# Dice: Wikis

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# Encyclopedia

For other uses, see either Die or Dice (disambiguation).
Two standard six-sided pipped dice with rounded corners.
A die or dice (plural dice, from Old French , from Latin datum "something given or played"[1]) is a small polyhedral object, usually cubic, 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.
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 set of dice, and are sized for ease of recognizing the pattern formed by the pips on a face. 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, though the word "dice" used without qualifiers refers to traditional dice by default. 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

Western-style, Asian-style, and casino dice.
Common dice are small cubes 1 to 3 cm along an edge (16 mm being the standard), whose faces are numbered from one to six (usually by patterns of dots called pips). It is traditional to combine pairs of numbers that total seven to opposite faces (it has been since at least classical antiquity)[citation needed]; 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. If the 1, 2 and 3 faces run counterclockwise around their common vertex, the die is called "right handed"; if they run clockwise it is called "left handed". Standard modern Western dice are right-handed, whereas Chinese dice are often left-handed.[2]
The pips on traditional European dice are arranged in specific circular patterns. The face with two usually has the dots in opposite corners, with the third face containing one between these two. The fourth face has one in each corner, and the fifth adds one in the center, forming a quincunx. The final face has two rows of three pips along opposite edges of the face. Pips on Asian-style dice are in a similar pattern, but are typically closer to the centre of the die; the "one" pip is larger than the others; and the "one" and "four" pips are coloured red: in some older sets the "one" pip is a colorless depression. It is suggested that an entirely black and white color combination on the one side would be unlucky and red (a lucky color in Chinese culture) would counteract this. Several legends[citation needed] also mention that the "four" side is colored red because a Chinese emperor (one legend said it was a Ming dynasty emperor, while another stated it was Chung Tsung) ordered it as "fours" helped him win a dice game (sugoroku) against his empress. This story, however, is questionable at best, as it is also probable that "red fours" are also of common Indian origin.[2][3] Another reason why the "four" side might be colored red is because in Asian cultures, the number four is seen as unlucky, like the number thirteen in Western culture, and as mentioned before, it is colored red so that the luckiness of the red counteracts the unluckiness of the four.
Typical facets of an Asian-style (top) and a Western-style die (bottom). Note the compactedness of the pips in the Asian-style die compared to those of the Western-style one.
Dice are thrown to provide random numbers for gambling and other games, and thus are a type of hardware random number generator. The result of a die roll is random in the sense of lacking predictability, not lacking cause. Exactly how dice are thrown determines how they will land according to the laws of classical mechanics. However, dice also can exhibit sensitive dependence on initial conditions, making it difficult to predict the outcome of a die roll even with good information about exactly how it is thrown. Some people claim that the pips on the face of certain styles of dice can cause a small bias, but there is no research to support this claim.[citation needed] The bias is reduced somewhat in the Japanese die with its oversized single pip (pictured).[citation needed] Casino dice have markings that are flush, offering the assurance that this brings them 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, usually by deciding the distance through which a piece will move along the board; examples of this are ludo and backgammon.

### Precision dice

Precision casino dice, used for the game of craps, 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 density as the acetate used for the dice, such that the dice remain in 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 with the pips filled in as with casino dice. While casino dice are noticeably larger than common dice, with sharp edges and corners, precision backgammon dice tend to be slightly smaller. Their corners and edges are rounded to allow better movement inside the dice cup and stop chaotic rolls from damaging the playing surface.

## Computer generated dice

Some computer games, such as clones of board games, must use computer generated dice. The values are usually determined by a random number generator, then displayed as a visual representation of a die. Some sites which show examples of computer generated random dice are GoTinker[4] and random.org [5]

## History

Bone die found at Cantonment Clinch (1823 - 1834), an American fort used in the American Civil War by both Confederate and Union forces at separate times. The fort was also used in 1898 in the Spanish-American War.
Knucklebones die, made of Steatite
A collection of historical dice from Asia
Dice have been used throughout Asia since before recorded history.
The oldest known dice were excavated as part of a 5000-year-old backgammon set, at Shahr-i Sokhta, the Burnt City, an archeological site in south-eastern Iran.[6]
Excavations from ancient tombs in the Harappan civilization,[7] seem to further indicate a South Asian origin. Dicing is mentioned as an Indian game in the Rig Veda, Atharva Veda[8] and Buddha games list. It also plays a critical role great Hindu epic, the Mahabharata, where Yudhisthira plays a game of dice against the Kauravas for the northern kingdom of Hastinapura, which becomes the trigger for a great war. There are several biblical references to "casting lots", as in Psalm 22, indicating that it had become commonplace in the region as of the time of King David. 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 symposia.
(However, the "dice game" mentioned in Rigveda book 10 hymn 34 (which is about the harm that gambling addiction causes) is a game played with about 150 nuts from a tree called Vibhīdaka (Terminalia bellerica).)
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. In addition to bone, ivory, wood, metal, and stone materials have been commonly used. Recently, the use of plastics, including cellulose acetate and Bakelite, is nearly universal. It is almost impossible to trace clearly the development of dice as distinguished from knucklebones, because ancient writers confused the two. It is certain, however, that both were used in prehistoric times.
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. Twenty-sided dice date back to Roman times, as far back as 2nd century AD [1].
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 favorite 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.

## 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"[9]; the others are the numbers 2–6 in old French. (The dice game marketed as Kismet uses ace, deuce, and trey.)

### Notation

In many modern gaming contexts, the count and number of sides of dice to be rolled at any given time is reduced to a common set of notations. Typically this involves the lower-case letter "d", preceded by a die count and followed by (optionally) the number of sides of the dice. For example, 6d8 or 2d6; the former meaning "six eight-sided dice," and the latter meaning "two six-sided dice." Addition or various other arithmetic operations are often added at the end as well, e.g. 3d6+4 "three six-sided dice plus four to the outcome thereof".

