Dice notation
Encyclopedia
Dice notation is a system to represent different combinations of dice
in role-playing games using simple algebra
-like notation such as
In most role-playing games, die rolls required by the system are given in the form AdX. A and X are variables, separated by the letter "d", which stands for die or dice. The letter "d" is most commonly lower-case
, but some notation uses upper-case "D" (non-english texts can use the equivalent form of the first letter of the given language's word for "dice", but also often use the English "d").
If the final number is omitted, it is typically assumed to be a six, but in some contexts, other defaults are used.
For example, if a game would call for a roll of
To this basic notation, an additive modifier can be appended, yielding expressions of the form, AdX+B. The plus is sometimes replaced by a minus sign ("−") to indicate subtraction. B is a number to be added to the sum of the rolls. So,
Rolling three or more dice gives a probability distribution
that is approximately Gaussian, in accordance with the central limit theorem
.
For example,
Multiplication can also mean repeating throws of similar setup (usually represented by x rather than multiplication symbol):
does exist, it is both more common and more uniformly random to use a combination of two ten-sided dice known as "percentile dice", where the "10" is replaced with a "0"; one die represents units and the other (typically distinguished by color) tens. Depending on the game, a roll of two zeroes can mean 100 or zero.
The d1000 (using three 10-sided dice) is occasionally also seen, although it is more common in wargames
than role-playing games.
Some games extend the standard notation to AdX(kY)+B where, in addition to the above, Y is the number of dice kept from the roll. Whether the dice omitted are the highest, lowest, or the player's choice depends on the game in question. 7th Sea only uses 10-sided dice, and uses notation of the form 8k6, meaning "Roll eight ten-sided dice, keep the highest six, and sum them."
An alternative notation used by the OpenRoleplaying.org die roller allows the use of a plus or minus followed by L or H instead of the modifier B, to denote dropping or re-adding the lowest or highest roll on a single die, respectively.
systems such as Necromunda
and Mordheim
use an anomalously named D66 roll, meaning d6×10+d6. There are (somewhat confusingly) 36, not 66, different possible results, ranging from 11 to 66. The D66 is a base six variant of the base ten Percentile die(d100). The D66 is generally a combination of two six-sided dice (or D6s), often made distinguishable from each other by colour, or simply one die rolled twice. The first die represents the tens digit, and the second die the ones digit. For example, a roll of 1 followed by a roll of 5 will give a total of 15, while a roll of 3 followed by a roll of 6 will give a total of 36.
Blood Bowl
, also a Games Workshop product, introduces the block die with special notation Xdb or roll block dice X times, blocker/defender chooses (if more than 1 die) with X being one of 3, 2, 1 (usually omitted), -2, -3. Alternatively words for & against can be used to describe a Xdb (in this case X > 0). As an example 2db against is equal to -2db which are both short ways of saying 2 dice block, defender chooses from the results rolled.
In Nomine
, a game about Angels and Demons from Steve Jackson Games
, uses a three-dice variation called the d666. However, this is actually a combination of 2d6 (for determining success / failure) and 1d6 (for determining degree of success / failure). The notation of d666 is a reference to The Number of the Beast.
The FUDGE role-playing system uses a die marked with minus signs, plus signs and blank sides, meaning -1, +1 and 0 respectively. The default is one third of each, usually represented by a six-sided die with two of each, known as dF.2 or just dF. Four of these (4dF) are rolled to determine results from -4 to +4, which is equivalent to 4d3-8.Variants include dF.1, which is a six-sided die with four blanks, one plus and one minus.
role-playing game by Gary Gygax
uses a dice mechanic called d10x. This is equivalent to d10×d10 and gives a non-linear distribution, with most results concentrated at the lower end of the range.
The 7th Sea and Legend of the Five Rings
role-playing games by Alderac Entertainment Group
uses a system called Roll and Keep. The notation is given as XkY, where X stands for the number of dice to roll, adding the values of Y dice you keep and comparing that to the TN (target number). This is coupled with exploding dice, a system where any die coming up 10 (normally) is rerolled, adding 10 to the result. Dice can "explode" indefinitely, making for extremely large rolls; for instance, if the same die comes up 10 three times in a row and then comes up 7, it counts as one die coming up 37.
The Talarius Gaming System
by Fantasy Makers Industries uses a system of open-ended, indefinite d100 (d%) roll, where one degree of success (or "one success") is accounted for every 10 points below a specified score or number. This is expanded with the open-ended roll, where any time the roll comes up within the 01–05 range, a reroll is allowed for additional successes. The open-ended roll will continue forever if a player were to continue rolling 01-05, gaining an extreme number of successes.
