# Dice

Two standard six-sided pipped dice with rounded corners.
Japanese die, with its distinctive oversized pip.
Typical role-playing dice, showing a variety of colors and styles. Note the older hand-inked green 12-sided die (showing an 11), manufactured before pre-inked dice were common. Many players collect or acquire a large number of mixed and unmatching dice.

A die ( Old French de, from Latin datum "something given or played" ) is a small polyhedral object, usually cubical, used for generating random numbers or other symbols. This makes dice suitable as gambling devices, especially for craps or sic bo, or for use in non-gambling tabletop games.

Traditionally, a die is seldom seen alone, and is rather one of a pair of identical dice that are sized to be comfortably rolled or thrown, together, from a user's hand. Because of this, the singular word "die" is rare, but treating "dice" as interchangeably singular or plural is less common; the plural form "dices" is rarer still.

A traditional die is a cube (often with corners slightly rounded), marked on each of its six faces with a different number of circular patches or pits called pips. All of these pips have the same appearance within a pair, or larger set of dice, and are sized for ease of recognizing the pattern the pips on one face form. The design as a whole is aimed at each die providing one randomly determined integer, in the range from one to six, with each of those values being equally likely.

More generally, a variety of analogous devices are often described as dice, but necessarily in a context, or with a word or two preceding "die" or "dice", that avoids the assumption that traditional dice are intended. Such specialized dice may have cubical or other polyhedral shapes, with faces marked with various collections of symbols, and be used to produce other random results than one through six. There are also "loaded" or "crooked" dice (especially otherwise traditional ones), meant to produce skewed or even predictable results, for purposes of deception or amusement.

## Ordinary dice

European-style, Chinese-style, and casino dice.

The common dice are small cubes 1 to 2 cm along an edge (16mm being the standard), whose faces are numbered from one to six (usually by patterns of dots called pips). It is traditional to assign pairs of numbers that total seven to opposite faces (it has been since at least classical antiquity); this implies that at one vertex the faces 1, 2 and 3 intersect. It leaves one other abstract design choice: the faces representing 1, 2 and 3 respectively can be placed in either clockwise or counterclockwise order about this vertex.

Dice are thrown to provide random numbers for gambling and other games and thus are a type of hardware random number generator. However, because the numbers on toy dice are marked with small indentations, slightly more material is removed from the higher numbered faces. This results in a small bias, and they do not provide fair (uniform) random numbers. The bias is reduced somewhat in the Japanese die with its oversized single pip (pictured). Casino dice have markings that are flush with the surface and come very close to providing true uniformly distributed random numbers.

Dice are thrown, singly or in groups, from the hand or from a cup or box designed for the purpose, onto a flat surface. The face of each die that is uppermost when it comes to rest provides the value of the throw. A typical dice game today is craps, wherein two dice are thrown at a time, and wagers are made on the total value of up-facing pips on the two dice. They are also frequently used to randomize allowable moves in board games such as Backgammon.

## History

Dice were probably originally made from the ankle bones (specifically the talus or "astragalus") of hoofed animals (such as oxen), colloquially known as " knucklebones", which are approximately tetrahedral. Modern Mongolians still use such bones, known as shagai, for games and fortunetelling. Even today in English, dice are sometimes colloquially referred to as "bones", as in "shake them bones". Ivory, bone, wood, metal, and stone materials have been commonly used, though the use of plastics is now nearly universal. It is almost impossible to trace clearly the development of dice as distinguished from knucklebones, because ancient writers confused the two games. It is certain, however, that both were played in prehistoric times.

A collection of historical dice from Asia

Dice have been used throughout Asia since time immemorial.

