, sq: list):
+    """
+    verifies whether or not the selected combination of cubes is valid,
+    by iterating through all square values.
+    @param cubes - tuple of cubes represented by numbers (having six 1 bits (0-9))
+    @param sq - list of squares to validate
+    """
+    for s in sq:
+        res = False
+        for p in permutations(s):
+            cur_res = True
+            for i, c in enumerate(map(int, p)):
+                cur_res = cur_res and is_bit_set(cubes[i], c)
+            res = res or cur_res
+        if not res:
+            return False
+    return True
+
+
+def solution(n: int = 9, m: int = 2) -> int:
+    """
+    returns the solution of problem 90 using helper functions
+    e.g 1217 for the default argument values
+
+    @param n - the number of squares to validate.
+    @param m - the number of dice to use.
+
+    >>> solution(3, 1)
+    55
+    >>> solution(7, 2)
+    2365
+    """
+    sq = [str(i ** 2).zfill(m) for i in range(1, n + 1)]
+    all_dices = [set_to_bit(c) for c in combinations(range(10), 6)]
+    dices = [p for p in combinations(all_dices, m)]
+
+    res = 0
+    for d in dices:
+        if validate_cubes(d, sq):
+            res += 1
+    return res
+
+
+if __name__ == "__main__":
+    print(f"{solution()}")