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# Dice

This article describes polyhedral objects used to randomize decisions.
For chopping food into small cubes, see cooking.
Rolling dice

Dice (the plural of the word die, probably from the Latin dare: to give) are typically cubes rolled out of a hand in games, so each one produces a randomly chosen number in the range from one to six; more generally, cubical or other polyhedral objects, with the faces marked with various kinds of symbols, used to produce other random results.

## Ordinary dice

The most common dice are small cubes 1 to 2 cm along an edge, whose faces are numbered from one to six (usually by patterns of dots called their pips). It is traditional to assign pairs of numbers that total seven to opposite faces; this leaves one other abstract design choice: the faces representing 1, 2 and 3 respectvely can be placed in either clockwise or anti-clockwise order about the vertex they have in common.

Dice are thrown to provide (supposedly uniformly distributed) 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. 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 spots on the two dice. They are also frequently used to randomize allowable moves in board games such as Backgammon.

### History

Dice probably evolved from knucklebones, which are approximately tetrahedral. Even today, 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, on account of the confusing of the two games by the ancient writers. It is certain, however, that both were played in times antecedent to those of which we possess any written records.

The fact that dice have been used throughout the Orient from time immemorial, as has been proved by excavations from ancient tombs, seems to point clearly to an Asiatic origin. Dicing is mentioned as an Indian game in the Rig-veda. 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 (symposium).

The Romans were passionate gamblers, especially in the luxurious days of the Roman Empire, and dicing was a favourite form, though it was forbidden except during the Saturnalia. Horace derided the 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 Japan, China, Korea, India, 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.

"Loaded" or "gaffed" dice can be made in many ways to cheat at such games. Weights can be added, or some edges made round while others are sharp, or some faces made slightly off-square, to make some outcomes more likely than would be predicted by pure chance. Dice used in casinos are often transparent to make loading more difficult.

A loaded 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 several types of loaded dice. 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 center 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 center 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.

Transparent acrylic dice, used in all reputable casinos, are harder to tamper with. It is unlikely for one to encounter loaded or other crooked dice at a licensed casino, since casinos make plenty of money on the percentages and they would not want to risk revocation of their licenses.

### Materials

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. 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 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. Lou Zocchi and his company Gamescience not only always guaranteed their high impact plastic dice to not wear down like other companies' dice did, but for years criticized major dice manufacturers of crafting unfair, loaded dice through sloppy polishing techniques and substandard materials.

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 unpainted and players took great care in painting their sets of dice. Many early d20's came with two sides with the numbers zero through nine on them; one side had to be painted a contrasting color to signify the "high" side.

## Cubical dice with faces representing other than 1 through 6

As noted, the faces of most dice are labelled to 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 colours of the playing pieces used in a game)
• Poker dice, with the following labels somewhat reminiscent of the names of standard playing cards:
• Nine (of spades; black)
• Ten (of diamonds; red)
• Jack (blue)
• Queen (blue)
• King (red)
• Ace (of clubs; black)
• dice with letters (e.g. in Boggle)
• doubling dice (2, 4, 8, 16, 32, 64)
• average dice (2, 3, 3, 4, 4, 5)
• 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
• so-called "3-sided dice", each a cubical die with each of its faces marked identically to exactly one of the other faces, yielding three equally likely distinguishable outcomes, for example:
• those (usually abbreviated d3) in some role-playing games, labelled 1, 2, and 3 respectively, or
• Fudge dice, with two minus (−) sides, two blank sides, and two plus (+) sides; a throw of n fudge dice yields an integer from −n to n, by reading "−" as "−1" and "+" as "+1" and summing the faces showing.

## Non-cubical dice

A twenty-sided die, commonly called a "d20".

Polyhedral dice are dice with more or fewer than six sides. They were once almost exclusively used by fortune-tellers and in other occult practices, but they have become popular lately 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).

Dice with various numbers of faces are often described by their numbers of sides, with a d6 being a six-sided die, a d10 a ten-sided die, and so forth.

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 10, 30, and other numbers of faces. (See Zocchihedron).

 20-sided die 10-sided die 4-sided die
Typical role-playing die set.
European‐style, Chinese, and casino dice.

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). Lou Zocchi for years charged that his dice were "fairer" than most made by major manufacturers. Casino dice are legally required to be fair; those used by all 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.

Cowry shells or coins may be used as a kind of two-sided dice ("d2"). (In the case of cowries it is questionable if they yield a uniform distribution.)

