Dice notation

Dice notation (also known as dice algebra, common dice notation, RPG dice notation, and several other titles) is a system to represent different combinations of dice in role-playing games using simple algebra-like notation such as 2d6+12.

Standard notation

Probability mass function of a roll of 3d6 and probability density function of the normal distribution with the same mean and standard deviation.

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"^[1] (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 that is approximately Gaussian, in accordance with the central limit theorem.

Variations and expansions

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 number (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 die 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.

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.^[2]

6-sided variations

Various Games Workshop 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.

10-sided variations

The Cyborg Commando 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.

References

1. ^ "Standard Dice Notation". dice-play. 2006-04-06. http://homepage.ntlworld.com/dice-play/Notation.htm. 
2. ^ OpenRoleplaying.org – Gaming Tools^{[dead link]}

• Winter, Steve (2007-03-02). "A Brief History of Dice". Wizards of the Coast. http://www.wizards.com/default.asp?x=dnd/alumni/20070302a. Retrieved 2007-03-06.