Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random numbers, commonly as part of

"Fair Dice"

''The American Mathematical Monthly'', 96(4):337–339, 1989 (Discussion of dice that are fair "by symmetry" and "by continuity".) * Iverson, G. R.; Longcour, W. H.; et al.; ''Bias and Runs in Dice Throwing and Recording: A Few Million Throws'', Psychometrika, vol. 36, no. 1, March 1971 * Reiner Knizia, Knizia, Reiner, ''Dice Games Properly Explained'', Elliot Right Way Books, 1999,

tabletop game
Tabletop games are game
with separate sliding drawer, from 1390–1353 BC, made of glazed faience, dimensions: 5.5 × 7.7 × 21 cm, in the Brooklyn Museum (New York City)
'', 1560, Pieter Bruegel the Elder
File:Paul Cézan ...

s, including dice games
Dice games are games that use or incorporate one or more dice as their sole or central component, usually as a random device.
The following are games which largely, if not entirely, depend on dice:
Collectible dice games
Patterned after the suc ...

, board game
Board games are tabletop games that typically use moved or placed on a pre-marked board (playing surface) and often include elements of Table game, table, Card game, card, Role-playing game, role-playing, and Miniatures game, miniatures games a ...

s, role-playing game
A role-playing game (sometimes spelled roleplaying game; abbreviated RPG) is a game in which players assume the roles of character
Character(s) may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch ...

s, and games of chance
A game of chance is a game
with separate sliding drawer, from 1390 to 1353 BC, made of glazed faience, dimensions: 5.5 × 7.7 × 21 cm, in the Brooklyn Museum (New York City)
'', 1560, Pieter Bruegel the Elder
File:Paul Cézanne, ...

.
A traditional die is a cube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

with each of its six faces marked with a different number of dots ( pips) from one to six. When thrown or rolled, the die comes to rest showing a random integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

from one to six on its upper surface, with each value being equally likely. Dice may also have polyhedral or irregular shapes and may have faces marked with numerals or symbols instead of pips. Loaded dice (see below) are designed to favor some results over others for cheating
Cheating generally describes various actions designed to subvert rules in order to obtain unfair advantages. This includes acts of bribery, cronyism and nepotism in any situation where individuals are given preference using inappropriate criter ...

or entertainment.
History

Dice have been used since before recorded history, and it is uncertain where they originated. It is theorized that dice developed from the practice of fortune-telling with the talus of hoofed animals, colloquially known asknucklebones
Knucklebones, also known as snobs, astragalus, tali, dibs, fivestones, jacks, or jackstones, among many other names, is a game of Fine motor skill, dexterity played with a number of small objects that are thrown up, caught, and manipulated in va ...

. The Egyptian
Egyptian describes something of, from, or related to Egypt.
Egyptian or Egyptians may refer to:
Nations and ethnic groups
* Egyptians, a national group in North Africa
** Egyptian culture, a complex and stable culture with thousands of years of r ...

game of senet
Senet or senat ( egy, znt, translation=passing; cf. Coptic
Coptic may refer to:
Afro-Asia
* Copts, an ethnoreligious group mainly in the area of modern Egypt but also in Sudan and Libya
* Coptic language, a Northern Afro-Asiatic language spoken ...

was played with flat two-sided throwsticks which indicated the number of squares a player could move, and thus functioned as a form of dice. Senet was played before 3000 BC and up to the 2nd century AD. Bone dice from Skara Brae
Skara Brae is a stone-built Neolithic
The Neolithic period is the final division of the Stone Age, with a wide-ranging set of developments that appear to have arisen independently in several parts of the world. It is first seen about 12 ...