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...
in role-playing games using simple algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
-like notation such as
2d6+12.Standard notation
Letter case
In orthography and typography, letter case is the distinction between the larger majuscule and smaller minuscule letters...
, but some notation uses upper-case "D" (non-english texts can use the equivalent form of the first letter of the given language's word for "dice", but also often use the English "d").
- A is the number of dice to be rolled (usually omitted if 1).
- X is the number of faces of the dice
If the final number is omitted, it is typically assumed to be a six, but in some contexts, other defaults are used.
For example, if a game would call for a roll of
d4 or 1d4 this would mean, "roll one 4-sided die."3d6 would mean, "roll three six-sided dice." Commonly, these dice are added together, but some systems could direct the player use them in some other way, such as choosing the best die rolled.To this basic notation, an additive modifier can be appended, yielding expressions of the form, AdX+B. The plus is sometimes replaced by a minus sign ("−") to indicate subtraction. B is a number to be added to the sum of the rolls. So,
1d20-10 would indicate a roll of a single 20-sided die with 10 being subtracted from the result. These expressions can also be chained (e.g. 2d6+1d8), though this usage is less common. Additionally, notation such as AdX-L is not uncommon, the "L" (or "H", less commonly) being used to represent "the lowest result" (or "the highest result"). For instance, 4d6-L means a roll of 4 six-sided dice, dropping the lowest result. This application skews the bell curve of probability towards the higher numbers, as a result of 3 can only occur when all four dice come up 1 (probability 1/1296), while a roll of 18 results if any three dice are 6 (probability 21/1296 = 7/432).Rolling three or more dice gives a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
that is approximately Gaussian, in accordance with the central limit theorem
Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
.
Multipliers
In some games, the above notation is expanded to allow for a multiplier, as in AdX×C or C×dX, where:- × denotes multiplication, and can be replaced by "/" or "÷" for division.
- C is a natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
(1 if omitted).
For example,
- 1d6×5 or 5×d6 means "roll one 6-sided die, and multiply the result by 5."
- 3d6×10+3 means "roll three 6-sided dice, add them together, multiply the result by 10, and then add 3."
Multiplication can also mean repeating throws of similar setup (usually represented by x rather than multiplication symbol):
- 3 x (2d6+4) means "roll two 6-sided dice adding four to result, repeat the roll 3 times adding the results together."
Percentile dice (d%)
Often, the variable X in the above notation will be 100, alternatively written "%". Although a 100-sided dieZocchihedron
Zocchihedron is the trademark of a 100-sided die invented by Lou Zocchi, which debuted in 1985. Rather than being a polyhedron, it is more like a ball with 100 flattened planes...
does exist, it is both more common and more uniformly random to use a combination of two ten-sided dice known as "percentile dice", where the "10" is replaced with a "0"; one die represents units and the other (typically distinguished by color) tens. Depending on the game, a roll of two zeroes can mean 100 or zero.
The d1000 (using three 10-sided dice) is occasionally also seen, although it is more common in wargames
Wargaming
A wargame is a strategy game that deals with military operations of various types, real or fictional. Wargaming is the hobby dedicated to the play of such games, which can also be called conflict simulations, or consims for short. When used professionally to study warfare, it is generally known as...
than role-playing games.
Selective results
A number of notational strategies exist for discarding only certain types of results.Some games extend the standard notation to AdX(kY)+B where, in addition to the above, Y is the number of dice kept from the roll. Whether the dice omitted are the highest, lowest, or the player's choice depends on the game in question. 7th Sea only uses 10-sided dice, and uses notation of the form 8k6, meaning "Roll eight ten-sided dice, keep the highest six, and sum them."
An alternative notation used by the OpenRoleplaying.org die roller allows the use of a plus or minus followed by L or H instead of the modifier B, to denote dropping or re-adding the lowest or highest roll on a single die, respectively.
6-sided variations
Various Games WorkshopGames Workshop
Games Workshop Group plc is a British game production and retailing company. Games Workshop has published the tabletop wargames Warhammer Fantasy Battle and Warhammer 40,000...
systems such as Necromunda
Necromunda
Necromunda is a skirmish tabletop war game produced by Specialist Games .In Necromunda, players control rival gangs battling each other in the Underhive, a place of anarchy and violence in the depths below the Hive City...
and Mordheim
Mordheim
Mordheim is a tabletop game produced by Games Workshop. It is a skirmish variant of the company's popular Warhammer Fantasy game set on a smaller scale with players fielding 15-20 models each, rather than the 150-200 models allowed in Warhammer Fantasy....
use an anomalously named D66 roll, meaning d6×10+d6. There are (somewhat confusingly) 36, not 66, different possible results, ranging from 11 to 66. The D66 is a base six variant of the base ten Percentile die(d100). The D66 is generally a combination of two six-sided dice (or D6s), often made distinguishable from each other by colour, or simply one die rolled twice. The first die represents the tens digit, and the second die the ones digit. For example, a roll of 1 followed by a roll of 5 will give a total of 15, while a roll of 3 followed by a roll of 6 will give a total of 36.
