Alea

An RPG dice notation language and dice rolling engine.

Dice Notation

Basics

Dice expressions let you describe rolling multiple dice of different sizes, and modifying the results. You can:

• Roll dice with any number of sides (dS)
• Add or subtract a constant value from the results (dS ± C)
• Roll multiple dice with the same number of sides and add their results together as a pool (NdS)
• Add or subtract multiple pools and constants together (NdS + MdT + C)

Operation	Expression	Interpretation	Example	Meaning	Notes
Roll	dS	Roll a die with S sides.	d20	Roll one d20.	Equivalent to Roll Many: 1dS. Equivalent to Range Die: d{1..20}.
Addition	dS + C	Roll a die with S sides, and add C to the result.	d20 + 1	Roll one d20 and add 1 to the result.
Subtraction	dS - C	Roll a die with S sides, and subtract C from the result.	d20 - 2	Roll one d20 and subtract 2 from the result.
Roll Many	NdS	Roll N dice with S sides and add the results.	3d6	Roll 3 d6 and add the results.
Roll Different	NdS + MdT + C	Roll N dice with S sides, M dice with T sides, and add all together plus C.	3d6 + d4 + 2	Roll 3 d6, one d4, and add 2 to the results.

Roll Modifiers

Roll modifiers let you describe rolling a pool of dice, and doing something with the results before factoring them into the calculation.

Available modifiers are:

• Advantage: (+) lets you treat each die rolled as if you had rolled it twice, and taken the better roll.
• Disadvantage (-): lets you treat each die rolled as if you had rolled it twice, and taken the worse roll.
• Maximize (M) lets you make the highest possible rolls for a given pool.
• Minimize (m) lets you make the lowest possible rolls for a given pool.
• Explode (!): lets you roll extra dice into a pool before adding them together, based on previous rolls.
• Keep (K): lets you only hold on to certain results in a pool before adding them together.
• Drop (D): lets you discard certain results from a pool before adding them together.
• Count (C) lets you change a dice pool's output from being a sum of dice rolls, to the count of rolled dice that meet a certain criteria.

Combining Modifiers

You can combine multiple modifiers in any order without restriction, except for Count, which must be last and can only be used at most once.

You can think of the modifiers as belonging these different groups:

• Advantage, Disadvantage, Maximize, and Minimize let you change how you roll the dice in a pool.
• Keep and Drop, and Explode let you look at what you've rolled so far and modify the pool before you proceed.
• Count lets you change how you combine results at the end of rolling a pool.

Since this is the order of resolution, this is the conventional order that they are written in expressions.

In practice, it is rare to use more than one modifier from each category.

Interactions

Maximize and Minimize override each other and Advantage and Disadvantage, so it never makes sense to repeat them or combine with Advantage and Disadvantage.

Exploded dice add new, unrolled dice to the pool. These dice do not keep the Advantage, Disadvantage, Maximize, and Minimize modifiers of the original pool; you must apply them again after the explosion modifier if you want them to apply to exploded die. For example:

• Maximize then Explode will maximize the originally rolled dice, but not the explosions.
• Explode then Maximize will only maximize the explosion dice.
• To maximize both in a d20 roll, you would need to write d20M!M.

Advantage and Disadvantage can be repeated, but use-cases for this are rare. For example, a d20+- rolled with advantage, then disadvantage, will:

• roll 2d20 and keep the highest result
• roll another 2d20 and keep the highest result
• then keep the lowest result of the two

Keep and Drop can be repeated in various orders to filter dice from the pool. Again, use-cases are rare.

Keep

Keep (K) lets you only hold on to certain dice in a pool before adding them together.

