# Dice

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Four coloured dice showing all six possible sides (on a right-handed, 6-sided die with pips)

A die (plural dice, which can also be used as the singular[1], from French , from Latin datum "something which is given or played")[2] is a small throwable object with multiple resting positions, used for generating random numbers. This makes dice suitable as gambling devices for games like craps, or for use in non-gambling tabletop games.

An example of a traditional die is a rounded cube, with each of its six faces showing a different number of dots (pips) from 1 to 6. When thrown or rolled, the die comes to rest showing on its upper surface a random integer from one to six, each value being equally likely. A variety of similar devices are also described as dice; such specialized dice may have polyhedral or irregular shapes and may have faces marked with symbols instead of numbers. They may be used to produce results other than one through six. Loaded and crooked dice are designed to favor some results over others for purposes of cheating or amusement.

## 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.
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 the Burnt City, an archeological site in south-eastern Iran.[3] Other excavations from ancient tombs in the Indus Valley civilization indicate a South Asian origin.[4] Dicing is mentioned as an Indian game in the Rigveda, Atharvaveda and Buddha games list;[5] it also plays a critical role in the great Hindu epic Mahabharata, where Yudhisthira plays a game of dice against the Kauravas for the northern kingdom of Hastinapura, which becomes the trigger for a war. There are several biblical references to "casting lots", as in Psalm 22, indicating that dicing was commonplace in the region during King David's reign. Knucklebones was a skill game played by women and children; a derivative form had the four sides of the bone receive different values and count as modern dice. Gambling with two or three dice was a very popular form of amusement in Greece, especially with the upper classes, and an invariable accompaniment to symposia.

Knucklebones die, made of steatite.

Dice were originally made from the talus of hoofed animals, colloquially known as "knucklebones". These are approximately tetrahedral, giving the term "bones" used for dice. Modern Mongolians still use such bones as shagai, for games and fortunetelling. Besides bone, materials like ivory, wood and recently plastics like cellulose acetate have been used. Dice are hard to differentiate from knucklebones because ancient writers confused the two, but both were used in prehistoric times.

The Romans were passionate gamblers, especially at the peak of the Roman Empire, and dicing was common though forbidden except during the Saturnalia. Horace derided what he thought as a typical youth of the period, who wasted time on dicing instead of horse-chasing. Throwing dice for money was the cause of many special laws in Rome; one of these stated that no lawsuit could be filed by a person who allowed gambling in his house, even if he had been cheated or assaulted. Professional gamblers were common; some of their loaded dice are preserved in museums. The public houses were the resorts of gamblers, and frescos exist showing two quarrelling dicers being ejected by the host. Twenty-sided dice date back to the 2nd century AD [6] and late BC.[7]

Tacitus stated that the Germans were passionately fond of dicing, so much that they would stake their personal liberty when bankrupt. During the Middle Ages, dicing became the favorite pastime of the knights, with dicing schools and guilds. After the downfall of feudalism, the landsknechts established a reputation as the most notorious dicing gamblers of their time; many of the dice then were curiously carved in the images of men and beasts. In France, dicing was played by both knights and ladies, despite repeated legislation, including interdictions on the part of St. Louis in 1254 and 1256. The markings on Chinese dominoes evolved from the markings on dice.

## Construction

Western, Asian and casino dice.
Chirality of dice. Faces may be placed counterclockwise (top) or clockwise.

Common dice are small cubes most commonly 1.6 cm across, whose faces are numbered from one to six, usually by patterns of round dots called pips. (While the use of Hindu-Arabic numerals is occasionally seen, such as in the game Pop-O-Matic Trouble, the ambigrammatic properties of the digits (for instance, 6 can look like a 9 upside down) make such dice uncommon.) Traditionally, opposite sides of a die add up to seven, implying that the 1, 2 and 3 faces share a vertex;[8] these faces may be placed clockwise or counterclockwise about this vertex. If the 1, 2 and 3 faces run counterclockwise, the die is called right-handed and vice versa. Western dice are normally right-handed and Chinese dice are normally left-handed.[9]

Typical facets of an Asian-style (top) and a Western-style die (bottom). The pips on the Asian-style facets are arranged more compactly than the Western-style facets.