Blood Bowl
Blood Bowl
Blood Bowl is a Fantasy Football game created by Jervis Johnson for the British games company Games Workshop as a parody of American Football. The game was first released in 1987 and has been re-released in new editions since...
, also a Games Workshop product, introduces the block die with special notation Xdb or roll block dice X times, blocker/defender chooses (if more than 1 die) with X being one of 3, 2, 1 (usually omitted), -2, -3. Alternatively words for & against can be used to describe a Xdb (in this case X > 0). As an example 2db against is equal to -2db which are both short ways of saying 2 dice block, defender chooses from the results rolled.
In Nomine
In Nomine (role-playing game)
In Nomine is a role-playing game designed by Derek Pearcy and published in 1997 by Steve Jackson Games, based on the French game In Nomine Satanis/Magna Veritas. Players typically assume the role of angels and demons in a setting that draws heavily on the traditional Christian mythos...
, a game about Angels and Demons from Steve Jackson Games
Steve Jackson Games
Steve Jackson Games is a game company, founded in 1980 by Steve Jackson, that creates and publishes role-playing, board, and card games, and the gaming magazine Pyramid.-History:...
, uses a three-dice variation called the d666. However, this is actually a combination of 2d6 (for determining success / failure) and 1d6 (for determining degree of success / failure). The notation of d666 is a reference to The Number of the Beast.
The FUDGE role-playing system uses a die marked with minus signs, plus signs and blank sides, meaning -1, +1 and 0 respectively. The default is one third of each, usually represented by a six-sided die with two of each, known as dF.2 or just dF. Four of these (4dF) are rolled to determine results from -4 to +4, which is equivalent to 4d3-8.Variants include dF.1, which is a six-sided die with four blanks, one plus and one minus.
10-sided variations
The Cyborg CommandoCyborg Commando
Cyborg Commando is a science fiction role-playing game published by New Infinities Productions, Inc in 1987 and designed by Kim Mohan and Frank Mentzer based on an outline by Gary Gygax, the creator of the original Dungeons & Dragons system....
role-playing game by Gary Gygax
Gary Gygax
Ernest Gary Gygax was an American writer and game designer best known for co-creating the pioneering role-playing game Dungeons & Dragons with Dave Arneson. Gygax is generally acknowledged as the father of role-playing games....
uses a dice mechanic called d10x. This is equivalent to d10×d10 and gives a non-linear distribution, with most results concentrated at the lower end of the range.
The 7th Sea and Legend of the Five Rings
Legend of the Five Rings
Legend of the Five Rings is a fictional setting created by the John Wick and published by Alderac Entertainment Group in 1995. The setting primarily involves the fictional country of Rokugan, though some additional areas and cultures have been discussed. Rokugan is based roughly on feudal Japan...
role-playing games by Alderac Entertainment Group
Alderac Entertainment Group
Alderac Entertainment Group, or AEG, is a publisher of role-playing game and collectible card game products. AEG was formed by Jolly Blackburn in 1993 and is based in the city of Ontario, California...
uses a system called Roll and Keep. The notation is given as XkY, where X stands for the number of dice to roll, adding the values of Y dice you keep and comparing that to the TN (target number). This is coupled with exploding dice, a system where any die coming up 10 (normally) is rerolled, adding 10 to the result. Dice can "explode" indefinitely, making for extremely large rolls; for instance, if the same die comes up 10 three times in a row and then comes up 7, it counts as one die coming up 37.
The Talarius Gaming System
Talarius gaming system
Published by Fantasy Makers Industries the d100 System is a role-playing game system. The system is primarily used for standard pen-and-paper role-playing games, it is also used in miniature battle games also published by Fantasy Makers Industries...
by Fantasy Makers Industries uses a system of open-ended, indefinite d100 (d%) roll, where one degree of success (or "one success") is accounted for every 10 points below a specified score or number. This is expanded with the open-ended roll, where any time the roll comes up within the 01–05 range, a reroll is allowed for additional successes. The open-ended roll will continue forever if a player were to continue rolling 01-05, gaining an extreme number of successes.