, Scotland have been dated to 3100–2400 BC. Excavations from graves at Mohenjo-daro
Mohenjo-daro (; sd, موئن جو دڙو'', ''meaning 'Mound of the Dead Men';

, an Indus Valley civilization
, c. 2500 BCE. Terracotta
Terracotta, terra cotta, or terra-cotta (; Italian language, Italian: "baked earth", from the Latin ''terra cocta''), a type of earthenware, is a clay-based ceramic glaze, unglazed or glazed ceramic, where the ...

settlement, unearthed terracotta
Terracotta, terra cotta, or terra-cotta (; Italian language, Italian: "baked earth", literally "cooked earth", from the Latin ''terra cocta''), a type of earthenware, is a clay-based ceramic glaze, unglazed or glazed ceramic, where the Pottery f ...

dice dating to 2500–1900 BC.
Games involving dice are mentioned in the ancient Indian ''Rigveda
The ''Rigveda'' or ''Rig Veda'' ( ', from ' "praise" and ' "knowledge") is an ancient Indian collection
Collection or Collections may refer to:
* Cash collection, the function of an accounts receivable department
* Collection agency, ag ...

'', ''Atharvaveda
The Atharva
The Atharva Veda (Sanskrit
Sanskrit (, attributively , ''saṃskṛta-'', nominalization, nominally , ''saṃskṛtam'') is a classical language of South Asia belonging to the Indo-Aryan languages, Indo-Aryan branch of t ...

,'' ''Mahabharata
The ''Mahābhārata'' (; sa, महाभारतम्, ', ) is one of the two major of , the other being the '. It narrates the struggle between two groups of cousins in the and the fates of the and the princes and their successors ...

'' and Buddhist games list. There are several biblical
The Bible (from Koine Greek
Koine Greek (, , Greek approximately ;. , , , lit. "Common Greek"), also known as Alexandrian dialect, common Attic, Hellenistic or Biblical Greek, was the koiné language, common supra-regional form of Greek ...

references to "casting lots" ( ''yappîlū ḡōrāl''), as in Psalm
The Book of Psalms ( or ; he, תְּהִלִּים, , lit. "praises"), commonly referred to simply as Psalms, the Psalter or "the Psalms", is the first book of the ("Writings"), the third section of the Tanakh, and a book of the Christianity ...

22, indicating that dicing (or a related activity) was commonplace when the psalm was composed. Knucklebones was a game of skill
A game of skill or game of wits is a game where the outcome is determined mainly by mental or physical skill, rather than chance.
Alternatively, a game of chance is one where its outcome is strongly influenced by some randomness, randomizing devic ...

played in ancient Greece
Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era wa ...

; a derivative form had the four sides of bones receive different values like modern dice.
Although gambling was illegal, many Romans
Roman or Romans usually refers to:
*Rome
, established_title = Founded
, established_date = 753 BC
, founder = King Romulus
, image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, ...

were passionate gamblers who enjoyed dicing, which was known as ''aleam ludere'' ("to play at dice"). 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. Twenty-sided dice date back to the 2nd century AD and from Ptolemaic Egypt
The Ptolemaic Kingdom (; grc-koi, Πτολεμαϊκὴ βασιλεία, Ptolemaïkḕ basileía) was an Ancient Greek
Ancient Greek includes the forms of the used in and the from around 1500 BC to 300 BC. It is often roughly divi ...

as early as the 2nd century BC.
Dominoes
Dominoes is a family of tile-based games played with gaming pieces, commonly known as dominoes. Each domino is a rectangular tile with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also called ''Pi ...

and playing card
A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a Pap ...

s originated in China as developments from dice. The transition from dice to playing cards occurred in China around the Tang dynasty
The Tang dynasty (, ; ), or Tang Empire, was an imperial dynasty of China that ruled from 618 to 907, with an interregnum
An interregnum (plural interregna or interregnums) is a period of discontinuity or "gap" in a government, organiza ...

(618–907 AD), and coincides with the technological transition from rolls of manuscripts to block printed books. In Japan, dice were used to play a popular game called sugoroku. There are two types of sugoroku. ''Ban-sugoroku'' is similar to backgammon
Backgammon is one of the oldest known board game
Board games are tabletop games that typically use moved or placed on a pre-marked board (playing surface) and often include elements of Table game, table, Card game, card, Role-playing game, ro ...

and dates to the Heian period
The is the last division of classical Japanese history
The first human habitation in the Japanese archipelago has been traced to prehistoric times around 30,000 BCE. The Jōmon period
The is the time in Japanese prehistory, traditi ...