The oldest known dice was excavated as part of a 5000 year old backgammon set, at the Burnt City archeological site in south-eastern Iran. Excavations from ancient tombs in the Harappan civilization, seem to further indicate a South Asian origin. Dicing is mentioned as an Indian game in the Rig Veda, Atharva Veda and Buddha games list. It is also mentioned in the great Hindu epic, the Mahabharata, where Yudhisthira plays a game of dice against the Kauravas for the northern kingdom of Hastinapura. In its primitive form knucklebones was essentially a game of skill played by women and children. In a derivative form of knucklebones, the four sides of the bones received different values and were counted as with modern dice. Gambling with three or sometimes two dice was a very popular form of amusement in Greece, especially with the upper classes, and was an almost invariable accompaniment to banquets (symposia).

The Romans were passionate gamblers, especially in the luxurious days of the Roman Empire, and dicing was a favorite form, though it was forbidden except during the Saturnalia. Horace derided what he presented as a typical youth of the period, who wasted his time amid the dangers of dicing instead of taming his charger and giving himself up to the hardships of the chase. Throwing dice for money was the cause of many special laws in Rome. One of these stated that no suit could be brought by a person who allowed gambling in his house, even if he had been cheated or assaulted. Professional gamblers were common, and some of their loaded dice are preserved in museums. The common public-houses were the resorts of gamblers, and a fresco is extant showing two quarrelling dicers being ejected by the indignant host.

Tacitus states that the Germans were passionately fond of dicing, so much so, indeed, that, having lost everything, they would even stake their personal liberty. Centuries later, during the middle ages, dicing became the favourite pastime of the knights, and both dicing schools and guilds of dicers existed. After the downfall of feudalism the famous German mercenaries called landsknechts established a reputation as the most notorious dicing gamblers of their time. Many of the dice of the period were curiously carved in the images of men and beasts. In France both knights and ladies were given to dicing. This persisted through repeated legislation, including interdictions on the part of St. Louis in 1254 and 1256.

In China, India, Japan, Korea, and other Asiatic countries, dice have always been popular and are so still. The markings on Chinese dominoes evolved from the markings on dice, taken two at a time.

## Materials

Precision backgammon dice

Dice have been made from a wide variety of materials throughout history, including stone, wood, and animal bones, and more recently, bakelite and plastic.

### Precision dice

Precision casino dice, used for the game of craps, are made from cellulose acetate. These dice may have a polished finish, making them transparent, or a sand finish, making them translucent. Casino dice have their pips drilled, and then filled flush with a paint of the same specific gravity as the acetate, such that the dice remain in perfect balance. In casino play, a stick of 5 dice are used, all stamped with a matching serial number to prevent a cheat from substituting a die.

Precision backgammon dice are also made from acetate, or a similar material, with the pips filled in as is done with casino dice. While casino dice are noticeably larger than common dice, with sharp edges and corners, precision backgammon dice tend to be somewhat smaller. Their corners and edges are beveled to allow greater movement inside the dice cup and prevent chaotic rolls from damaging the playing surface.

### Polyhedral dice

It is unknown of what material the earliest polyhedral dice were made. A pair of icosahedral (20-sided) dice dating from Roman times are on display at the British Museum.

Roughly cubical six-sided Roman dice made of wood, bone, ivory and lead have been discovered. It is possible that polyhedral dice were used by even earlier cultures.

Polyhedral dice are usually made of plastic, though infrequently metal, wooden, and semi-precious stone dice can be found. Early polyhedral dice from the 1970s and 1980s were made of a soft plastic that would easily wear as the die was used. Typical wear and tear would gradually round the corners and edges of the die until it was unusable. Modern polyhedral dice are typically made of high-impact plastic and can withstand years of use without visible wear.

Polyhedral dice can be purchased at most hobby stores in numerous combinations. In the early days of role-playing games, most dice came with the numbers uninked and players took great care in painting their sets of dice. Some twenty-sided dice of this era came numbered zero through nine twice; half of the numbers had to be painted a contrasting colour to signify the "high" faces. Such a die could also double as a ten-sided die by ignoring the distinguishing coloring.

## Terms

While the terms ace, deuce, trey, cater, cinque and sice are hardly common today having been replaced with the ordinary names of the numbers one to six, they are still used by some professional gamblers to describe the different sides of the dice. Ace is from the Latin as, meaning "a unit" ; the others are the numbers 2–6 in old French.

