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For other uses, see either Die or Dice (disambiguation).

A die (Old French de, from Latin datum "something given or played" ) is a small polyhedral object (usually a cube) suitable as a gambling device (especially for craps or sic bo).

Traditionally, a die is seldom seen alone, rather than as one of a pair of identical dice that are sized to be comfortably rolled or thrown, together, from a user's hand. (The singular word "die" is therefore rare, and treating "dice" as interchangeably singular or plural is not uncommon.) 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

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 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 anti-clockwise order about this vertex.

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.

Terms

While the terms ace, deuce, cater, cinq and sice are hardly common today having been replaced with one, two, three, four, five, and six they are still used by some professional gamblers to describe the different sides of the dice.

Probability

For a single roll, the probability of rolling each value, 1 through 6, is exactly 1 in 6. For a double roll, however, the total of both rolls is not evenly distributed, but is distributed in a triangular curve, as follows:

For the total of rolls of three or more dice, the curve becomes more bell-shaped with each additional die (according to the central limit theorem).

The probability of rolling the same random number repeatedly goes down by /₆ with each additional die:

The above list only applies, if the number you want to throw multiple times in a row, is randomly chosen. If you want to throw a chosen number more than once in a row, the probability is lower:

History

Dice probably evolved from the ankle bones of hoofed animals (such as oxen), colloquially known as "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, because ancient writers confused the two games. It is certain, however, that both were played in prehistoric times.

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 (symposia).

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 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 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 dice

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

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.

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. While red is the most common color, they are also seen in casinos in green, amber, blue, or other colors. 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.

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 that their high-impact plastic dice would not wear down the way other companies' dice did, but for years criticized major dice manufacturers for 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 dice came with two sides with the numbers zero through nine on them; half of the sides had to be painted a contrasting color to signify the "high" side.

Cubical dice with faces representing values other than digits 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.
• 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.

Non-cubical dice

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. When more than one die is used, the standard terminology is to have two numbers separated by the 'd' - Number of Dice 'd' Number of sides on each die. Hence 2d6 is simply Two Six-Sided Dice, suitable for games of Monopoly or Craps.

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 5, 7, 10, 16, 24, 30, 34, 50, or 100 sides, but other than the 10 sided, they are rarely used. (See Zocchihedron.)

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 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"). (Because of their shape, cowry shells probably do not yield a uniform distribution.)

Standard variations

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.

Type	Shape		Platonic?	Notes
d4	tetrahedron		Yes	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 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.
d6	cube		Yes	A common die. The sum of the numbers on opposite faces is seven.
d8	octahedron		Yes	Each face is triangular; looks something like two Egyptian pyramids attached at the base.
d10	pentagonal trapezohedron		No	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, and often all odd numbered faces converge at the same sharp corner, and the even ones at the other.
d12	dodecahedron		Yes	Each face is a regular pentagon.
d20	icosahedron		Yes	Faces are equilateral triangles. Typically, opposite faces add to twenty-one.

Rarer variations

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)

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

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.

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, 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. When tossing these dice, the player indicates which die is "high" (representing the tens position).

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.

See also

• Fuzzy dice
• Craps - on the casino game.
• Threes - street dice game.

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.

External links

• MathWorld: Dice Analysis of dice probabilities.
• Fair Dice is an illustrated Math Games column about all the possible fair dice, and the mathematical reasons why other shapes are not fair.
• Roman Board Games (See, in particular, Tali and Tesserae.)
• Properties of Dice is a page describing all solids that make for provably fair dice.
• Openroleplaying.org's automated Die Roller - allows rolling any combination of any die using standard RPG 'Dice Equations')
• Worlds Largest Dice Collection Links, Photos, Information about dice

 This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). Encyclopædia Britannica (11th ed.). Cambridge University Press. {{cite encyclopedia}}: Missing or empty |title= (help)

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