Dice Notation
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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 Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing g ...
in
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s and
tabletop role-playing game A tabletop role-playing game (typically abbreviated as TRPG or TTRPG), also known as a pen-and-paper role-playing game, is a form of role-playing game (RPG) in which the participants describe their characters' actions through speech. Participa ...
s using simple
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
-like notation such as 2d6+12.


Standard notation

In most
tabletop role-playing game A tabletop role-playing game (typically abbreviated as TRPG or TTRPG), also known as a pen-and-paper role-playing game, is a form of role-playing game (RPG) in which the participants describe their characters' actions through speech. Participa ...
s, 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 Letter case is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written representation of certain languages. The writing ...
, but some forms of notation use 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 dice. For example, if a game calls for a roll of d4 or 1d4, it means "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 sign 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 = ). Rolling three or more dice gives a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
that is approximately
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
, in accordance with the
central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
.


History

Miniatures wargamers began using dice in the shape of
Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
in the late 1960s and early ’70s, to obtain results that could not easily be produced on a conventional six-sided die. ''
Dungeons & Dragons ''Dungeons & Dragons'' (commonly abbreviated as ''D&D'' or ''DnD'') is a fantasy tabletop role-playing game (RPG) originally designed by Gary Gygax and Dave Arneson. The game was first published in 1974 by TSR (company)#Tactical Studies Rules ...
'' emerged in this milieu, and was the first game with widespread commercial availability to use such dice. In its earliest edition (1974), 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 30d10. D&D player Ted Johnstone introduced standard dice notation as a way to discuss
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
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 Ernest Gary Gygax ( ; July 27, 1938 – March 4, 2008) was an American game designer and author best known for co-creating the pioneering role-playing game ''Dungeons & Dragons'' (''D&D'') with Dave Arneson. In the 1960s, Gygax created an ...
would adopt it as a commonplace in the first edition of ''
Advanced Dungeons & Dragons Several different editions of the ''Dungeons & Dragons'' (''D&D'') fantasy role-playing game have been produced since 1974. The current publisher of ''D&D'', Wizards of the Coast, produces new materials only for the most current edition of the ga ...
'' (1977-1979). The close association between D&D fandom and standard dice notation is reflected in the name of the
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version of the D&D rules: the "'' d20 System''."


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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
(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 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."


Percentile dice (d%)

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; however, it is rare for the 0 on the ones die to be read as 10, making a roll of zero on both dice equal to 10 (0 tens, and 10 ones). The d1000 (using three 10-sided dice) is occasionally also seen, although it is more common in
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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.)


Selective results

A number of notational strategies exist for discarding only certain types of results. Some games extend the standard notation to A\textX\text(Y)+B or A\textX\textY+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; "k" can be replaced with "kh", "kl", or "kc", respectively, to represent this, which also allows for combining them. In this case, A\textX\textH\textL\textC+B, which means "roll A dice with X sides each, keeping the H highest, the L lowest, and C of the player's choice". Just as "B" is normally not shown when it is 0, terms where you would keep ''all'' of the dice are normally omitted for ease of reading. While "k" can be replaced with "d" to denote number of dice dropped instead of kept, this will result in two ds in the notation, so care must be kept (in particular, the number of sides can no longer be omitted unambiguously): A\textX\textY+B (here, the Y term is omitted when it is 0, meaning not dropping any dice), such as 4\text6\text1\text1, which means "roll 4 6-sided dice, dropping the highest and lowest"; this is preferable to keep notation, which has no way to convey keeping middle dice. 7th Sea and
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use only 10-sided dice, so it omits the number of sides, using notation of the form 8\text6, meaning "roll eight ten-sided dice, keep the highest six, and sum them."Although using a roll and keep system,
Cortex Plus The Cortex Plus System is a toolkit RPG system that evolved from Margaret Weis Productions, Ltd's Cortex System. It has been used for four published games and one published preview to date, and the design principles are in the ''Cortex Plus Hack ...
games all use roll all the dice of different sizes and keep two (normally the two best), although a
Plot Point In television and film, a plot point is any incident, episode, or event that "hooks" into the action and spins it around into another direction. Noted screenwriting teacher Syd Field discusses plot points in his paradigm, popularized in his book ...
may be spent to keep an additional die, and some abilities let you keep a third automatically. An alternative notation used by the OpenRoleplaying.org die roller uses "L" and "H" to denote the lowest and highest die, respectively, allowing ''B'' above to be replaced with this letter rather than a number; when used with a minus, this is equivalent to dropping the lowest or highest die, respectively, while when used with a plus, this lets you add the lowest or highest die to the total again, which is equivalent to doubling the lowest or highest die.