#### Dice Notation

Often the names of the dice appear in formulas for calculating game parameters: e.g., hit points. "6d8+10", for example, will yield a number between 16 (6×1+10) and 58 (6×8+10), as it means "Roll an eight-sided die six times and add ten to the total." Occasionally they may be written "1d6×10+20"; this means "Roll one six-sided die. Multiply it by ten and add twenty."

### "Crooked" dice

"Crooked dice" refers to dice that have been altered in some way to change the distribution of the dice's outcome.

A loaded or gaffed die is a die that has been tampered with to land with a selected side facing upwards more often than it would simply by chance. There are methods of creating loaded dice, including having some edges round and other sharp and slightly off square faces. If the dice are not transparent, weights can be added to one side or the other. They can be modified to produce winners ("passers") or losers ("miss-outs"). "Tappers" have a drop of mercury in a reservoir at the centre of the cube, with a capillary tube leading to another mercury reservoir at the side of the cube. The load is activated by tapping the die on the table so that the mercury leaves the centre and travels to the side. Often one can see the circle of the cut used to remove the face and bury the weight. In a professional die, the weight is inserted in manufacture; in the case of a wooden die, this can be done by carving the die around a heavy inclusion, like a pebble around which a tree has grown.

A variable loaded die is hollow with a small weight and a semi-solid substance inside, usually wax, whose melting point is just lower than the temperature of the human body. This allows the cheater to change the loading of the die by breathing on it or holding it firmly in hand, causing the wax to melt and the weight to drift down, making the chosen opposite face more likely to land up. A less common type of variable die can be made by inserting a magnet into the die and embedding a coil of wire in the game table. Then, either leave the current off and let the die roll unchanged or run current through the coil to increase the likelihood that the north side or the south side will land on the bottom depending on the direction of the current.

Plastic dice can be biased to roll a certain number by heating them (for example in an oven) with the desired face upward, so that the plastic will soften slightly and "pool" at the opposite (bottom) side of the die without showing much, if any, visible distortion.

Transparent acetate dice, used in all reputable casinos, are harder to tamper with.

#### Cheat dice

Cheat dice (see below) are often sold as loaded dice but usually are not technically loaded.

#### Shaved dice

A die can be "shaved" on one side i.e. slightly shorter in one dimension, making it slightly rectangular and thus affecting its outcome. One countermeasure employed by casinos against shaved dice is to measure the dice with a micrometer.

## Variants

### Dice with faces other than digit sequences

As noted, the faces of most dice are labelled using an unbroken series of whole numbers, starting at one (or zero), expressed with either pips or digits. Common exceptions include:

• colour dice (e.g., with the colors of the playing pieces used in a game)
• Poker dice, with labels reminiscent of playing cards. Several varieties exist, but the most common contain the following pattern: 9♣, 10♦, Jack (blue), Queen (green), King (red), A♠
• dice with letters (e.g. in Boggle)
• average dice (2, 3, 3, 4, 4, 5) (In some war games, units are identified as regulars or irregulars. Because regulars are more predictable, the strength of a regular unit is multiplied by an average die. For this reason, average dice are jocularly called regular dice.)
• cheat dice, such as:
• one face each with two through five, and two with sixes, or
• for craps, a pair of dice in which one die has five on each face, and its mate has a mixture of twos and sixes, guaranteeing rolls of seven or 11.
• dice with a single sequence of markings repeated multiple times, for example:
• a cubical die numbered twice from 1 to 3, or thrice from 1 to 2.
• icosahedral dice numbered twice from 1 to 10 (commonly used in Dungeons & Dragons before the popularization of ten-sided dice).
• Fudge dice, numbered twice from −1 to 1, represented as −, blank, +.
Doubling cube
• random direction dice, also known as scatter dice. The dice have arrows on each side; the outcome of a roll is a random direction. Scatter dice are used in tabletop wargames such as Warhammer Fantasy Battle to determine random movements of troops, wind direction or direction of misfired arms. Note that this is an unusual case where the die is read not according to which symbol is shown on its uppermost face, but its compass orientation.
• A doubling cube with the numbers 2, 4, 8, 16, 32, and 64 is used in backgammon and some other boardgames. This die is not actually rolled; it is used to denote the current stakes of the game.
• Some board games use dice with positive and negative numbers for use in gain or loss of something.
• Sicherman dice, a pair having the same odds of rolling a given sum as a pair of standard six-sided dice, but with different markings: one die has 1, 3, 4, 5, 6, and 8, and the other has 1, 2, 2, 3, 3, and 4. Sicherman dice are the only such alternative arrangement if positive numbers are used.
• I Ching dice such as
• Eight-sided dice bearing the eight trigrams
• Six-sided dice bearing yin and yang twice each, and old yin and old yang once each
• "Projector dice" which are clear and marked only on one of each pair of opposing faces. For a "six"-sided die, e.g., a clear twelve sided-shape is used. Rolled on an overhead projector such a die will have the top or bottom marking equally readable.