### Crooked dice

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

A loaded or gaffed or cogged die is one that has been tampered with to land with a selected side facing upwards more often than it otherwise 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.
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.[10]

#### Iced Dice

Iced dice have lead in them, making them land on the 6 side more often. The "ice" refers to the lead in the dice.

## Variants

### Dice with faces other than digit sequences

As noted, the faces of most dice are labeled using an unbroken series of whole numbers, starting at one (rarely zero), expressed with either pips or digits. Common exceptions include:
• color 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, Queen, King, A♠; the face cards have only colors attached to them and not suits
• 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:
• With its faces numbered 2 3 4 5 6 6
• 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 1 2 3 1 2 3, or 1 2 1 2 1 2
• cubical dice numbered 0 1 2 0 1 2. Dice rolls with these dice have the same expected value as the number of dice thrown.
• 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, +, −, blank, +.
Backgammon 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 majority of the time 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. There is also a doubling octahedron with 1, 2, 4, 8, 16, 32, 64, and 128.
• 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 whole 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.
• Sex Dice. The game generally uses two die, one for actions and one for body part. Instead of numbers, each face on the die contains a word. When rolled, a random combination of words are formed to create a sexual instruction such as "kiss lips" which a player must perform.

### Non-cubical dice

Some dice are polyhedral other than cubical in shape. Both seven– and eight-sided dice of modern format are stated in the 13th century Libro de los juegos to have been invented by Alfonso X in order to speed up play in chess variants.[11][12]
In more recent times around the early 1950s,[citation needed] they have become popular among players of wargames and have since been employed extensively in role-playing games, German-style board games, and trading card 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 discovery of two icosahedral dice dating from the days of ancient Rome, currently on display in the British Museum). In modern times, 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 other numbers of faces 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.
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 uniform distribution of random percentages; they avoid number base conversions and are more convenient. Summing multiple dice produces 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" or "honest" dice (dice that roll with even distributions over their number span). Casino dice are legally required to be fair; those used by others are not subject to legally required standards.
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 cause the dice to keep rolling.
Cowry shells, Yut sticks or coins may be used as a kind of two-sided dice. Because of their lack of symmetry, cowry shells and Yut sticks are not likely to yield a uniform distribution, and the angle and speed of the throw may possibly affect the result.

#### Standard variations

A matched Platonic solids set of five dice, (from left) tetrahedron (4 sides), cube (6), octahedron (8), dodecahedron (12), and icosahedron (20).
Dice are often sold in sets, matching in color, of five or six different shapes: the five Platonic solids, whose faces are regular polygons, and optionally the pentagonal trapezohedron, whose faces are ten kites, each with two different edge lengths and three different angles; the die's vertices also are of two different kinds.
Normally opposing faces of dice will add up to one more than the number of faces, but in the case of the d4, d5, and standard d10 (among others), this is simply not possible.
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 like two square pyramids attached base-to-base. Usually, the sum of the opposite faces is 9.
10 pentagonal trapezohedron Each face is a kite. The die has two sharp corners, where five kites meet, and ten blunter corners, where three kites meet. The ten faces usually bear numbers from zero to nine, rather than one to ten (zero being read as "ten" in many applications). Often all odd numbered faces converge at one 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 2nd century AD Roman icosahedron die is in the collection of the British Museum, though the game it was used for is not known.[13]

#### Rarer variations

Sides Shape Notes
1 sphere Most commonly a joke die[citation needed], this is just a sphere with a 1 marked on it. About spherical dice that may produce more than one result, see the section Non-cubical dice above.
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, the game rules usually simply call for the use of a coin rather than requiring the use of 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 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 prism's top edge. 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 a rhombus.
14 heptagonal trapezohedron Each face is a kite.
16 octagonal dipyramid Each face is an isosceles triangle.
24 tetrakis hexahedron Each face is an isosceles triangle.
24 deltoidal icositetrahedron Each face is a kite.
30 rhombic triacontahedron Each face is a rhombus. Although not included in most dice kits, it can be found in most hobby and game stores.
34 heptadecagonal trapezohedron Each face is a kite.
50 icosakaipentagonal trapezohedron Similar to the 14- and 16-sided dice, the faces of the 50-sided die are kites, although very narrow.
100 Zocchihedron 100-sided dice can be found in hobby and game stores, and such a die is used in some narrative role-playing games such as Dungeons & Dragons. They are not, however, a true polyhedron. A 100-sided die is made by flattening 100 facets on a sphere. The name Zocchihedron was taken from its creator, Lou Zocchi. A typical d100 will be hollow and filled with small plastic objects to dampen the die's momentum when rolled (lest it take off like a golf ball). A 100-sided die is equivalent to a pair of ten-sided dice, and so, even in role-playing games, the Zocchihedron is rarely seen and is generally more of a novelty item.
The full geometric set of "uniform fair dice" (face-transitive) are:
• Platonic solids, the five regular polyhedra: 4, 6, 8, 12, 20 sides
• Catalan solids, the duals of the 13 Archimedean solids: 12, 24, 30, 48, 60, 120 sides
• Bipyramids, the duals of the infinite set of prism, with triangle faces: any even number above 4
• Trapezohedrons, the duals of the infinite set of antiprisms, with kite faces: any even number above 4
• Disphenoids, an infinite set of tetrahedra made from congruent non-regular triangles: 4 sides
• "Rolling-pin style dice" (also called "rolling logs"[14]) are the only way to make dice with an odd number of flat faces.[15] They are based on an infinite set of prisms. All the (rectangular) faces they may actually land on are congruent, so they are equally fair. (The other 2 sides of the prism are rounded or capped with a pyramid, designed so that the dice never actually rests on those faces.)