The pips on dice are arranged in specific patterns as shown in the picture to the left. Asian style dice bear similar patterns to Western ones, but the pips are closer to the centre of the face; the one's pip is larger than the others, with that and the four's pips 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's and four's sides would be unlucky; red, a lucky color in Chinese culture, would counteract this. The word for four in Chinese is a homophone of the word for death and is considered unlucky; it is probable that red fours are of Indian origin.[9][10]

The result of a die roll is determined by the way it is thrown, according to the laws of classical mechanics; they are made random by uncertainty due to factors like movements in the thrower's hand. Thus, they are a type of hardware random number generator. Perhaps to militate against concerns that the pips on the faces of certain styles of dice cause a small bias,[citation needed] casinos use dice with flush markings.

Dice are thrown, singly or in groups, from the hand or from a container designed for this (such as a dice cup), onto a flat surface; the face of the die that is uppermost when it comes to rest provides the value of the throw. A typical dice game today is craps, where two dice are thrown at a time and wagers are made on the total value of the two dice. Dice are frequently used to randomize moves in board games, usually by deciding the distance through which a piece will move along the board; examples of this are backgammon and Monopoly.

Clones of board games must use computer generated dice; the values are usually determined by a pseudorandom number generator, then displayed as a visual representation of a die. A notable example is the Mario Party video game series. The reverse is also possible, with bar coded dice shuffling as a source of true random data for computers.[11]

### Manufacturing

Non-precision dice are manufactured via the plastic injection molding process. The pips or numbers on the dice are a part of the mold. The coloring for numbering is achieved by submersing the dice entirely in paint, which is allowed to dry and then polished via a tumble finishing process, similar to rock polishing. The abrasive agent scrapes off all of the paint except for the indents of the numbering. A finer abrasive is then used to polish the die. This process also creates the smoother, rounded edges on the dice.[12]

### Precision dice

Precision casino dice may have a polished or sand finish, making them transparent or translucent respectively. Casino dice have their pips drilled, then filled flush with a paint of the same density as the material used for the dice, such that the center of gravity of the dice is as close to the geometric center as possible. All such dice are stamped with a serial number to prevent potential cheaters from substituting a die.[citation needed] Precision backgammon dice are made the same way; they tend to be slightly smaller and have rounded corners and edges, to allow better movement inside the dice cup and stop forceful rolls from damaging the playing surface.

## Terms

While the terms ace, deuce, trey, cater, cinque and sice have been made obsolete by one to six, they are still used by some professional gamblers to designate different sides of the dice. Ace is from the Latin as, meaning "a unit";[13] the others are 2 to 6 in old French.[citation needed]

### Notation

Using Unicode characters, the faces ⚀ ⚁ ⚂ ⚃ ⚄ ⚅, can be shown in text using the range U+2680–U+2685 or using decimal &9856;–&9861;.[14]

In many gaming contexts, especially tabletop RPGs, the count and number of sides of dice to be rolled at any given time is reduced to a common set of notations; typically these involve the letter "d" for dice. Hence, 6d8 means six eight-sided dice and 2d6 two common dice. Various arithmetic operations are often added at the end; 3d6+4 is three six-sided dice plus four to the outcome.

A loaded, weighted or crooked die is one that has been tampered with to land with a specific side facing upwards more often than it normally would. There are several methods for creating loaded dice; these include round and off-square faces and (if not transparent) weights. Tappers have a mercury drop in a reservoir at the center, with a capillary tube leading to another reservoir at a side; the load is activated by tapping the die so that the mercury travels to the side.

Another type of loaded die is hollow with a small weight and a semi-solid substance inside whose melting point is just lower than the temperature of the human body, allowing the cheater to change the loading of the die by applying body heat, causing the semi-solid to melt and the weight to drift down, making the chosen opposite face more likely to land up. A less common type of loaded die can be made by inserting a magnet into the die and embedding a coil of wire in the game table; running current through the coil increases the likelihood of a certain side landing on the bottom, depending on the direction of the current. Transparent acetate dice, used in all reputable casinos, are harder to tamper with than other dice.

### Shaved dice

A die may be shaved on one side, making it slightly shorter in one dimension, thus affecting its outcome. One countermeasure employed by casinos against shaved dice is to measure the dice with a micrometer before playing.[15]

## Variants

### Non-numeric

The faces of most dice are labelled using sequences of whole numbers, usually starting at one, expressed with either pips or digits. However, there are some applications that require results other than numbers. Examples include letters for Boggle, directions for Warhammer Fantasy Battle, playing card symbols for poker dice, and instructions for sexual acts using sex dice.

### Non-cubic

A Platonic solids set of five dice.
10-sided dice are used in various games, often for generating percentages.

Seven- and eight-sided dice 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.[16][17] Around the end of 1960s, non-cubical dice became popular among players of wargames,[18] and since have been employed extensively in RPGs and TCGs. Reciprocally symmetric numerals like 6 and 9 are distinguished with a dot or by being underlined.