(794–1185), while ''e-sugoroku'' is a racing game
Racing games are a video game genre
A video game genre is a classification assigned to a video game based on its core gameplay (type of interaction) rather than Computer graphics, visual or narrative features. A video game genre is normally ...

.Use

Dice are thrown onto a surface either from the hand or from a container designed for this (such as a cup or tray). The face of the die that is uppermost when it comes to rest provides the value of the throw. 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 maderandom
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. I ...

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
In computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm
of an algorithm (Euclid's algorithm) for calculating the greatest comm ...

.
One typical contemporary dice game
Dice games are games that use or incorporate one or more dice as their sole or central component, usually as a random device.
The following are games which largely, if not entirely, depend on dice:
Collectible dice games
Patterned after the suc ...

is craps
Craps is a dice game
Dice games are games that use or incorporate one or more dice as their sole or central component, usually as a random device.
The following are games which largely, if not entirely, depend on dice:
Collectible dice gam ...

, where two dice are thrown simultaneously and wagers are made on the total value of the two dice. Dice are frequently used to introduce randomness into board game
Board games are tabletop games that typically use moved or placed on a pre-marked board (playing surface) and often include elements of Table game, table, Card game, card, Role-playing game, role-playing, and Miniatures game, miniatures games a ...

s, where they are often used to decide the distance through which a piece will move along the board (as in backgammon
Backgammon is one of the oldest known board game
Board games are tabletop games that typically use moved or placed on a pre-marked board (playing surface) and often include elements of Table game, table, Card game, card, Role-playing game, ro ...

and ''Monopoly
A monopoly (from Greek el, μόνος, mónos, single, alone, label=none and el, πωλεῖν, pōleîn, to sell, label=none) is as described by Irving Fisher, a market with the "absence of competition", creating a situation where a specific ...

'').
Construction

Arrangement

Common dice are smallcube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

s, most often across, whose faces are numbered from one to six, usually by patterns of round dots called pips. (While the use of Arabic numerals
Arabic numerals are the ten numerical digit
A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a Positional notation, positional numeral sy ...

is occasionally seen, such dice are less common.)
Opposite sides of a modern die traditionally add up to seven, requiring the 1, 2, and 3 faces to share a vertex
Vertex (Latin: peak; plural vertices or vertexes) means the "top", or the highest geometric point of something, usually a curved surface or line, or a point where any two geometric sides or edges meet regardless of elevation; as opposed to an Apex ( ...

. The faces of a die may be placed clockwise or counterclockwise about this vertex. If the 1, 2, and 3 faces run counterclockwise, the die is called "right-handed". If those faces run clockwise, the die is called "left-handed". Western dice are normally right-handed, and Chinese dice are normally left-handed.
The pips on standard six-sided 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. Red fours may be of Indian origin.
Manufacturing

Non-precision dice are manufactured via the plasticinjection molding
Injection moulding (U.S. spelling: injection molding) is a manufacturing process for producing parts by injecting molten material into a Molding (process), mould, or mold. Injection moulding can be performed with a host of materials mainly inc ...

process, often made of polymethyl methacrylate (PMMA). The pips or numbers on the die are a part of the mold. Different pigments can be added to the dice to make them opaque or transparent, or multiple pigments may be added to make the dice speckled or marbled.
The coloring for numbering is achieved by submerging the die entirely in paint, which is allowed to dry. The die is then polished via a tumble finishing
Tumble finishing, also known as tumbling or rumbling, is a technique for smoothing and polishing a rough surface on relatively small parts. In the field of metalworking
Metalworking is the process of shaping and reshaping metal
A metal (fro ...

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 produces the smoother, rounded edges on the 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
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

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. This mitigates concerns that the pips will cause a small bias. All such dice are stamped with a serial number
A serial number is a unique identifier
A unique identifier (UID) is an identifier that is guaranteed to be unique among all identifiers used for those objects and for a specific purpose. The concept was formalized early in the development o ...