## Probability

Probability distribution for the sum of two six-sided dice
For a single roll of a fair 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 two six-sided dice, for example, the probability distribution is as follows:
Sum
2
3
4
5
6
7
8
9
10
11
12
Probability
136
236
336
436
536
636
536
436
336
236
136
Probability (simplified)
136
118
112
19
536
16
536
19
112
118
136
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,n of a sum of n s-sided dice can be calculated as the repeated convolution of the single-die probability distribution with itself.
$F_{s,n}(k) = \sum_{i=1}^{k-n+1} {F_{s,1}(i) F_{s,n-1}(k - i)} \,$
where $F_{s,1}(k) = \frac{1}{s}$ for all $1\leq k \leq s$ and 0 otherwise.
A fastest algorithm would adapt the exponentiation by squaring algorithm, using $F_{s,x+y}(k) = \sum_i {F_{s,x}(i) F_{s,y}(k - i)} \,$.
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:
$F_{s,n}(k)=\frac{1}{s^n}\sum_{i=0}^{\left \lfloor \frac{k-n}{s} \right \rfloor} (-1)^i {n \choose i} {k-si-1 \choose n-1}$
The probability of rolling any exact sequence of numbers is simply $\frac{1}{s^n}$ . 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 $\frac{1}{216}$.
The article Sampling equiprobably with dice describes the probabilities of sampling with dice from any range.

## Application in role-playing games

Full set of matching dice used in role-playing games: a d4, d6, d8, d12, d20, and two d10s for percentile: ones and tens.
While polyhedral dice had previously been used in teaching basic arithmetic, the fantasy role-playing game Dungeons & Dragons is largely credited with popularizing their use in roleplaying games. Some games use only one type, such as Exalted which uses only ten-sided dice, while others use numerous types for different game purposes, such as Dungeons & Dragons, which make use of 20-, 12-, 10-, 8- and 4-sided dice in addition to the traditional 6-sided die. Unlike the common six-sided die, these dice often have the numbers engraved on them rather than a series of dots.
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.
Roleplaying games generally use dice to determine the outcome of events, such as the success or failure of actions which are difficult to perform. A player may have to roll dice for combat, skill use, or magic use, amongst other things, generally referred to as a "check". This is generally considered fairer than decision by game master fiat, since success and failure are decided randomly based on a flat probability. Games typically determine success as either a total on one or more dice above (Dungeons & Dragons third edition) or below (Call of Cthulhu) a target number, or a certain number of rolls above a certain number (such as 8 or higher on a d10) on one or more dice (White Wolf's World of Darkness series). The player may gain a bonus or penalty due to circumstances or character skill, usually either by a number added to or subtracted from the final result, or by having the player roll extra or fewer dice. For example, a character trying to climb a sheer wall may subtract from their dice roll (known as a penalty) if the wall is slippery, which simulates the increased difficulty of climbing a slickened surface, while a character using a rope may add to the roll (known as a bonus) to simulate that the rope makes the act of climbing easier.
Dice can also be used by a game master for other purposes, such as to randomly generate game content or to make arbitrary decisions. Some games use dice to determine what attributes the player's character has when created, such as how strong he or she is.
In Dungeons & Dragons and some other roleplaying games which use more than one kind of die, dice notation is used for clarity and conciseness. For example, a six-sided die is referred to as a d6, and the notation for rolling two such dice is 2d6. A constant bias may be added or subtracted by ordinary arithmetic: for example, 2d6+4 adds a 4 point bonus, while 2d6-2 subtracts a 2 point penalty. Games which use only one type of dice rarely require complex dice notation.
A common special case is percentile rolls, referred to in dice notation as 1d100 or 1d%. Since actual hundred-sided dice are large, almost spherical, and difficult to read, percentile rolls are usually handled by rolling two ten-sided dice together, using one as the "tens" and the other as the "units". A roll of ten or zero on either die is taken as a zero, unless both are zeros or tens, in which case this is 100 (rather than zero). To avoid this confusion, some sets of percentile dice exist where one is marked in tens (00, 10, 20... up to 90) and the other from 0 to 9. White and black percentile dice are also used, and are commonly found in color-coded dice sets, or sold separately.
Dice for role-playing games are usually made of plastic, though infrequently metal, wood, 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. Many early dice were unmarked and players took great care in painting their sets of dice. Some twenty-sided dice of this era were numbered zero through nine twice; half of the numbers had to be painted a contrasting color to signify the "high" faces. Such a die could also double as a ten-sided die by ignoring the distinguishing coloring.

## Use of dice for divination

Dice can be used for divination. Using dice for such a purpose is called cleromancy. A pair of standard 6-sided dice is usual though other forms of polyhedra can be used. Tibetan Buddhists sometimes use this method of divination.
It is uncertain if the Pythagoreans used the "Platonic Solids" as dice, but it is highly likely. They referred to these perfect geometries as "The Dice of the Gods". Julia E. Diggins, writer of String, Straightedge, and Shadow (Viking Press, New York, 1965) writes how the Pythagorean Brotherhood sought to understand the mysteries of the Universe through an understanding of geometry in polyhedra. It is recorded that the dodecahedron (12-sided platonic solid) was discovered by Pythagoras. (Guthrie: The Pythagorean Sourcebook)
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.