The other four Platonic solids are the most common non-cubical dice; these can make for 4, 8, 12, and 20 faces. The only other common non-cubical die is the 10-sided die. The 4-sided platonic solid is difficult to roll, and a few games like Dayakattai and Daldøs use a 4-sided rolling pin instead. More uncommon dice include 3, 5, 7, 14, 16, 18, 24, 30, 50, and the 100 sided Zocchihedron. Using these dice in various ways, like 10-sided dice in pairs to produce a uniform distribution of random percentages and summing multiple dice to produce approximations to normal distributions, games can closely approximate the real probability distributions of the events they simulate.

The tetrahedral die can be marked in one of two ways – either the numbers are printed around the points, so that when it settles, the numbers at the vertex pointing up are the same and the one counted; or, the numbers can be placed at the middle of the edges, in which case the numbers around the base are read.

A die can be constructed in the shape of a sphere, with the addition of an internal cavity in the shape of the dual polyhedron of the desired die shape and an internal weight. The weight will settle in one of the points of the internal cavity, causing it to settle with one of the numbers uppermost. For instance, a sphere with an octahedral cavity and a small internal weight will settle with one of the 6 points of the cavity held downwards by the weight.

#### Standard variations

Dice are often sold in sets, matching in color, of five or six different shapes (frequently also with a second 10-sided die of a complementary or contrasting color): 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, the faces on a die will be numbered sequentially beginning with 1, and opposite faces will thus add up to one more than the number of faces (but in the case of the d4 and dice with an odd-number of faces, this is simply not possible). Some dice, such as d10, are usually numbered sequentially beginning with 0, in which case the opposite faces will add to one less than the number of faces.

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. The sum of the numbers on opposite faces is usually 9 (numbered 0–9) or 11 (number 1–10).
12 dodecahedron Each face is a regular pentagon. The sum of the numbers on opposite faces is usually 13.
20 icosahedron Faces are equilateral triangles. Icosahedrons have already been made in Roman/ Ptolemaic times, but it is not known if they were used as gaming dice. Modern d20 are sometimes numbered 0–9 twice as an alternative to 10-sided dice. The sum of the numbers on opposite faces is 21 if numbered 1–20.

#### 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. See also: non-cubical dice, Monostatic polytope, and Gömböc.
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 The faces of the 50-sided die are kites, although very narrow.
60 pentakis dodecahedron
60 Pentagonal hexecontahedron Each face is a pentagon
100 Zocchihedron 100-sided dice can be found in hobby and game stores. They are not, however, a true polyhedron. A 100-sided die is made by flattening 100 facets on a sphere.

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 prisms, 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" are the only way to make dice with an odd number of flat faces.[19] 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 die never actually rests on those faces.)

## Probability

Probability density function, p(S), for the sum of two fair six-sided dice, S.

For a single roll of a fair s-sided die, the probability of rolling each value is exactly 1/s; this is an example of a discrete uniform distribution. For n multiple rolls, with a s-sided die the possibility space is equal to sn. So, for n rolls of an s-sided die the probability of any result is 1/sn.

However, if we are rolling two dice and adding the result together, as in the game craps, the total is distributed in a triangular curve; the case for common dice follows:

 Sum Probability 2 3 4 5 6 7 8 9 10 11 12 1/36 2/36 =1/18 3/36 =1/12 4/36 =1/9 5/36 6/36 =1/6 5/36 4/36 =1/9 3/36 =1/12 2/36 =1/18 1/36
Comparison of probability density functions, p(k) for the sum of n fair 6-sided dice to show their convergence to a normal distribution with increasing n, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).

As the number of dice increases, the distribution of the sum of all numbers tends to normal distribution by the central limit theorem; the exact value $F_{s,n}(k)$ of a sum of n s-sided dice, k, is

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

where Fs,1(k) = 1/s for 1 ≤ k ≤ s and 0 otherwise.

A faster algorithm would adapt the exponentiation by squaring algorithm:

$F_{s,x+y}(k) = \sum_i {F_{s,x}(i) F_{s,y}(k - i)} \,$.

In the triangular curve described above,

\begin{align} 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) + \\ & \qquad \cdots + F_{6,1}(5) F_{6,1}(1) \\[6pt] & = 5\cdot\frac{1}{6}\cdot\frac{1}{6}=\frac{5}{36}\approx0.14 \end{align}

Equivalently, the probability can be calculated using combinations:

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

where $\lfloor {x} \rfloor$ is the floor function. The probability of rolling an exact sequence of numbers is 1/sn.