Dice pools

A number of games including the original ''Ghostbusters'' role-playing game, the
Storyteller system The ''Storytelling System'' is a role-playing game system created by White Wolf, Inc. for the Chronicles of Darkness (formerly known as the New World of Darkness), a game world with several pen and paper games tied in. The Storytelling System ...
, and
Fantasy Flight Games Fantasy Flight Games (FFG) is a game company based in Roseville, Minnesota, United States, that creates and publishes role-playing, board, card, and dice games. As of 2014, it is a subsidiary of Asmodée Éditions. History Fantasy Flight Publish ...
' 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
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. 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
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systems such as '' Necromunda'' and ''
Mordheim ''Mordheim'' is a tabletop game published by Games Workshop in 1999. It is a variant of the company's Warhammer Fantasy game set on a warband or "skirmish" scale. Mordheim was designed by Alessio Cavatore, Tuomas Pirinen, and Rick Priestley. Besi ...
'' use an anomalously-named D66 roll, meaning d6×10+d6. This sort of roll originated in the
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(no relation) game, '' Traveller'', to roll on various tables and charts, usually involving encounters, but did not use the notation. 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
average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
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 ''X''db, which is shorthand for , X, \text6\text1 (roll the absolute value of X in 6-sided dice and keep 1), where the sign of X specifies whether the attacker (if positive) or defender (if negative) chooses which die to keep; X is usually omitted when 1, and as -1 and 1 are equal, -1 is never used. Common values are 3, 2, 1, -2, or -3. Alternatively, the words "for" and "against" can be used to replace X's sign, where "against" means negative and "for" means positive; the word is placed after the rest of the formula. As an example, 2\text is equal to \text2\text, which both mean "roll 2 6-sided dice, defender chooses from the results rolled". ''
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'', a game about Angels and Demons from
Steve Jackson Games Steve Jackson Games (SJGames) is a game company, founded in 1980 by Steve Jackson, that creates and publishes role-playing, board, and card games, and (until 2019) the gaming magazine ''Pyramid''. History Founded in 1980, six years after the cr ...
, 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. One other commonly used variant of the 6-sided dice roll is the d3, which is a 6-sided die roll, with the result divided by 2. The average result is 2, and the standard deviation is 0.816.


10-sided variations

The '' Cyborg Commando'' role-playing game by
Gary Gygax Ernest Gary Gygax ( ; July 27, 1938 – March 4, 2008) was an American game designer and author best known for co-creating the pioneering role-playing game ''Dungeons & Dragons'' (''D&D'') with Dave Arneson. In the 1960s, Gygax created an ...
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 ''Savage Worlds'' is a role-playing game written by Shane Lacy Hensley and published by Pinnacle Entertainment Group. The game emphasizes speed of play and reduced preparation over realism or detail. The game received the 2003 Origin Gamers' Choi ...
''. 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: ''A''d''X''! or 6d6!. The
Storyteller system The ''Storytelling System'' is a role-playing game system created by White Wolf, Inc. for the Chronicles of Darkness (formerly known as the New World of Darkness), a game world with several pen and paper games tied in. The Storytelling System ...
combines exploding dice with a dice pool threshold and target number. Diana: Warrior Princess explodes all successes, and
Hackmaster HackMaster is a fantasy role-playing game produced by Kenzer & Company. It began as a fictional game, a parody of ''Dungeons & Dragons'' played by the characters of the ''Knights of the Dinner Table'' comic strip by Jolly R. Blackburn. The ...
uses a variant called ''dice penetration'' by which 1 is subtracted from the total of the rerolled dice.


References

*{{cite web , last =Winter , first =Steve , title =A Brief History of Dice , publisher =
Wizards of the Coast Wizards of the Coast LLC (often referred to as WotC or simply Wizards) is an American publisher of games, primarily based on fantasy and List of science fiction themes, science fiction themes, and formerly an operator of retail stores for ga ...
, date =2007-03-02 , url =http://archive.wizards.com/default.asp?x=dnd/alumni/20070302a , archive-url =https://web.archive.org/web/20141110193757/http://archive.wizards.com/default.asp?x=dnd/alumni/20070302a , url-status =dead , archive-date =November 10, 2014 , access-date =2007-03-06 Role-playing game terminology Notation Dice