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.
Etymology and terms

The word die comes fromOld French
Old French (, , ; Modern French
French ( or ) is a Romance language
The Romance languages, less commonly Latin or Neo-Latin languages, are the modern languages that evolved from Vulgar Latin
Vulgar Latin, also known as Popular o ...

'' dé''; from Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

''datum
Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sense, data are a set of values of qualitative property, qualitative or quantity, quantitative variable (research), variables about one or m ...

'' "something which is given or played".
While the terms ''ace
An ace is a playing card
A playing card is a piece of specially prepared , heavy paper, thin cardboard, , cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a to ...

'', ''deuce'', ''trey'', ''cater'', ''cinque'' and ''sice'' are generally obsolete, with the names of the numbers preferred, they are still used by some professional gamblers to designate different sides of the dice. ''Ace'' is from the Latin ''as'', meaning "a unit"; the others are 2 to 6 in Old French
Old French (, , ; Modern French
French ( or ) is a Romance language
The Romance languages, less commonly Latin or Neo-Latin languages, are the modern languages that evolved from Vulgar Latin
Vulgar Latin, also known as Popular o ...

.
When rolling two dice, certain combinations have slang names. The term ''snake eyes'' is a roll of one pip on each die. The ''Online Etymology Dictionary
The ''Online Etymology Dictionary'' is a free online dictionary
In computer technology and , online indicates a state of connectivity and offline indicates a disconnected state. In modern terminology, this usually refers to an , but (especia ...

'' traces use of the term as far back as 1919.
The US term ''boxcars'', also known as ''midnight'', is a roll of six pips on each die. The pair of six pips resembles a pair of boxcar
A boxcar is the North America
North America is a continent entirely within the Northern Hemisphere and almost all within the Western Hemisphere. It can also be described as the northern subcontinent of the Americas. It is bordered to ...

s on a freight train. Many rolls have names in the game of craps.
Unicode representation

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

to `⚅`

.
Loaded dice

A loaded, weighted, cheat, 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 making loaded dice, including rounded faces, off-square faces, and weights. Casinos and gambling halls frequently use transparentcellulose acetate
Cellulose acetate refers to any acetate
An acetate is a salt formed by the combination of acetic acid with a base (e.g. alkaline, earthy, metallic, nonmetal
image:Simple Periodic Table Chart-en.svg, upright=1.75, Nonmetals (and metalloids ...

dice as tampering is easier to detect than with opaque dice.
Variants

Polyhedral dice

Various shapes like two-sided or four-sided dice are documented in archaeological findings; for example, from Ancient Egypt and the Middle East. While the cubical six-sided die became the most common type in many parts of the world, other shapes were always known, like 20-sided dice in Ptolemaic and Roman times. The modern tradition of using ''sets'' of polyhedral dice started around the end of the 1960s when non-cubical dice became popular among players ofwargames
''WarGames'' is a 1983 American Cold War science-fiction film, science fiction techno-thriller film written by Lawrence Lasker and Walter Parkes, Walter F. Parkes and directed by John Badham. The film, which stars Matthew Broderick, Dabney Cole ...

, and since have been employed extensively in role-playing game
A role-playing game (sometimes spelled roleplaying game; abbreviated RPG) is a game in which players assume the roles of character
Character(s) may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch ...

s and trading card games
A collectible card game (CCG), also called a trading card game (TCG), among other names, is a type of card game that mixes strategy game, strategic deck building elements with features of trading cards, introduced with ''Magic: The Gathering'' in 1 ...

. Dice using both the numerals 6 and 9, which are reciprocally symmetric through rotation, typically distinguish them with a dot or underline.
Common variations

Dice are often sold in sets, matching in color, of six different shapes. Five of the dice are shaped like thePlatonic solid
In three-dimensional space, a Platonic solid is a Regular polyhedron, regular, Convex set, convex polyhedron. It is constructed by Congruence (geometry), congruent (identical in shape and size), regular polygon, regular (all angles equal and all sid ...

s, whose faces are regular polygon
In , a regular polygon is a that is (all angles are equal in measure) and (all sides have the same length). Regular polygons may be either or . In the , a sequence of regular polygons with an increasing number of sides approximates a , if the ...

s. Aside from the cube, the other four Platonic solids have 4, 8, 12, and 20 faces, allowing for those number ranges to be generated. The only other common non-cubical die is the 10-sided die, a pentagonal trapezohedron
The ''n''-gonal trapezohedron, antidipyramid, antibipyramid, or deltohedron is the dual polyhedron of an ''n''-gonal antiprism. The 2''n'' faces of the ''n''-trapezohedron are congruent and symmetrically staggered; they are called ''twisted ki ...

die, whose faces are ten kites
A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...