### Non-cubical dice

Barrel Dice

Some dice are polyhedra other than cubes in shape. They were once almost exclusively used by fortune-tellers and in other occult practices, but they have become popular lately (at least since the early 1950s) among players of wargames, trading card games, German-style board games, and role-playing games. Although polyhedral dice are a relative novelty during modern times, some ancient cultures appear to have used them in games (as evidenced by the presence of two icosahedral dice dating from the days of ancient Rome on display in the British Museum). Such dice are typically plastic, and have faces bearing numerals rather than patterns of dots. Reciprocally symmetric numerals are distinguished with a dot in the lower right corner (6. vs 9.) or by being underlined (6 vs 9).

The platonic solids are commonly used to make dice of 4, 6, 8, 12, and 20 faces. Other shapes can be found to make dice with 2, 5, 7, 10, 16, 24, 30, 34, 50, or 100 sides, but other than the 10 sided, they are rarely used. (See Zocchihedron.) The 4 sided platonic solid is difficult to roll, and a few games like Daldøs use a 4 sided rolling pin instead.

 20-sided die 10-sided die 4-sided die

A large number of different probability distributions can be obtained using these dice in various ways; for example, 10-sided dice (or 20-sided dice labeled with single digits) are often used in pairs to produce a linearly-distributed random percentage. Summing multiple dice approximates a normal distribution (a "bell curve"), while eliminating high or low throws can be used to skew the distribution in various ways. Using these techniques, games can closely approximate the real probability distributions of the events they simulate.

There is some controversy over whether manufacturing processes create genuinely "fair" dice (dice that roll with even distributions over their number span). Casino dice are legally required to be fair; those used by others hold no such requirement.

Spherical dice also exist; these function like the plain cubic dice, but have an octahedral internal cavity in which a weight moves which causes them to settle in one of six orientations when rolled. However, these dice are somewhat awkward in use because they require a flat and level surface to roll properly — an uneven surface often causes them to stop partway between two numbers, while a sloped surface will obviously cause the dice to keep rolling.

Cowry shells or coins may be used as a kind of two-sided dice. (Because of their shape, cowry shells probably do not yield a uniform distribution.)

#### Standard variations

A matched Platonic-solids set of five dice, (from left) tetrahedron, cube, octahedron, dodecahedron and icosahedron.

The most common non-cubical dice — often sold in sets of five or six that are each differently shaped but with the same pair of background and marking colors — include one each of the five Platonic solids, which are highly symmetrical. The six-die versions add the pentagonal trapezohedron, in which the faces (identical to one another as to angles and edge lengths) each have two different lengths of side, and three different sizes of angle; the corners at which multiple faces meet are also of two different kinds.

