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Subject: Expected value, Board game, Ludo (board game), Randomness, Dominoes
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Four differently colored traditional dice showing all six different sides

Dice (singular die or dice;[1] from Old French ; from Latin datum "something which is given or played";[2] plural dice or occasionally dices) are small throwable objects with multiple resting positions, used for generating random numbers. Dice are suitable as gambling devices for games like craps and are also used in non-gambling tabletop games.

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.

A dice tray, a tray used to contain thrown dice, is sometimes used for gambling or board games, in particular to allow dice throws which do not interfere with other game pieces.


The Royal Game of Ur, a Mesopotamian board game played with dice
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 various regions of Asia

Dice likely originated in ancient Near East. One of the oldest known dice games was excavated from a Mesopotamian tomb, dating back to around 24th century BCE.[3] British archaeologist Leonard Woolley discovered the dice in the Royal Cemetery at Ur with a board game known as the Royal Game of Ur. Two types of dice, stick dice and tetrahedral dice (dice with four triangular surfaces), were found with the board game.[4] Unlike modern dice, the numbers on the opposite sides of Mesopotamian dice were consecutive numbers rather than numbers that add up to seven.[5]

Die made of baked clay from from the Akkadian Period (2350-2150 BC)

The Egyptian game of Senet was played with dice. Senet was played before 3000 BC and up to the 2nd century AD. It was likely a racing game, but there is no scholarly consensus on the rules of Senet.[6] Dicing is mentioned as an Indian game in the Rigveda, Atharvaveda and Buddha games list.[7] There are several biblical references to "casting lots", as in Psalm 22, indicating that dicing (or a related activity) was commonplace when the psalm was composed. It is theorized that dice developed from the practice of fortunetelling with the talus of hoofed animals, colloquially known as "knucklebones", but knucklebones is not the oldest divination technique that incorporates randomness.[3] Knucklebones was a game of skill played by women and children; a derivative form had the four sides of the bone receive different values and count as modern dice.[8]

Knucklebones die, made of steatite

Although gambling was illegal, many Romans were passionate gamblers that enjoyed dicing, which was known as aleam ludere ("to play at dice"). Dicing was even a popular past time of emperors. Letters by Augustus to Tacitus and his daughter recount his hobby of dicing. There were two sizes of Roman dice. Tali were large dice inscribed with one, three, four, and six on four sides. Tesserae were smaller dice with sides numbered from one to six.[9] Twenty-sided dice date back to the 2nd century AD [10] and from Ptolemaic Egypt as early as 2nd century BC.[11]

Dominoes and playing cards originated in China as developments from dice. The transition from dice to playing cards occurred in China around the Tang dynasty, and coincides with the technological transition from rolls of manuscripts to block printed books.[12] In Japan, dice were used to play a popular game called sugoroku. There are two types of sugoroku. Ban-sugoroku is similar to backgammon and dates back to the Heian period, while e-sugoroku is a racing game.[13]


Dice are thrown onto a flat surface either from the hand or from a container designed for this (such as a dice cup). The face of the die that is uppermost when it comes to rest provides the value of the throw. One 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.

The result of a die roll is determined by the way it is thrown, according to the laws of classical mechanics. A die roll is made random by uncertainty in minor factors such as tiny movements in the thrower's hand; They are thus a crude form of hardware random number generator. Perhaps to mitigate against concerns that the pips on the faces of certain styles of dice cause a small bias, casinos use precision dice with flush markings.


A net for a normal 6 sided die
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 centimetres (0.63 in) 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 dice are less common.) Opposite sides of a die traditionally add up to seven, implying that the 1, 2 and 3 faces share a vertex;[14] 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 if those faces run clockwise, the die is called "left-handed". Western dice are normally right-handed, and Chinese dice are normally left-handed.[15]

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. Asian style dice bear similar patterns to Western ones, but the pips are closer to the center of the face; in addition, the pips are differently sized on Asian style dice, and the pips are colored red on the 1 and 4 sides. One possible explanation is that red fours are of Indian origin.[15][16] In some older sets, the "one" pip is a colorless depression.


