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Dice notation: the story on HearLore | HearLore
Common questions
When did dice notation become standardized in role-playing games?
The standardization of dice notation began in 1975 when Ted Johnstone published an article titled Dice as Random Number Generators in the fanzine Alarums & Excursions. This system used the letter d to separate the number of dice from the number of sides, creating expressions like 3d6 to mean roll three six-sided dice. By 1979, Gary Gygax had adopted this notation as the standard in the first edition of Advanced Dungeons & Dragons.
What does the letter d represent in dice notation?
In standard dice notation, the letter d stands for die or dice and is followed by the number of faces on the die. The faces are numbered from 1 to that number, though the d10 is labeled from 0 to 9 with the 0 read as 10 depending on game rules. When a final number is omitted, it is typically assumed to be a d6.
How are percentile dice used to generate numbers from 1 to 100?
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. The modern standard uses a combination of two ten-sided dice known as percentile dice, where one die represents units and the other tens. A roll of 0 on both dice may be interpreted as either 0 or 100, depending on the game rules.
What is the mathematical effect of rolling multiple dice in dice notation?
Rolling three or more dice creates a probability distribution that approximates a Gaussian curve in accordance with the central limit theorem. This means that results cluster around the average, making extreme outcomes less likely and allowing game designers to balance challenges and rewards. For instance, rolling four six-sided dice and dropping the lowest result skews the probability curve towards higher numbers.
Dice notation
In the late 1960s, a quiet revolution began in the basements and garages of American hobbyists, where miniatures wargamers started reaching for polyhedral shapes that defied the familiar six-sided cube. These were the Platonic solids, geometric forms that offered possibilities impossible to generate with standard equipment. By 1974, Dungeons & Dragons emerged as the first commercially available game to embrace these shapes, yet it lacked a standardized language to describe the rolls required by its rules. The original text often relied on vague verbal instructions or ambiguous ranges, such as the spell sticks to snakes which conjured between 2 and 16 snakes without specifying the exact method of generation. A random encounter might list a village of 30 to 300 orcs, leaving players to wonder if that meant rolling 3d10 multiplied by 10 or simply 30d10. This ambiguity created a chaotic environment where the same text could yield vastly different outcomes depending on the Dungeon Master's interpretation.
The standardization of this chaotic system began in 1975, when a fan named Ted Johnstone published an article titled Dice as Random Number Generators in the inaugural issue of the fanzine Alarums & Excursions. Johnstone introduced a notation system that used the letter d to separate the number of dice from the number of sides, creating expressions like 3d6 to mean roll three six-sided dice. This simple algebraic approach allowed players to discuss probability distributions with precision, a concept previously absent from the hobby. Another writer named Barry Gold used the same notation in that same issue, and the idea quickly spread through the fan community. By the time Gary Gygax published the first edition of Advanced Dungeons & Dragons between 1977 and 1979, this notation had become so deeply ingrained in the culture that it was adopted as the standard. The legacy of this shift is still visible today in the name of the Open Game version of the rules, known as the d20 System.
The Mechanics of Probability
The core of dice notation relies on the relationship between the number of dice rolled and the number of sides on each die, creating a mathematical framework that governs the outcome of the game. In the standard form, the letter d stands for die or dice, followed by the number of faces, which are numbered from 1 to that number. A notable exception exists for the d10, which is labeled from 0 to 9, though the 0 can be read as a 10 depending on the game rules. When a final number is omitted, it is typically assumed to be a d6, but some contexts use other defaults. For example, 1d4 means roll one 4-sided die, while 3d6 means roll three six-sided dice and add them together. The addition of modifiers allows for further complexity, such as 1d20 minus 10, which indicates a roll of a single 20-sided die with 10 subtracted from the result.
The mathematical implications of rolling multiple dice create a probability distribution that approximates a Gaussian curve, in accordance with the central limit theorem. This means that rolling three or more dice produces results that cluster around the average, making extreme outcomes less likely. For instance, a roll of 4d6 minus L, which means rolling four six-sided dice and dropping the lowest result, skews the probability curve towards higher numbers. A roll of 3 can only occur when all four dice come up 1, a probability of one in 1296, while a roll of 18 results if any three dice are 6, a probability of one in 1296. This mathematical precision allows game designers to balance challenges and rewards, ensuring that the randomness of the dice serves the narrative rather than undermining it.