### Standard variations

A matched Platonic-solids set of five dice, (from left) tetrahedron (d4), cube (d6), octahedron (d8), dodecahedron (d12) and icosahedron (d20). 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 — are based on the Platonic solids and the pentagonal trapezohedron.
TypeShapePlatonic?Notes
d4tetrahedron YesEach face has three numbers: they are arranged such that the upright number (which counts) is the same on all three visible faces. This die does not roll well and thus it is usually thrown into the air instead.
d6cube YesA common die. The sum of the numbers on opposite faces is seven.
d8octahedron YesEach face is triangular; looks something like two Egyptian pyramids attached at the base.
d10pentagonal trapezohedron NoEach face is kite-shaped; the smallest angle of five faces point to one edge, the smallest angle of the other points to the opposite. Often, all odd numbers are on one half of the die and all even numbers are on the other half. Additionally, on most currently-manufactured dice, faces on opposite halves of the die meet at a right angle. There is usually a face marked "0" but no face marked "10".
d12dodecahedron YesEach face is a regular pentagon.
d20icosahedron YesFaces are equilateral triangles. Typically, opposite faces add to twenty-one.

### Rarer variations

TypeShapeNotes
d7 pentagonal prism A rare die type, thick enough to land either on its "edge" or "face". When landing on an edge, the topmost edge has pips for 1 through 5. The pentagonal faces are labeled with the digits 6 and 7. Such dice are used in a seven-player variant of backgammon.
d12 rhombic dodecahedron Each face is in the shape of a rhombus.
d24 tetrakis hexahedron Each face is in the shape of an isosceles triangle.
d24 deltoidal icositetrahedron
d30 rhombic triacontahedron Each face is in the shape of a rhombus (diamond-shaped).
d100
d%
ZocchihedronTrue d100s are rare; they are nicknamed death stars due to a passing resemblance to the Star Wars structure. Two d10s can substitute for a d100, one of which may have sides labeled 00, 10, 20, … 90. Use of this die, (or a replacement such as two different-colored d10s with there being a convention among players as to which of them will count as "tens" and which as "ones") is referred to as a percentile roll.

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 of all the rolls'. Occasionally they may be written '10×d6+20' or '1d6×10+20'; this means 'roll one six-sided die. Multiply it by ten and add twenty', and avoids boring repetitive dice‐rolling at the expense of reducing the number of possible results (i.e., 30, 40, 50, 60, 70, and 80 are the only possible outcomes) compared to rolling the die 10 times (yielding any number between 30 and 80).

### Application in role-playing games

Dungeons & Dragons is noted for introducing the use of polyhedral dice during modern times. While the game uses traditional six-sided dice from time to time, other types of dice are used more frequently. The Third Edition of Dungeons & Dragons and the offshoot d20 System uses the d20 as a basic mechanic. Most types of dice are used for many different purposes (such as weapon damage, spell damage, saving throws, character generation, etc.).

Players use polyhedral dice together in a number of ways. For example, often a d10 is used in conjunction with a d6 instead of using a d20. If the d6 displays a 1–3, the number on the d10 is resolved as 1–10. If the d6 displays a 4–6, 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". However, d6 are preferred as many players think they have the best "rolling" action (they don't roll too much, such as d20, d12, d10 or d8's may) and they actually roll, whereas d4's usually just sit where they are dropped. Hence a d8 is sometimes used in place of a d4 (1–2 on the d8 gives "1", 3–4 gives "2" etc.)

Almost any die can be used for a throw where a binary result (true or false) is needed. In these cases, the player calls the meaning of the result as the die is thrown, "One to three is true, four to six is false", or simply flips a coin. Some companies produce "binary dice" for just this niche — typically a d6 printed with plus and minus signs, or the words "even" and "odd".

Two d10 (or two d20) are used for probability throws where a 1–100 result is needed. When tossing these dice, the player indicates which dice is "high" (representing the tens position). For example, "red is high".

The Earthdawn game system pioneered "step die mechanics" through the use of its action step table. Generally speaking, a low skill is represented by a low die size, and a high skill is represented by a high die size. The Earthdawn table lists combinations of dice that are expected to produce average rolls from 1 to 40, and is used for almost all die-rolling in the game. Deadlands and The Window also make use of similar step die mechanics, though low abilities in The Window are represented by higher die types.

Both Vampire and Shadowrun use a "success test" mechanic, whereby the player rolls a certain number of the same kind of die (d6s in Shadowrun, d10s in Vampire), and only the dice that roll higher than a certain number are counted towards a successful test.

Several game systems allow dice to "open‐end", whereby if a die shows the highest value, the player may roll the die again and add- sometimes without limit. Usually, the game system uses colorful lingo to describe this mechanic: In the swashbuckling RPG 7th Sea, dice explode; in the western-horror RPG Deadlands, such a die is said to be an ace. In Vampire, this is called the "ten-again" rule.

## References

• Persi Diaconis and Joseph B. Keller. "Fair Dice". The American Mathematical Monthly, 96(4):337–339, 1989. (Discussion of dice that are fair "by symmetry" and "by continuity".)
• Bias and Runs in Dice Throwing and Recording: A Few Million Throws. G. R. Iverson. W. H. Longcour, et al. Psychometrika, Vol. 36, No. 1, March 1971
• Knizia, Reiner (1999). Dice Games Properly Explained. Elliot Right Way Books. ISBN 0716021129.