## Notes

1. ^ AskOxford: die2
2. ^ a b Standard Dice from dice-play
3. ^ Chinese Dice from the Elliott Avedon Museum & Archive of Games
4. ^ Gotinker.com, Project site
5. ^ Random.org - True Random Number Service, Random.org Random number generation
6. ^ http://www.presstv.ir/detail.aspx?id=5668&sectionid=351020108
7. ^ Possehl, Gregory. "Meluhha". In: J. Reade (ed.) The Indian Ocean in Antiquity. London: Kegan Paul Intl. 1996a, 133–208
8. ^ 2.3, 4.38, 6.118, 7.52, 7.109
9. ^ AskOxford: ace
10. ^ http://www.fullbooks.com/The-Art-of-Iugling-or-Legerdemaine.html
11. ^ http://games.rengeekcentral.com/tc4.html
12. ^ http://wwmat.mat.fc.ul.pt/~jnsilva/HJT2k9/AlfonsoX.pdf
13. ^ Thompson, Clive (December 2, 2003). "Ancient Roman dungeonmastering". Collision Detection. Retrieved 2006-06-26.
14. ^ The International Bone Rollers' Guild
15. ^ Properties of Dice

## 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 0-7160-2112-9.

This article incorporates text from the Encyclopædia Britannica, Eleventh Edition, a publication now in the public domain.

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

A die (plural dice, from Old French , from Latin datum "something given or played"[1]) is a small throwable object with multiple resting attitudes, 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.

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 set of dice, and are sized for ease of recognizing the pattern formed by the pips on a face. 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, though the word "dice" used without qualifiers refers to traditional dice by default. Such specialized dice may have polyhedral or irregular 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

dice.]]


Common dice are small cubes 1 to 3 cm along an edge (16 mm being most common), whose faces are numbered from one to six (usually by patterns of round dots called pips). Since classical antiquity[2] it is traditional to arrange the numbers so that opposite faces add to seven; this implies that the faces 1, 2 and 3 meet at a common vertex. This constraint leaves one more abstract design choice: the faces representing 1, 2 and 3 respectively can be placed in either clockwise or counterclockwise order about this vertex. If the 1, 2 and 3 faces run counterclockwise around their common vertex, the die is called "right-handed"; if they run clockwise it is called "left-handed". Standard modern Western dice are right-handed, whereas Chinese dice are often left-handed.[3]

The pips on traditional European dice are arranged in specific patterns. These can be conveniently described by pairs: one pair of pips in opposite corners; the next pair in the remaining corners; and a last pair in the middles of two opposite sides. For odd numbers, a dot is added in the center of the face; thus "three" is a diagonal row, and "five" a quincunx. Asian-style dice bear similar patterns, but the pips are typically closer to the centre of the face; the "one" pip is larger than the others; and the "one" and "four" pips are coloured red. In some older sets the "one" pip is a colorless depression. It is suggested[who?] that an entirely black and white color combination on the "one" side would be unlucky and red (a lucky color in Chinese culture) would counteract this. In some Asian languages, the word "four" sounds similar to the word "death" and is considered unlucky. Several legends[citation needed] also mention that the "four" side is colored red because a Chinese emperor (one legend said it was a Ming dynasty emperor, while another stated it was Chung Tsung) ordered it as "fours" helped him win a dice game (sugoroku) against his empress. This story, however, is questionable at best, as it is also probable that "red fours" are also of common Indian origin.[3][4] Another reason why the "four" side might be colored red is because in Asian cultures, the number four is seen as unlucky, like the number thirteen in Western culture, and as mentioned before, it is colored red so that the luckiness of the red counteracts the unluckiness of the four.

Dice are thrown to provide random numbers for gambling and other games, and thus are a type of hardware random number generator. The result of a die roll is random in the sense of lacking predictability, not lacking cause. Exactly how dice are thrown determines how they will land according to the laws of classical mechanics. However, dice also can exhibit sensitive dependence on initial conditions, making it difficult to predict the outcome of a die roll even with good information about exactly how it is thrown. Some people claim that the pips on the face of certain styles of dice can cause a small bias, but there is no research to support this claim.[citation needed] The supposed bias is reduced somewhat in the Japanese die with its oversized single pip (pictured).[citation needed] Casino dice have markings that are flush, offering the assurance that this brings them very close to providing true uniformly distributed random numbers.[citation needed]

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, usually by deciding the distance through which a piece will move along the board; examples of this are ludo and backgammon.

### Precision dice

Precision casino dice, used for the game of craps, 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 density as the acetate used for the dice, such that the dice remain in balance. In casino play, a stick of 5 dice is used, all stamped with a matching serial number to prevent a cheater from substituting a die.

Precision backgammon dice are also made with the pips filled in as with casino dice. While casino dice are noticeably larger than common dice, with sharp edges and corners, precision backgammon dice tend to be slightly smaller. Their corners and edges are rounded to allow better movement inside the dice cup and to stop chaotic rolls from damaging the playing surface.

## Computer generated dice

Some computer games, such as clones of board games, must use computer generated dice. The values are usually determined by a random number generator, then displayed as a visual representation of a die.

## History

by both Confederate and Union forces at separate times. The fort was also used in 1898 in the Spanish-American War.]]
die, made of Steatite]]


]]Dice have been used throughout Asia since before recorded history.

The oldest known dice were excavated as part of a 5000-year-old backgammon set, at Shahr-i Sokhta, the Burnt City, an archeological site in south-eastern Iran.[5]

Excavations from ancient tombs in the Harappan civilization,[6] seem to further indicate a South Asian origin. Dicing is mentioned as an Indian game in the Rig Veda, Atharva Veda[7] and Buddha games list. It also plays a critical role in the great Hindu epic, the Mahabharata, where Yudhisthira plays a game of dice against the Kauravas for the northern kingdom of Hastinapura, which becomes the trigger for a great war. There are several biblical references to "casting lots", as in Psalm 22, indicating that it had become commonplace in the region as of the time of King David. 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 symposia.

(However, the "dice game" mentioned in Rigveda book 10 hymn 34 (which is about the harm that gambling addiction causes) is a game played with about 150 nuts from a tree called Vibhīdaka (Terminalia bellerica).)