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 submerging 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.[17]

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. 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.


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";[18] the others are 2 to 6 in old French.[19]


Using Unicode characters, the faces ⚀ ⚁ ⚂ ⚃ ⚄ ⚅, can be shown in text using the range U+2680 to U+2685 or using decimal to .[20]

In many gaming contexts, especially tabletop role-playing games, it is common to see shorthand notations representing different dice rolls. A "d" or "D" is used to indicate a die with a specific number of sides, d4 indicating a four-sided die, for example. If several dice of the same type are to be rolled, this is indicated by a leading number specifying the number of dice. Hence, 6d8 means the player should roll six eight-sided dice. Modifiers to a die roll can also be indicated as desired. For example, 3d6+4 instructs the player to roll three six-sided dice, calculate the total, and add four to it.

Loaded dice

A loaded, weighted or crooked die is one that has been tampered with so that it will land with a specific side facing upwards more or less often than a fair die would. There are several methods for creating loaded dice, including round faces, off-square faces and 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.

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.[21]



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.

Polyhedral dice

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

Around the end of the 1960s, non-cubical dice became popular among players of wargames,[22] and since have been employed extensively in role-playing games and trading card games. Reciprocally symmetric numerals like 6 and 9 are distinguished with a dot or underline.

The other four Platonic solids are the most common non-cubical dice; these can have 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 Chaupur[23] and Daldøs[24] use 4-sided long dice instead. Using these dice in various ways, games can closely approximate the real probability distributions of the events they simulate. For instance, 10-sided dice can be rolled in pairs to produce a uniform distribution of random percentages; and summing the values of multiple dice will produce approximations to normal distributions.[25]

Unlike other common dice, a tetrahedral die does not have a side that faces upward when it is at rest on a surface, so it has to be read in a different way. Many such dice have the numbers printed around the points, so that when it settles, the numbers at the vertex pointing up are the same and the one counted. Less commonly, the numbers on a tetrahedral die can be placed at the middle of the edges, in which case the numbers around the base are read.

A spherical die.

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. They are also sold frequently with a second 10-sided die of a complementary or contrasting color. Sometimes, dice are sold additionally with a die resembling the five Platonic solids, whose faces are regular polygons, or the pentagonal trapezohedron die, whose faces are ten kites, each with two different edge lengths, three different angles, and two different kinds of vertices.

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 a dice with 4 sides and dice with an odd-number of faces, this is simply not possible). Some dice, such as those with 10 sides, are usually numbered sequentially beginning with 0, in which case the opposite faces will add to one less than the number of faces.

Shape Notes
4 Tetrahedron Tetrahedron Each face has three numbers, arranged such that upright number, placed either near the vertex or near the opposite edge, is the same on all three visible faces. The upright numbers represent the value of the roll. This die does not roll well and thus it is usually thrown into the air instead.
6 Cube Cube A common die. The sum of the numbers on opposite faces is seven.
8 octahedron Octahedron Each face is triangular and looks like two square pyramids attached base-to-base. Usually, the sum of the opposite faces is 9.
10 Pentagonal trapezohedron 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 Dodecahedron Each face is a regular pentagon. The sum of the numbers on opposite faces is usually 13.
20 Icosahedron Icosahedron Faces are equilateral triangles. Icosahedrons have been found dating to Roman/Ptolemaic times, but it is not known if they were used as gaming dice. Modern dice with 20 sides 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