, each with two different edge lengths, three different angles, and two different kinds of vertices. Such sets frequently include a second 10-sided die either of contrasting color or numbered by tens, allowing the pair of 10-sided dice to be combined to generate numbers between 1 and 100.
Using these dice in various ways, games can closely approximate a variety of probability distribution
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

s. 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 distribution
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by ex ...

s.
Unlike other common dice, a four-sided (tetrahedral) die does not have a side that faces upward when it is at rest on a surface, so it must be read in a different way. On some four-sided dice, each face features multiple numbers, with the same number printed near each vertex on all sides. In this case, the number around the vertex pointing up is used. Alternatively, the numbers on a tetrahedral die can be placed at the middles of the edges, in which case the numbers around the base are used.
Normally, the faces on a die will be placed so opposite faces will add up to one more than the number of faces. (This is not possible with 4-sided dice and dice with an odd-number of faces.) 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.
Some twenty-sided dice have a different arrangement used for the purpose of keeping track of an integer that counts down, such as health points. These ''spindown dice'' are arranged such that adjacent integers appear on adjacent faces, allowing the user to easily find the next lower number. They are commonly used with collectible card games
A collectible card game (CCG), also called a trading card game (TCG), among other names, is a type of card game
A card game is any game using playing cards as the primary device with which the game is played, be they traditional or game-specif ...

.
Rarer variations

"Uniform fair dice" are dice where all faces have equal probability of outcome due to the symmetry of the die as it isface-transitive
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

. Theoretically, these include:
* Bipyramid
A (symmetric) ''n''-gonal bipyramid or dipyramid is a polyhedron
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the old ...

s, the duals of the infinite set of prisms
An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical shape o ...

, with triangle faces: any multiple of 4 (so that a face will face up) above 8
* Catalan solid
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...

s, the duals
''Duals'' is a compilation album by the Irish Rock music, rock band U2. It was released in April 2011 to u2.com subscribers.
Track listing
:* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose ...

of the 13 Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex polytope, convex Uniform polyhedron, uniform polyhedra composed of regular polygons meeting in identical vertex (geometry), vertices, ...

s: 12, 24, 30, 48, 60, 120 sides
* Disphenoid
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

s, an infinite set of tetrahedra made from congruent non-regular triangles: 4 sides. This is a less symmetric tetrahedron than the Platonic tetrahedron, but still sufficiently symmetrical to be face-transitive. Similarly, and tetartoid
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

s are less symmetrical but still face-transitive dodecahedra: 12 sides.
* Platonic solid
In three-dimensional space, a Platonic solid is a Regular polyhedron, regular, Convex set, convex polyhedron. It is constructed by Congruence (geometry), congruent (identical in shape and size), regular polygon, regular (all angles equal and all sid ...

s, the five regular polyhedra: 4, 6, 8, 12, 20 sides
* Trapezohedra, the duals of the infinite set of antiprism
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

s, with kite faces: any even number not divisible by 4 (so that a face will face up) above 6
Two other types of polyhedra are technically not face-transitive, but are still fair dice due to symmetry:
* antiprism
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

s: the basis of barrel dice
* prism
A prism
An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical ...

s: the basis of long dice and teetotums
Long dice and teetotum
A teetotum (or T-totum) is a form of gambling
File:A photo of a gambling stand in Paris.jpg, A gambling stand in Paris
Gambling (also known as betting) is the wagering of money or something of Value (economics), value (referred to as "the st ...

s can in principle be made with any number of faces, including odd numbers. Long dice are based on the infinite set of prisms
An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical shape o ...