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. In addition to bone, ivory, wood, metal, and stone materials have been commonly used. Recently, the use of plastics, including cellulose acetate and Bakelite, is nearly universal. It is almost impossible to trace clearly the development of dice as distinguished from knucklebones, because ancient writers confused the two. It is certain, however, that both were used in prehistoric times.

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. Twenty-sided dice date back to Roman times, as far back as 2nd century AD.[8]

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 favorite 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.

## 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";[9] the others are the numbers 2–6 in old French. (The dice game marketed as Kismet uses ace, deuce, and trey.)

### Notation

In many modern gaming contexts, the count and number of sides of dice to be rolled at any given time is reduced to a common set of notations. Typically this involves the lower-case letter "d", preceded by a die count and followed by (optionally) the number of sides of the dice. For example, 6d8 or 2d6; the former meaning "six eight-sided dice," and the latter meaning "two six-sided dice." Addition or various other arithmetic operations are often added at the end as well, e.g. 3d6+4 "three six-sided dice plus four to the outcome thereof".

### Crooked dice

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

A loaded (or gaffed or cogged or weighted or crooked or gag) die is one that has been tampered with to land with a selected side facing upwards more often than it otherwise 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.

#### 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.[10]

#### Iced dice

Iced dice have lead in them, making them land on the 6 side more often. The "ice" refers to the lead in the dice.

#### Heated dice

If a die is heated in an oven for 10–15 minutes, with the desired face upward, the plastic will soften slightly and "pool" at the opposite (bottom) side without showing much, if any, visible distortion.

## Variants

### Dice with faces other than digit sequences

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

• color 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, Queen, King, A♠; the face cards have only colors attached to them and not suits
• 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:
• With its faces numbered 2 3 4 5 6 6
• 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 (or, in another scenario, an all-six die with another die that rolls one and five), guaranteeing rolls of seven or 11.
• dice with a single sequence of markings repeated multiple times, for example:
• a cubical die numbered 1 2 3 1 2 3, or 1 2 1 2 1 2
• cubical dice numbered 0 1 2 0 1 2. Dice rolls with these dice have the same expected value as the number of dice thrown.
• 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, +, −, 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 majority of the time 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. There is also a doubling octahedron with 1, 2, 4, 8, 16, 32, 64, and 128.
• 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 whole 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.
• Sex Dice. The game generally uses two die, one for actions and one for body part. Instead of numbers, each face on the die contains a word. When rolled, a random combination of words are formed to create a sexual instruction such as "kiss lips" which a player must perform.
• Money Dice. Six-sided dice used for randomly generating coin values with facings: 1¢, 5¢, 10¢, 25¢, 50¢ and \$, representing the Cent, Nickel, Dime, Quarter, Half dollar and Dollar coins of the United States, Canada and other countries with dollar-based economies and similar coinage denominations.

### Non-cubical dice

]] ]] Some dice are polyhedral other than cubical in shape. Both seven– and eight-sided dice of modern format are stated in the 13th century Libro de los juegos to have been invented by Alfonso X in order to speed up play in chess variants.[11][12]

In more recent times around the early 1950s,[citation needed] they have become popular among players of wargames and have since been employed extensively in role-playing games, German-style board games, and trading card 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 discovery of two icosahedral dice dating from the days of ancient Rome, currently on display in the British Museum). In modern times, 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 other numbers of faces 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 Dayakattai and Daldøs use a 4-sided rolling pin instead.

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 uniform distribution of random percentages; they avoid number base conversions and are more convenient. Summing multiple dice produces approximations to normal distributions ("bell curves"), 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" or "honest" dice (dice that roll with even distributions over their number span). Casino dice are legally required to be fair; those used by others are not subject to legally required standards.

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 cause the dice to keep rolling.

Cowry shells, Yut sticks or coins may be used as a kind of two-sided dice. Because of their lack of symmetry, cowry shells and Yut sticks are not likely to yield a uniform distribution, and the angle and speed of the throw may possibly affect the result.

#### Standard variations

s set of five dice, (from left) tetrahedron (4 sides), cube (6), octahedron (8), dodecahedron (12), and icosahedron (20).]]

Dice are often sold in sets, matching in color, of five or six different shapes: the five Platonic solids, whose faces are regular polygons, and optionally the pentagonal trapezohedron, whose faces are ten kites, each with two different edge lengths and three different angles; the die's vertices also are of two different kinds.

Normally opposing faces of dice will add up to one more than the number of faces, but in the case of the d4, d5, and standard d10 (among others), this is simply not possible.

Sides Shape Notes
4tetrahedron 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.
6cubeA common die. The sum of the numbers on opposite faces is seven.
8octahedronEach face is triangular; looks like two square pyramids attached base-to-base. Usually, the sum of the opposite faces is 9.
10pentagonal trapezohedronEach face is a kite. The die has two sharp corners, where five kites meet, and ten blunter corners, where three kites meet. The ten faces usually bear numbers from zero to nine, rather than one to ten (zero being read as "ten" in many applications). Often all odd numbered faces converge at one sharp corner, and the even ones at the other.
12dodecahedronEach face is a regular pentagon.
20icosahedronFaces are equilateral triangles. Typically, opposite faces add to twenty-one. A 2nd century AD Roman icosahedron die is in the collection of the British Museum, though the game for which it was used is not known.[13]