How does the d66 roll work in Games Workshop systems?
The D66 roll is a base-six variant of the base ten percentile die that uses two six-sided dice to generate numbers from 11 to 66. The first die represents the tens digit and the second die the ones digit, with an average result of 38.5 and a standard deviation of about 17.16. This roll originated in the Game Designers' Workshop game Traveller to roll on various tables and charts involving encounters.
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. Gary Gygax, in the Advanced Dungeons & Dragons Dungeon Masters Guide on page 10, described the practice of assuming the use of a standard d20 which is numbered 0-9 twice. This historical quirk allowed players to generate numbers from 1 to 100 using a single die, with half of the faces given a distinct color to indicate the addition of ten. The modern standard uses a combination of two ten-sided dice known as percentile dice, where one die represents units and the other tens. These are typically distinguished by color, but dice marked with multiples of ten are also available for use as the tens die. A roll of 0 on both dice may be interpreted as either 0 or 100, depending on the game rules, though 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.
The d1000, using three 10-sided dice, is occasionally seen, although it is more common in wargames than role-playing games. The notation for these rolls often uses the variable X as 100, alternatively written as %. This system allows for the generation of large numbers without the need for a single 100-sided die, which can be unwieldy. The use of percentile dice has become a staple of the hobby, appearing in countless games from Vampire: The Requiem to the original Ghostbusters role-playing game. The flexibility of this system has allowed designers to create complex mechanics that rely on precise probability distributions, ensuring that the randomness of the dice serves the narrative rather than undermining it.
Selective Results and Dice Pools
A number of notational strategies exist for discarding only certain types of results, allowing players to manipulate the outcome of a roll. Some games extend the standard notation to include the letter k, which represents 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. The letter k can be replaced with kh, kl, or kc to represent keeping the highest, lowest, or the player's choice, respectively. For example, 4d6k2 means roll four 6-sided dice, keeping the two highest. This notation allows for the creation of complex mechanics that rely on the manipulation of probability distributions. The use of dice pools, where a fixed number of dice are rolled and the total number of dice which meet a fixed condition are recorded as the result, has become a popular mechanic in games such as the Storyteller system and Fantasy Flight Games' Star Wars Roleplaying Games.
The Fudge role-playing system uses a set of dice which are each marked with minus signs, plus signs and blank sides, meaning minus 1, plus 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 are rolled to determine results from minus 4 to plus 4, which is equivalent to 4d3 minus 8. Variants include dF.1, which is a six-sided die with four blanks, one plus and one minus. The use of dice pools has allowed designers to create mechanics that rely on the manipulation of probability distributions, ensuring that the randomness of the dice serves the narrative rather than undermining it.
Six-Sided Variations
Various Games Workshop systems such as Necromunda and Mordheim use an anomalously-named D66 roll, meaning d6 times 10 plus d6. This sort of roll originated in the Game Designers' Workshop 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. 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 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, which is shorthand for the absolute value of X in 6-sided dice and keep 1. 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 minus 1 and 1 are equal, minus 1 is never used. Common values are 3, 2, 1, minus 2, or minus 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, 2db is equal to 2d6, which both mean roll 2 6-sided dice, defender chooses from the results rolled. The use of these variations has allowed designers to create mechanics that rely on the manipulation of probability distributions, ensuring that the randomness of the dice serves the narrative rather than undermining it.
Ten-Sided and Open-Ended Systems
The Cyborg Commando role-playing game by Gary Gygax uses a dice mechanic called d10x. This is equivalent to d10 times 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 simply called explode. Notational shorthand for exploding dice is to suffix the roll with an exclamation point, 6d6!, asterisk 6d6*, or the letter X 6d6X.
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. The use of these systems has allowed designers to create mechanics that rely on the manipulation of probability distributions, ensuring that the randomness of the dice serves the narrative rather than undermining it. The flexibility of these systems has allowed for the creation of complex mechanics that rely on the manipulation of probability distributions, ensuring that the randomness of the dice serves the narrative rather than undermining it.