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

File:Roll of 3d6.svg

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" (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 each die.

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 probability curve towards the higher numbers, as a result a roll 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 dice, 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 the letter, x, rather than the 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%)

File:D100.jpg

A typical pair of percentile dice including a "tens" die

Often, the variable X in the above notation will be 100, alternatively written "%". Although 100-sided dice exist, it is more common to use a combination of two ten-sided dice known as "percentile dice". One die represents units and the other tens; typically these are distinguished by color, but dice marked with multiples of ten are also available for use as the "tens" die. Twenty-sided dice intended specifically for use as percentile dice typically have no tens notation (the faces are numbered such that there are two complete sequences of 0 through 9). A roll of 0 on both dice may be interpreted as either 0 or 100, depending on the game rules.

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 (k) from the roll. Whether the dice omitted are the highest, lowest, or the player's choice depends on the game in question. 7th Sea and Legend of the Five Rings use only 10-sided dice, with notation of the form 8k6, meaning "Roll eight ten-sided dice, keep the highest six, and sum them." Although using a Roll and keep system, Cortex Plus games all use roll all the dice of different sizes and keep two (normally the two best) although a Plot Point may be spent to keep an additional dice, and some abilities let you keep a third automatically.

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.

Dice pools

Main article: Dice pool

File:White Fudge Dice.jpg

Fudge dice

A number of games including the original Ghostbusters role-playing game, the Storyteller system, and Fantasy Flight Games' Star Wars Roleplaying Games use a system where a dice pool consisting of an indicated number of dice are rolled and the total number of dice which meet a fixed condition are recorded as the result. For example, Vampire: the Requiem has players roll a pool of ten sided dice and note the number that come up as 8 or higher as "successes". Some companies produce custom dice, marked with successes and failures, for use in games which use this mechanic.

The Fudge role-playing system uses a set of dice which are each 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.

6-sided variations

Various Games Workshop systems such as Necromunda and Mordheim use an anomalously named D66 roll, meaning d6×10+d6. There are 36 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, often made distinguishable from each other by color, 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. The mean result of the D66 is 38.5, and the standard deviation about 17.16.

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.

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 mean result of d10x is 30.25 and its standard deviation is about 23.82.

Open-ended variations

Several games use mechanics that allow one or more dice to be rerolled (often a die that rolls the highest possible number), with each successive roll being added to the total. Terms for this include open-ended rolling, exploding dice, and penetration rolls. Games that use such a system include Feng Shui, and Savage Worlds. On Anydice the function to make dice explode on their highest value is called quite simply explode. Another common notation shorthand for exploding dice is to use an exclamation point: AdX! or 6d6!.

The Storyteller system combines exploding dice with a dice pool threshold and target number. Diana: Warrior Princess explodes all successes, and Hackmaster uses a variant called dice penetration by which 1 is subtracted from the total of the rerolled dice.