#### Rarer variations

SidesShapeNotes
1 sphere Most commonly a joke die[citation needed], this is just a sphere with a 1 marked on it. About spherical dice that may produce more than one result, see the section non-cubical dice above.
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, the game rules usually simply call for the use of a coin rather than requiring the use of 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 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. A third variety features faces that resemble warped squares.
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 prism's top edge. 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 a rhombus.
14 heptagonal trapezohedron Each face is a kite.
16 octagonal dipyramid Each face is an isosceles triangle.
24 tetrakis hexahedron Each face is an isosceles triangle.
24 deltoidal icositetrahedron Each face is a kite.
30 rhombic triacontahedron Each face is a rhombus. Although not included in most dice kits, it can be found in most hobby and game stores.
34 heptadecagonal trapezohedron Each face is a kite.
50 icosakaipentagonal trapezohedron Similar to the 14- and 16-sided dice, the faces of the 50-sided die are kites, although very narrow.
100 Zocchihedron 100-sided dice can be found in hobby and game stores, and such a die is used in some narrative role-playing games such as Dungeons & Dragons. They are not, however, a true polyhedron. A 100-sided die is made by flattening 100 facets on a sphere. The name Zocchihedron was taken from its creator, Lou Zocchi. A typical d100 will be hollow and filled with small plastic objects to dampen the die's momentum when rolled (lest it take off like a golf ball). A 100-sided die is equivalent to a pair of ten-sided dice, and so, even in role-playing games, the Zocchihedron is rarely seen and is generally more of a novelty item.

The full geometric set of "uniform fair dice" (face-transitive) are:

• Platonic solids, the five regular polyhedra: 4, 6, 8, 12, 20 sides
• Catalan solids, the duals of the 13 Archimedean solids: 12, 24, 30, 48, 60, 120 sides
• Bipyramids, the duals of the infinite set of prism, with triangle faces: any even number above 4
• Trapezohedrons, the duals of the infinite set of antiprisms, with kite faces: any even number above 4
• Disphenoids, an infinite set of tetrahedra made from congruent non-regular triangles: 4 sides
• "Rolling-pin style dice" (also called "rolling logs"[14]) are the only way to make dice with an odd number of flat faces.[15] They are based on an infinite set of prisms. All the (rectangular) faces they may actually land on are congruent, so they are equally fair. (The other 2 sides of the prism are rounded or capped with a pyramid, designed so that the dice never actually rests on those faces.)

## Probability

For a single roll of a fair 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 two six-sided dice, for example, the probability distribution is as follows:

Sum
2
3
4
5
6
7
8
9
10
11
12
Probability
136
236
336
436
536
636
536
436
336
236
136
Probability (simplified)
136
118
112
19
536
16
536
19
112
118
136

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 $F_\left\{s,n\right\}$ of a sum of n s-sided dice can be calculated as the repeated convolution of the single-die probability distribution with itself.

$F_\left\{s,n\right\}\left(k\right) = \sum_\left\{i=1\right\}^\left\{k-n+1\right\} \left\{F_\left\{s,1\right\}\left(i\right) F_\left\{s,n-1\right\}\left(k - i\right)\right\} \,$

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

A fastest algorithm would adapt the exponentiation by squaring algorithm, using $F_\left\{s,x+y\right\}\left(k\right) = \sum_i \left\{F_\left\{s,x\right\}\left(i\right) F_\left\{s,y\right\}\left(k - i\right)\right\} \,$.

For example, in the triangular curve described above,

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

Equivalently, one can calculate the probability using combinations:

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

The probability of rolling any exact sequence of numbers is simply $\frac\left\{1\right\}\left\{s^n\right\}$ . For example, the chance of rolling 1, 2, and 3 in that order with three rolls of a six-sided die is $\frac\left\{1\right\}\left\{6^3\right\}$, or $\frac\left\{1\right\}\left\{216\right\}$.

The article Sampling equiprobably with dice describes the probabilities of sampling with dice from any range.

## Application in role-playing games

s: a d4, d6, d8, d12, d20, and two d10s for percentile: ones and tens.]] While polyhedral dice had previously been used in teaching basic arithmetic, the fantasy role-playing game Dungeons & Dragons is largely credited with popularizing their use in roleplaying games. Some games use only one type, such as Exalted which uses only ten-sided dice, while others use numerous types for different game purposes, such as Dungeons & Dragons, which make use of 20-, 12-, 10-, 8- and 4-sided dice in addition to the traditional 6-sided die. Unlike the common six-sided die, these dice often have the numbers engraved on them rather than a series of dots.

Roleplaying games generally use dice to determine the outcome of events, such as the success or failure of actions which are difficult to perform. A player may have to roll dice for combat, skill use, or magic use, amongst other things, generally referred to as a "check". This is generally considered fairer than decision by game master fiat, since success and failure are decided randomly based on a flat probability. Games typically determine success as either a total on one or more dice above (Dungeons & Dragons third edition) or below (Call of Cthulhu) a target number, or a certain number of rolls above a certain number (such as 8 or higher on a d10) on one or more dice (White Wolf's World of Darkness series). The player may gain a bonus or penalty due to circumstances or character skill, usually either by a number added to or subtracted from the final result, or by having the player roll extra or fewer dice. For example, a character trying to climb a sheer wall may subtract from their dice roll (known as a penalty) if the wall is slippery, which simulates the increased difficulty of climbing a slickened surface, while a character using a rope may add to the roll (known as a bonus) to simulate that the rope makes the act of climbing easier.

Dice can also be used by a game master for other purposes, such as to randomly generate game content or to make arbitrary decisions. Some games use dice to determine what attributes the player's character has when created, such as how strong he or she is.

In Dungeons & Dragons and some other roleplaying games which use more than one kind of die, dice notation is used for clarity and conciseness. For example, a six-sided die is referred to as a d6, and the notation for rolling two such dice is 2d6. A constant bias may be added or subtracted by ordinary arithmetic: for example, 2d6+4 adds a 4 point bonus, while 2d6-2 subtracts a 2 point penalty. Games which use only one type of dice rarely require complex dice notation.