Shape Image Notes
1 Sphere Most commonly a joke die, this is just a sphere with a 1 marked on it. See also: non-cubical dice, Monostatic polytope, and Gömböc. A one-sided die can also be shaped like a Möbius strip.
2 Cylinder This is a die with 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. Seven-sided dice are described in the 13th century Libro de los juegos as having been invented by Alfonso X in order to speed up play in chess variants.[26][27]
12 Rhombic dodecahedron Each face is a rhombus.
Triakis tetrahedron Each face is an isosceles triangle.
14 Heptagonal trapezohedron Each face is a kite.
16 Octagonal bipyramid Each face is an isosceles triangle.
18 Rounded rhombicuboctahedron 18 faces are squares; the 8 triangular faces are rounded and cannot be landed on.
24 Triakis octahedron Each face is an isosceles triangle.
Tetrakis hexahedron Each face is an isosceles triangle.
Deltoidal icositetrahedron Each face is a kite.
Pentagonal icositetrahedron Each face is an irregular pentagon.
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.
48 Disdyakis dodecahedron Each face is a scalene triangle.
50 Icosakaipentagonal trapezohedron The faces of the 50-sided die are kites, although very narrow, but numbered 0-49 (0 read as 50 in many applications).
60 Deltoidal hexecontahedron Each face is a kite.
Pentakis dodecahedron Each face is an isosceles triangle.
Pentagonal hexecontahedron Each face is an irregular pentagon.
Triakis icosahedron Each face is an isosceles triangle.
100 Zocchihedron 100-sided dice can be found in hobby and game stores. They are made by flattening 100 facets on a sphere, but are not "uniform fair dice" as described below this table.
120 Disdyakis triacontahedron Each face is a scalene triangle.

"Uniform fair dice" are dice where equal probability of the faces follow from the symmetry of the die (as it is face-transitive), and include:

Dice with an odd number of flat faces can be made as "rolling-pin style dice".[28] 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.)

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 with use, which would gradually render them 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 board games

Many board games use dice to randomize how far pieces move or to settle conflicts. Typically, this has meant that rolling higher numbers is better. Some games, such as Axis & Allies, have inverted this system by making the lower values more potent. In the modern age, a few games and game designers have approached dice in a different way by making each side of the die similarly valuable. In Castles of Burgundy, players spend their dice to take actions based on the die's value. In this game, a six is not better than a one, or vice-versa. In Quarriors (and its descendent, Dicemasters), different sides of the dice can offer completely different abilities. Several sides often give resources while others grant the player useful actions. [29]

Application in divination

Dice can be used for divination and 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.[30]

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.

See also

Medieval People at Play. The Walters Art Museum, Baltimore


  1. ^ Definition of dice in English, Oxford Dictionaries
  2. ^
  3. ^ a b
  4. ^
  5. ^
  6. ^
  7. ^ 2.3, 4.38, 6.118, 7.52, 7.109
  8. ^
  9. ^
  10. ^
  11. ^
  12. ^
  13. ^
  14. ^ 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."
  15. ^ a b
  16. ^ Chinese Dice from the Elliott Avedon Museum & Archive of Games
  17. ^ How Dice Are Made from Awesome Dice
  18. ^
  19. ^
  20. ^
  21. ^
  22. ^
  23. ^
  24. ^
  25. ^
  26. ^
  27. ^
  28. ^
  29. ^
  30. ^


  • 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.

External links

  • Weisstein, Eric W., "Dice", MathWorld. Analysis of dice probabilities, also features Uspenski's work on rolling multiple dice.
  • Mathematically "Fair Dice"
    • "Fair Dice" is an illustrated Math Games column about all the possible fair dice, and the mathematical reasons why other shapes are not fair.
    • A complete list of all possible Fair Dice which has nice illustrations
  • Animation clearly demonstrating the probability space of dice
  • World's Largest Dice Collection Links, Photos, Information about dice
  • Computer Simulation of Irregular Dice
  • "A Pair of Dice Which Never Roll 7"
  • The oldest backgammon set found in Iran
  • "A Brief History of Dice" (in Dungeons & Dragons games)
  • "How do you make loaded dice?", The Straight Dope, July 14, 2009
  • A discussion linking dice and Tarot cards
  • "Why Dice Behave the Way They Do", Popular Science July 1945
  • Dice size chart shows common dice dimensions
  • "Dice – A Dicey Love Affair" A list of board games with special dice

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