Sides Shape Notes
4 tetrahedron Each face has three numbers: they are arranged such that the upright number (which counts) is the same on all three visible faces. Alternatively, all of the sides have the same number in the lowest edge and no number on the top. This die does not roll well and thus it is usually thrown into the air instead.
6 cube A common die. The sum of the numbers on opposite faces is seven.
8 octahedron Each face is triangular; looks something like two Egyptian pyramids attached at the base. Usually, the sum of the opposite faces is 9.
10 pentagonal trapezohedron Each face is kite-shaped; five of them meet at the same sharp corner (as at the top of the diagram in this row), and five at another equally sharp one; about halfway between them, a different group of three faces converges at each of ten blunter corners. The ten faces usually bear numbers from zero to nine, rather than one to ten (zero being read as "ten" in many applications), and often all odd numbered faces converge at the same sharp corner, and the even ones at the other.
12 dodecahedron Each face is a regular pentagon.
20 icosahedron Faces are equilateral triangles. Typically, opposite faces add to twenty-one. A Roman icosahedron die from the 2nd century AD has been found, though the game it was used for is not known.

#### Rarer variations

Sides Shape Notes
2 cylinder This is nothing more than a coin shape with 1 marked on one side and 2 on the other. While some tasks in roleplaying require flipping a coin, it is usually referred to as such, and not as rolling a two-sided die. It is possible, however, to find dice of this sort for purchase, but they are rare, and can typically be found among other joke dice.
3 Rounded-off triangular prism This is essentially a rounded-off triangular prism, intended to be rolled like a rolling-pin style die. The die is rounded-off at the edges to make it impossible for it to somehow land on the triangular sides, which makes it look a bit like a jewel. When the die is rolled, one edge (rather than a side) appears facing upwards. On either side of each edge the same number is printed (from 1 to 3). The numbers on either side of the up-facing edge are read as the result of the die roll. Another possible shape is the "American Football" or " Rugby ball" shape, where the ends are pointed (with rounded points) rather than just rounded.
5 Triangular prism This is a prism that is thin enough to land either on its "edge" or "face". When landing on an edge, the result is displayed by digits (2–4) close to the "pyramid"'s top. The triangular faces are labeled with the digits 1 and 5.
7 Pentagonal prism Similar in constitution to the 5-sided die. When landing on an edge, the topmost edge has pips for 1–5. The pentagonal faces are labeled with the digits 6 and 7. This kind of die is particularly odd since it has pips for five of its results and digits for two of them. Seven sided dice are used in a seven-player variant of backgammon. Some variants have heptagonal ends and rectangular faces.
12 rhombic dodecahedron Each face is in the shape of a rhombus.
14 heptagonal dipyramid Each face is in the shape of an isosceles triangle.
16 octagonal dipyramid Each face is in the shape of an isosceles triangle.
24 tetrakis hexahedron Each face is in the shape of an isosceles triangle.
24 deltoidal icositetrahedron Each face is in the shape of a kite (geometry).
30 rhombic triacontahedron Each face is in the shape of a rhombus (diamond-shaped).
50 icosakaipentagonal dipyramid Just like the 14- and 16-sided dice, the faces of the 50-sided die are isosceles triangles, although very narrow.
100 Dice of this sort are rare. See main article.

The full geometric set of "uniform fair dice" (with all congruent sides) are:

• Platonic solids: 5 regular polyhedra: (4, 6, 8, 12, 20 sides)
• Catalan solids: 13 Archimedean duals: (12, 24, 30, 48, 60, 120 sides)
• Bipyramids: infinite set of prism duals, triangle faces: (6, 8, 10, 12, ... sides)
• Trapezohedrons: infinite set of antiprism duals, kite faces: (6, 8, 10, 12, ... sides)

Rolling-pin style dice are usually made so that all the faces they may actually land on are congruent, so they are equally fair.