A common special case is percentile rolls, referred to in dice notation as 1d100 or 1d%. Since actual hundred-sided dice are large, almost spherical, and difficult to read, percentile rolls are usually handled by rolling two ten-sided dice together, using one as the "tens" and the other as the "units". A roll of ten or zero on either die is taken as a zero, unless both are zeros or tens, in which case this is 100 (rather than zero). To avoid this confusion, some sets of percentile dice exist where one is marked in tens (00, 10, 20... up to 90) and the other from 0 to 9. White and black percentile dice are also used, and are commonly found in color-coded dice sets, or sold separately.

Dice for role-playing games are usually made of plastic, though infrequently metal, wood, 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. Many early dice were unmarked and players took great care in painting their sets of dice. Some twenty-sided dice of this era were numbered zero through nine twice; half of the numbers had to be painted a contrasting color to signify the "high" faces. Such a die could also double as a ten-sided die by ignoring the distinguishing coloring.

## Use for divination

Dice can be used for divination. Using dice for such a purpose is called cleromancy. A pair of standard 6-sided dice is usual though other forms of polyhedra can be used. Tibetan Buddhists sometimes use this method of divination.

It is uncertain if the Pythagoreans used the platonic solids as dice, but it is highly likely.[citation needed] They referred to these perfect geometries as "the dice of the gods". Julia E. Diggins writes that the Pythagorean Brotherhood sought to understand the mysteries of the Universe through an understanding of geometry in polyhedra. (Guthrie: The Pythagorean Sourcebook)

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.

## Notes

1. ^ AskOxford: die
2. ^ Cf. Greek Anthology Book 14, §8: "The Opposite Pairs of Numbers on a Die. The numbers on a die run so: six one, five two, three four."
3. ^ a b Standard Dice from dice-play
4. ^ Chinese Dice from the Elliott Avedon Museum & Archive of Games
5. ^ http://www.presstv.ir/detail.aspx?id=5668&sectionid=351020108
6. ^ Possehl, Gregory. "Meluhha". In: J. Reade (ed.) The Indian Ocean in Antiquity. London: Kegan Paul Intl. 1996a, 133–208
7. ^ 2.3, 4.38, 6.118, 7.52, 7.109
8. ^ http://www.christies.com/Lotfinder/lot_details.aspx?intObjectID=4205385
9. ^ AskOxford: ace
10. ^ http://www.fullbooks.com/The-Art-of-Iugling-or-Legerdemaine.html
11. ^ http://games.rengeekcentral.com/tc4.html
12. ^ http://wwmat.mat.fc.ul.pt/~jnsilva/HJT2k9/AlfonsoX.pdf
13. ^ Thompson, Clive (December 2, 2003). "Ancient Roman dungeonmastering". Collision Detection. Retrieved 2006-06-26.
14. ^ The International Bone Rollers' Guild
15. ^ Properties of Dice

## 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 0-7160-2112-9.

This article incorporates text from a publication now in the public domainChisholm, Hugh, ed (1911). Encyclopædia Britannica (Eleventh ed.). Cambridge University Press.

# 1911 encyclopedia

Up to date as of January 14, 2010

### From LoveToKnow 1911

DICE (plural of die, O. Fr. de, derived from Lat. dare, to give), small cubes of ivory, bone, wood or metal, used in gaming. The six sides of a die are each marked with a different number of incised dots in such a manner that the sum of the dots on any two opposite sides shall be 7. Dice seem always to have been employed, as is the case to-day, for gambling purposes, and they are also used in such games as backgammon. There are many methods of playing, from one to five dice being used, although two or three are the ordinary numbers employed in Great Britain and America. The dice are thrown upon a table or other smooth surface either from the hand or from a receptacle called a dice-box, the latter method having been in common use in Greece, Rome and the Orient in ancient times. Dice-boxes have been made in many shapes and of various materials, such as wood, leather, agate, crystal, metal or paper. Many contain bars within to ensure a proper agitation of the dice, and thus defeat trickery. Some, formerly used in England, were employed with unmarked dice, and allowed the cubes to fall through a kind of funnel upon a. board marked off into six equal parts numbered from, to 6. It is a remarkable fact, that, wherever dice have been found, whether in the tombs of ancient Egypt, of classic Greece, or of the far East, they differ in no material respect from those in use to-day, the elongated ones with rounded ends found in Roman graves having been, not dice but tali, or knucklebones. Eightsided dice have comparatively lately been introduced in France as aids to children in learning the multiplication table. The teetotum, or spinning die, used in many modern games, was known in ancient times in China and Japan. The increased popularity of the more elaborate forms of gaming has resulted in the decline of dicing. The usual method is to throw three times with three dice. If one or more sixes or fives are thrown the first. time they may be reserved, the other throws being made with the dice that are left. The object is to throw three sixes= 18 or as, near that number as possible, the highest throw winning, or, when drinks are to be paid for, the lowest throw losing. (For other methods of throwing consult the Encyclopaedia of Indoor Games,. by R. F. Foster, 1903.) The most popular form of pure gambling with dice at the present day, particularly with the lower classes in America, is Craps, or Crap-Shooting, a simple form of Hazard, of French origin. Two dice are used. Each player puts up a stake and the first caster may cover any or all of the bets. He then" followed the lead of its emperors, and, in spite of the severity of shoots, i.e. throws the dice from his open hand upon the table.
If the sum of the dice is 7 or II the throw is a nick, or natural, and the caster wins all stakes. If the throw is either 2, 3 or 12 it is a crap, and the caster loses all. If any other number is thrown it is a point, and the caster continues until he throws the same number again, in which case he wins, or a 7, in which case he loses. The now practically obsolete game of Hazard was much more complicated than Craps. (Consult The Game of Hazard Investigated, by George Lowbut.) Poker dice are marked with ace, king, queen, jack and ten-spot. Five are used and the object is, in three throws, to make pairs, triplets, full hands or fours and fives of a kind, five aces being the highest hand. Straights do not count. In throwing to decide the payment of drinks the usual method is called horse and horse, in which the highest throws retire, leaving the two lowest to decide the loser by the best two in three throws. Should each player win one throw both are said to be horse and horse, and the next throw determines the loser. The two last casters may also agree to sudden death, i.e. a single throw. Loaded dice, i.e. dice weighted slightly on the side of the lowest number, have been used by swindlers from the very earliest times to the present day, a fact proved by countless literary allusions. Modern dice are often rounded at the corners, which are otherwise apt to wear off irregularly.