## Application in role-playing games

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.

The fantasy role-playing game Dungeons & Dragons (D&D) is largely credited with popularizing dice in such games. Some games use only one type, like Exalted which uses only ten-sided dice. Others use numerous types for different game purposes, such as D&D, which makes use of all common polyhedral dice.

Dice are used to determine the outcome of events; such usage is called a check. Games typically determine results either as a total on one or more dice above or below a fixed number, or a certain number of rolls above a certain number on one or more dice. Due to circumstances or character skill, the initial roll may have a number added to or subtracted from the final result, or have the player roll extra or fewer dice. To keep track of rolls easily, dice notation is frequently used.

A common special case is percentile rolls, referred to as 1d100 or 1d%. Since actual hundred-sided dice are large, almost spherical, and difficult to read, percentile rolls are instead 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. Some sets of percentile dice explicitly mark one die in tens and the other in units to avoid ambiguity.

Dice for role-playing games are usually plastic; early polyhedral dice from the 1970s and 1980s were made of a soft plastic that would easily wear as the die was used, and wear would gradually render the die unusable. Many early dice were unmarked, and players took great care in painting them. Some twenty-sided dice then were numbered zero through nine twice; half of the numbers had to be painted a contrasting color to differentiate faces. These could double as a ten-sided die by ignoring the distinguishing coloring.

## Application in divination

Dice can be used for divination; using dice for such a purpose is called cleromancy. A pair of common dice is usual, though other forms of polyhedra can be used. Tibetan Buddhists sometimes use this method of divination. It is highly likely that the Pythagoreans used the Platonic solids as dice; they referred to such dice as "the dice of the gods" and they sought to understand the universe through an understanding of geometry in polyhedra.[20]

Astrological dice are a specialized set of three 12-sided dice for divination; the first die represents planets, the Sun, the Moon, and the nodes of the Moon; the second die represents the 12 zodiac signs; and the third represents the 12 houses. An icosahedron provides the answers of the Magic 8-Ball, conventionally used to provide answers to yes-or-no questions.

Medieval People at Play. The Walters Art Museum, Baltimore

## Notes

1. ^ http://oxforddictionaries.com/definition/english/dice
2. ^ "die". AskOxford. Retrieved 2012-06-18.
3. ^ "presstv.ir". presstv.ir. Retrieved 2012-06-18.
4. ^ Possehl, Gregory. "Meluhha". In: J. Reade (ed.) The Indian Ocean in Antiquity. London: Kegan Paul Intl. 1996a, 133–208
5. ^ 2.3, 4.38, 6.118, 7.52, 7.109
6. ^ "christies.com". christies.com. Retrieved 2012-06-18.
7. ^ "metmuseum.org (Dice)". metmuseum.org. Retrieved 2012-11-06.
8. ^ 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."
9. ^ a b Standard Dice from dice-play
10. ^ Chinese Dice from the Elliott Avedon Museum & Archive of Games
11. ^ "Bar Coded Dice for Digital Entropy Collection". Pudec.connotech.com. Retrieved 2012-06-18.
12. ^ How Dice Are Made from Awesome Dice
13. ^ "ace". AskOxford. Retrieved 2012-06-18.
14. ^ "Dice faces in block Miscellaneous Symbols". The Unicode standard.
15. ^ "fullbooks.com". fullbooks.com. Retrieved 2012-06-18.
16. ^ "games.rengeekcentral.com". games.rengeekcentral.com. Retrieved 2012-06-18.
17. ^ "wwmat.mat.fc.ul.pt" (PDF). Retrieved 2012-06-18.
18. ^ Jon Peterson (July 2012). Playing at the World: A History of Simulating Wars, People and Fantastic Adventures, from Chess to Role-Playing Games. Unreason Press. pp. 315–318. ISBN 978-0-615-64204-8.
19. ^ "Properties of Dice". http://www.aleakybos.ch. Retrieved 2012-10-07.
20. ^ Guthrie, Kenneth (1988). The Pythagorean sourcebook and library : an anthology of ancient writings which relate to Pythagoras and Pythagorean philosophy. Grand Rapids, Michigan: Phanes Press. ISBN 978-0-933999-50-3. OCLC 255212063.

## 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, and others. Psychometrika, Vol. 36, No. 1, March 1971
• Knizia, Reiner (1999). Dice Games Properly Explained. Elliot Right Way Books. ISBN 0-7160-2112-9.