By default, random dice are kept. This can be further modified to:

• Keep the highest rolls (Keep Highest: KH)
• Keep the lowest rolls (Keep Lowest: KL)
• Explicitly keep random rolls Keep Random: (KR)

Operation	Expression	Interpretation	Example	Meaning	Notes
Keep	NdSK	Roll N dice with S sides, and keep a random die.	2d20K	Roll 2 d20 and keep one at random.	Equivalent to Keep Random: NdSK1R.
Keep Many	NdSKM	Roll N dice with S sides, and keep M random dice.	3d20K2	Roll 3 d20 and keep 2 at random.	Equivalent to Keep Random: NdSKMR.
Keep Random	NdSKMR	Roll N dice with S sides, and keep M random dice.	3d20K2R	Roll 3 d20 and keep 2 at random.	Equivalent to Keep Many: NdSKM.
Keep Highest	NdSKH	Roll N dice with S sides, and keep the highest result.	2d20KH	Roll 2 d20 and keep the highest result.
Keep Many Highest	NdSKMH	Roll N dice with S sides, and keep the highest M results.	3d20K2H	Roll 3 d20 and keep the 2 highest.
Keep Lowest	NdSKL	Roll N dice with S sides, and keep the lowest result.	2d20KL	Roll 2 d20 and keep the lowest result.
Keep Many Lowest	NdSKML	Roll N dice with S sides, and keep the lowest M results.	3d20K2L	Roll 3 d20 and keep the 2 lowest.

Drop

Drop (D) lets you discard certain dice from a pool before adding them together.

By default, random dice are dropped. This can be further modified to:

• Drop the highest rolls (Drop Highest: DH)
• Drop the lowest rolls (Drop Lowest: DL)
• Explicitly drop random rolls (Drop Random: DR)

Operation	Expression	Interpretation	Example	Meaning	Notes
Drop	NdSD	Roll N dice with S sides, and drop a random die.	2d20D	Roll 2 d20 and drop one at random.	Equivalent to Drop Random: NdSD1R.
Drop Many	NdSDM	Roll N dice with S sides, and drop M random dice.	3d20D2	Roll 3 d20 and drop 2 at random.	Equivalent to Drop Random: NdSDMR.
Drop Random	NdSDMR	Roll N dice with S sides, and drop M random dice.	3d20D2R	Roll 3 d20 and drop 2 at random.	Equivalent to Drop Many: NdSKM.
Drop Highest	NdSDH	Roll N dice with S sides, and drop the highest result.	2d20DH	Roll 2 d20 and drop the highest result.
Drop Many Highest	NdSDMH	Roll N dice with S sides, and drop the highest M results.	3d20D2H	Roll 3 d20 and drop the 2 highest.
Drop Lowest	NdSDL	Roll N dice with S sides, and drop the lowest result.	2d20DL	Roll 2 d20 and drop the lowest result.
Drop Many Lowest	NdSDML	Roll N dice with S sides, and drop the lowest M results.	3d20D2L	Roll 3 d20 and drop the 2 lowest.

Explode

Explode (!) lets you roll extra dice into a pool before adding them together, based on previous rolls.

• Only the maximum possible roll triggers an explosion. This can be further modified by providing an explicit threshold at which to explode (Explode At: !N).
• An explosion adds another die of the same sides to the pool. This can be further modified by specifying a die to explode with (Explode With: !dT)
• Re-rolls do not trigger explosions. This can be further modified by specifying that re-rolls can keep exploding (Keep Exploding: !!, Keep Exploding At: !!N).

NOTE:

Keep Exploding (!!) will not trigger on 1-sided dice, or dice with a single value on all sides.

Keep Exploding At (!N) will not explode dice where no side is less than N.

Keep Exploding (!!) chains cannot be Maximized (M).

Other safeguards exist to ensure no explosion chain is allowed to run forever.

Operation	Expression	Interpretation	Example	Meaning	Notes
Explode	dS!	Roll one die with S sides, and if the die rolls S, roll another dS to the pool.	d20!	Roll one d20 and add another d20 roll to the pool if the first rolls 20.	Equivalent to Explode At: dS!S.
Explode At	dS!N	Roll one die with S sides, and if the die rolls higher than or equal to N, roll another dS to the pool.	d20!18	Roll one d20 and add another d20 roll to the pool if the first rolls 18 or higher.
Keep Exploding	dS!!	Roll one die with S sides, and if the die rolls S, roll another dS to the pool. If that die rolls S, repeat...	d20!!	Roll one d20 and add another d20 roll to the pool if the first rolls 20, and another if that rolls 20...	Equivalent to Keep Exploding At: dS!!S.
Keep Exploding At	dS!!N	Roll one die with S sides, and if the die rolls higher than or equal to N, roll another dS to the pool. If that die rolls S, repeat...	d20!!18	Roll one d20 and add another d20 roll to the pool if the first rolls 18 or higher, and another if that rolls 18 or higher...