## Probability

For a single roll of an s-sided die, the probability of rolling each value, 1 through s, is exactly 1/s. This is an example of a discrete uniform distribution. For a double roll, however, the total of both rolls is not evenly distributed, but is distributed in a triangular curve. For a six-sided die, for example, the probability distribution is as follows:

Probability distribution for the sum of two six-sided dice
Sum
2
3
4
5
6
7
8
9
10
11
12
Probability
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
Probability (simplified)
1/36
1/18
1/12
1/9
5/36
1/6
5/36
1/9
1/12
1/18
1/36

For three or more die rolls, the curve becomes more bell-shaped with each additional die (according to the central limit theorem). The exact probability distribution Fs,i for any number of s-sided dice i can be calculated as the repeated convolution of the single-die probability distribution with itself.

$F_{s,i}(k) = \sum_n {F_{s,1}(n) F_{s,i-1}(k - n)} \,$

where $F_{s,1}(k) = \frac{1}{s}$ for all $1\leq k \leq s$ and 0 otherwise.

For example, in the triangular curve described above,

 $F_{6,2}(6)\,$ $=\sum_n {F_{6,1}(n) F_{6,1}(6 - n)}\,$ $=F_{6,1}(1) F_{6,1}(5) + F_{6,1}(2) F_{6,1}(4) + \ldots + F_{6,1}(5) F_{6,1}(1)\,$ $=5\cdot\frac{1}{6}\cdot\frac{1}{6}=\frac{5}{36}\approx0.14\,$

Equivalently, one can calculate the probability using combinations:

$F_{s,i}(k)=\frac{1}{s^i}\sum_{n=0}^{\left \lfloor \frac{k-i}{s} \right \rfloor} (-1)^n {i \choose n} {k-sn-1 \choose i-1}$

The probability of rolling any exact sequence of numbers is simply $\frac{1}{s^i}$. For example, the chance of rolling 1, 2, and 3 in that order with three rolls of a six-sided die is $\frac{1}{6^3}$, or $\frac{1}{216}$. Rolling any single number i times in a row, regardless of which number, is s times more likely, at a $\frac{s}{s^i}$ chance.

## Application in role-playing games

Full set of matching dice used in roleplaying: a d4, d6, d8, d12, d20, and two d10s for percentile: ones and tens.

The fantasy role-playing game Dungeons & Dragons introduced the use of polyhedral dice during modern times and paved the way for their use in other role-playing games, using 20-, 12-, 10-, 8- and 4-sided dice in addition to the traditional 6 sided die. Such dice are often sold in sets.

Types of polyhedral dice are distinguished by prefixing a "d" to the number of faces; for example, a ten-sided die is a d10.

Players use polyhedral dice together in a number of ways. For example, a d10 can be used in conjunction with a d6 instead of using a d20. If the d6 displays a 1, 2 or 3, the number on the d10 is resolved as 1–10. If the d6 displays a 4, 5 or 6, the number shown on the d10 is resolved to 11–20 ("1" is 11, "2" is 12, etc.). In cases like this, almost any sided die can be used as a "resolver".

Two d10 are often used to generate a number between 1 and 100, with one die representing the tens position. The tens die may be distinguished by colour, by using a custom die marked with multiples of ten, or any other means the player can indicate. Similar methods can be used for additional digits.

## Use of dice for divination

Some people believe that dice can be used for divination. Using dice for such a purpose is called cleromancy. A pair of standard 6-sided dice is generally used.

Astrological dice are a specialized set of three 12-sided dice for divination, using the concepts of astrology and containing astrological symbols for the planets, the zodiac signs and the astrological houses. The first die represents planets, the Sun, the Moon, and two nodes (North Node and South Node). The second die represents the 12 zodiac signs, and the third represents the 12 houses. In simplified terms, the planets, etc. could represent the 'actor'; the zodiac signs could represent the 'role' being played by the actor; and the house could represent the 'scene' in which the actor plays.

Rune dice are a specialized set of dice for divination ( runecasting), using the symbols of the runes printed on the dice.

An icosahedron is used to provide the answers of a Magic 8-Ball, which is conventionally used to provide advice on yes-or-no questions.