## History

Dice were probably evolved from knucklebones. The antiquary Thomas Hyde, in his Syntagma, records his opinion that the game of " odd or even," played with pebbles, is nearly coeval with the creation of man. 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. Sophocles, in a fragment, ascribed their invention to Palamedes, a Greek, who taught them to his countrymen during the siege of Troy, and who, according to Pausanias (on Corinth, xx.), made an offering of them on the altar of the temple of Fortune. Herodotus (Clio) relates that the Lydians, during a period of famine in the days of King Atys, invented dice, knucklebones and indeed all other games except chess. 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, while dice were used for gambling, and it was doubtless the gambling spirit of the age which was responsible for the derivative form of knucklebones, in which four sides of the bones received different values, which were then counted, like dice. Gambling with three, sometimes two, dice (idu ot) was a very popular form of amusement in Greece, especially with the upper classes, and was an almost invariable accompaniment to the symposium, or drinking banquet. The dice were cast from conical beakers, and the highest throw was three sixes, called Aphrodite, while the lowest, three aces, was called the dog. Both in Greece and Rome different modes of counting were in vogue. Roman dice were called tesserae from the Greek word for four, indicative of the four sides. The Romans were passionate gamblers, especially in the luxurious days of the Empire, and dicing was a favourite form, though it was forbidden except during the Saturnalia. The emperor Augustus wrote in a letter to Suetonius concerning a game that he had played with his friends: " Whoever threw a dog or a six paid a denarius to the bank for every die, and whoever threw a Venus (the highest) won everything." In the houses of the rich the dice-beakers were of carved ivory and the dice of crystal inlaid with gold. Mark Antony wasted his time at Alexandria with dicing, while, according to Suetonius, the emperors Augustus, Nero and Claudius were passionately fond of it, the last named having written a book on the game. Caligula notoriously cheated at the game; Domitian played it, and Commodus set apart special rooms in his palace for it. The emperor Verus, adopted son of Antonine, is known to have thrown dice whole nights together. Fashionable society the laws, fortunes were squandered at the dicing-table. 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, according to one of which 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. Virgil, in the Copa generally ascribed to him, characterizes the spirit of that age in verse, which has been Englished as follows: " What ho ! Bring dice and good wine !
Who cares for the morrow?

## Liv

so calls grinning Death Live, for I come to you soon!" That the barbarians were also given to gaming, whether or not they learned it from their Roman conquerors, is proved by Tacitus, who 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 (scholae deciorum) and gilds 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, which repeated legislation, including interdictions on the part of St Louis in 1254 and 1256, did not abolish. In Japan, China, Korea, India and other Asiatic countries dice have always been popular and are so still.
See Foster's Encyclopaedia of Indoor Gaines (1903); Raymond's Illustriertes Knobelbrevier (Oranienburg, 1888); Les Jeux des Anciens, by L. Becq de Fouquieres (Paris, 1869); Das KnOchelspiel der Alten, by Bolle (Wismar, 1886); Die Spiele der Griechen and Ramer, by W. Richter (Leipzig, 1887); Raymond's Alte and neue Wiurfelspiele; Chinese Games with Dice, by Stewart Culin (Philadelphia, 1889); Korean Games, by Stewart Culin (Philadelphia, 1895).

# Gaming

Up to date as of February 01, 2010

### From Wikia Gaming, your source for walkthroughs, games, guides, and more!

For the game developer see: DICE, and for the game developer conference see: D.I.C.E. Summit.
A large variety of polyhedral dice can be found in both role-playing and tabletop gaming games.

## Dice notation

In gaming rulebooks and documents, dice are usually refered to as d followed by the number of sides on the die. So d6 is shorthand for a common six-sided die. Often, another number is placed before the d to indicate how many dice are to be thrown. So 2d6 means two six-sided dice should be thrown and their results totalled. 3d6+3 means to roll three six-sided dice and add three to the result.
Many computer games use virtual implementations of physical dice rolls behind the scenes to determine the outcome of an action.

## Standard die variations

Commonly used numbered dice include d4, d6, d8, d10, d12, d20, d100. The d100, or percentile dice, is often simulated by throwing two ten-side dice after specifying which die represents which digit.

d20 Dice Bag
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# Simple English

[[File:|thumb|right|250px|Different types of dice]] Dice are objects used in games when something random needs to be done, such as moving a piece a random number of places on a board. Most dice are cubes that have the numbers 1, 2, 3, 4, 5 and 6 on the faces. Dice are sometimes made in other shapes than cubes, but the 6-sided die (die is the single of dice) is the most common. A die is usually rolled by the player’s hand.

Some games are played with two dice and some use only one. Some games that use dice are Monopoly, Yahtzee, Risk, Dungeons and Dragons, and many more. Dice with different numbers of sides are called polyhedral dice.

Weights can be put inside a die to make it only land on a certain number once it is rolled.

# Citable sentences

Up to date as of December 16, 2010

Here are sentences from other pages on D12, which are similar to those in the above article.