Advantage/Disadvantage

Advantage (dS+) lets you treat each die rolled as if you had rolled it twice, and taken the better roll. Conversely, Disadvantage (dS-) takes the worse roll.

NOTE

This works differently from just rolling twice as many dice with Keep High/Keep Low, when rolling more than one die with Advantage/Disadvantage!

KH/KL will remove the worst/best rolls from the entire pool, whereas Advantage/Disadvantage treats each roll as a pool of two before proceeding, allowing more good/poor rolls to contribute to the total.

In practice, this means that NdS+ will have more variance and average lower results than 2NdSKHN, and NdS- will have more variance and average higher results than 2NdSKLN.

Operation	Expression	Interpretation	Example	Meaning	Notes
Advantage	dS+	Roll two dice with S sides, and take the better roll.	d20+	Roll two d20 and take the better result.	Equivalent to Keep Highest: 2dSKH.
Multiple Advantage	NdS+	Roll two dice with S sides, and take the better roll N times.	4d20+	Roll two d20 and take the better result, 4 times.	NOT equivalent to Keep Highest: 2NdSKHN! See NOTE above.
Disadvantage	dS-	Roll two dice with S sides, and take the worse roll.	d20-	Roll two d20 and take the worse result.	Equivalent to Keep Lowest: 2dSKL.
Multiple Disadvantage	NdS-	Roll two dice with S sides, and take the worse roll N times.	4d20-	Roll two d20 and take the worse result, 4 times.	NOT equivalent to Keep Highest: 2NdSKLN! See NOTE above.

Maximize/Minimize

Maximize (M) lets you make the highest possible rolls for a given pool. Conversely, Minimize (m) ensures every roll is the lowest possible.

Operation	Expression	Interpretation	Example	Meaning	Notes
Maximize	dSM	Roll one die with S sides, and assume you rolled S.	d20M	Ignore the dice roll and assume you rolled a 20.
Minimize	dSm	Roll one die with S sides, and assume you rolled a 1.	d20m	Ignore the dice roll and assume you rolled a 1.

Count

Count (C) lets you change a dice pool's output from being a sum of dice rolls, to the count of rolled dice that meet a certain criteria.

By default, all dice are counted. This can be further modified to:

• Count rolls higher than a number (Count Greater Than: C>)
• Count rolls higher than a number (Count Greater Than Or Equal To: C>=)
• Count rolls lower than a number (Count Lower Than: C<)
• Count rolls lower than a number (Count Lower Than Or Equal To: C<=)
• Count how many rolls were the maxiumum possible value (Count Maximum: CM)
• Count how many rolls were the minimum possible value (Count Minimum: Cm)
• Count how many rolls exploded (Count Explosions: C!)

Operation	Expression	Interpretation	Example	Meaning	Notes
TBD

Other Dice

Custom Numeric Dice

While dS syntax describes a die with S sides, with numbers 1-S on those sides, you can in fact model dice with any number of sides, with any values on those sides.

Die	Expression	Interpretation	Example	Meaning	Notes
Range Die	d[X..Y]	Roll one die with Y - X + 1 sides, with the integers Y, X, and all integers in between them on those sides.	d[-1..1]	Roll a 3-sided die, with numbers -1, 0, and 1 on those sides.	Equivalent to List Die: d{-1, 0, 1}.
List Die	d{X, Y, Z}	Roll a die with the integers specified in the list on each side.	d{1, 1, 100}	Roll a die with a 2/3 chance of coming up 1, and 1/3 chance coming up 100.

Alias Dice

Common custom die have shortcut aliases for easier use.

Die	Expression	Interpretation	Notes
Percent Die	d%	Roll a d100 to represent a probabilistic outcome.	Equivalent to Roll: d100.
Standard Fudge Die	dF	Roll a 3-sided die with values -1, 0, and 1 on the sides.	Equivalent to Range Die: d[-1..1].
Fudge Die	dFN	Roll a 2N + 1-sided die with values between -S and S, including 0.	Equivalent to Range Die: d[-S..S].