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 wargames and 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").
For example, if a game would call for a roll of
1d4 this would mean, "roll one 4-sided die."
If the final number is omitted, it is typically assumed to be a six, but in some contexts, other defaults are used.
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 ), while a roll of 18 results if any three dice are 6 (probability = ).
Miniatures wargamers began using dice in the shape of Platonic solids in the late 1960s and early '70s, to obtain results that could not easily be produced on a conventional six-sided die. Dungeons & Dragons emerged in this milieu, and was the first game with widespread commercial availability to use such dice. In its earliest edition, D&D had no standardized way to call for polyhedral die rolls or to refer to the results of such rolls. In some places the text gives a verbal instruction; in others, it only implies the roll to be made by describing the range of its results. For example, the spell sticks to snakes says, "From 2-16 snakes can be conjured (roll two eight-sided dice)." When only a range is listed, the exact method of rolling can be ambiguous. For example, a typical random wilderness encounter might be a village of "30-300" orcs. A number in that range might be generated by rolling
3d10×10, or alternately by rolling
D&D player Ted Johnstone introduced standard dice notation as a way to discuss probability distribution in an article, "Dice as Random Number Generators," in the inaugural issue of fanzine Alarums & Excursions (1975). The notation was also used by another writer, Barry Gold, in the same issue, and quickly spread throughout the fan community. Eventually, standard dice notation became so deeply ingrained in D&D fan culture that Gary Gygax would adopt it as a commonplace in the first edition of Advanced Dungeons & Dragons. The close association between D&D fandom and standard dice notation is reflected in the name of the Open Game version of the D&D rules: the "d20 System."
In some games, the above notation is expanded to allow for a multiplier, as in
5×d6means "roll one 6-sided die, and multiply the result by 5."
3d6×10+3means "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):
3x(2d6+4)means "roll two 6-sided dice adding four to result, repeat the roll 3 times adding the results together."
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. Ten-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.
d1000 (using three 10-sided dice) is occasionally also seen, although it is more common in wargames than role-playing games.
Before the introduction of ten-sided dice around 1980, twenty-sided dice were commonly manufactured with two copies of each digit 0 to 9 for use as percentile dice. (Half could be given a distinct color, indicating the addition of ten, for use when randomizing numbers from 1 to 20.) 
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, as in
4d6-L, which means roll 4 times a 6-sided dice and drop the lowest value,respectively.
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.
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 (
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" and "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 or failure) and
1d6 (for determining degree of success or failure). The notation of
d666 is a reference to The Number of the Beast.
Planet Mercenary calls its variation
d6³, to indicate that in addition to using the conventional sum of
3d6 to check for success or failure, various secondary effects are determined via comparison of the individual numbers rolled. These include whether a specially marked die (called the Mayhem die) has rolled highest, the lowest number rolled, and whether any two dice show the same number.
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.
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:
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.