. All the rectangular faces are mutually face-transitive, so they are equally probable. The two ends of the prism may be rounded or capped with a pyramid, designed so that the die cannot rest on those faces. 4-sided long dice are easier to roll than tetrahedra, and are used in the traditional board games dayakattai
Dayakattai or Dayaboss is a Tamil dice game played by 2 or 4 people (or multiples) by forming teams. It originated in Tamil Nadu
Tamil Nadu () is a States and union territories of India, state in southern India. Its capital and largest city is ...

and daldøs.
Non-numeric dice

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 forBoggle
''Boggle'' is a word game
Word games (also called word game puzzles or word search games) are spoken or board games often designed to test ability with language
A language is a structured system of communication used by humans, includin ...

, directions for ''Warhammer Fantasy Battle
''Warhammer'' (formerly ''Warhammer Fantasy Battle'' or just ''Warhammer Fantasy'') is a tabletop miniature wargame
Miniature wargaming is a form of wargaming in which players enact battles between opposing military forces
A mili ...

'', , playing card symbols for poker dice, and instructions for sexual acts using sex dice.
Alternatively-numbered dice

Dice may have numbers that do not form a counting sequence starting at one. One variation on the standard die is known as the "average" die. These are six-sided dice with sides numbered`2, 3, 3, 4, 4, 5`

, which have the same arithmetic mean
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

as a standard die (3.5 for a single die, 7 for a pair of dice), but have a narrower range of possible values (2 through 5 for one, 4 through 10 for a pair). They are used in some table-top wargames
''WarGames'' is a 1983 American Cold War science-fiction film, science fiction techno-thriller film written by Lawrence Lasker and Walter Parkes, Walter F. Parkes and directed by John Badham. The film, which stars Matthew Broderick, Dabney Cole ...

, where a narrower range of numbers is required. Other numbered variations include Sicherman dice and intransitive diceA set of dice is intransitive (or nontransitive) if it contains three dice, ''A'', ''B'', and ''C'', with the property that ''A'' rolls higher than ''B'' more than half the time, and ''B'' rolls higher than ''C'' more than half the time, but it is ...

.
Spherical dice

A die can be constructed in the shape of a sphere, with the addition of an internal cavity in the shape of thedual polyhedron
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

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.
Applications

Polyhedral dice are commonly used in role-playing games. The fantasy role-playing game ''Dungeons & Dragons
''Dungeons & Dragons'' (commonly abbreviated as ''D&D'' or ''DnD'') is a fantasy tabletop role-playing game (RPG) originally designed by Gary Gygax and Dave Arneson. It was first published in 1974 by TSR (company)#Tactical Studies Rules, Tacti ...

'' (D&D) is largely credited with popularizing dice in such games. Some games use only one type, like ''Exalted
''Exalted'' is a role-playing game published by White Wolf Publishing. The game is classified as high fantasy and it was inspired by a mixture of world mythologies as well as Japanese anime. The game is currently in its third edition. First Edition ...

'' 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 usually used to determine the outcome of events. 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 notationDice notation (also known as dice algebra, common dice notation, RPG dice notation, and several other titles) is a system to represent different combinations of dice in wargames and tabletop role-playing games using simple algebra-like notation such ...

is frequently used.
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 descendant, ''Dicemasters''), different sides of the dice can offer completely different abilities. Several sides often give resources while others grant the player useful actions.
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 Mo (divination), 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.
Astrological dice are a specialized set of three 12-sided dice for divination; the first die represents the planets, the Sun, the Moon, and the orbital node, nodes of the Moon, the second die represents the 12 Astrological sign, zodiac signs, and the third represents the House (astrology), 12 houses. A specialized icosahedron die provides the answers of the Magic 8-Ball, conventionally used to provide answers to yes-or-no questions.
Dice can be used to generate random numbers for use in passwords and cryptography applications. The Electronic Frontier Foundation describes a method by which dice can be used to generate passphrases. Diceware is a method recommended for generating secure but memorable passphrases, by repeatedly rolling five dice and picking the corresponding word from a pre-generated list.
Notation

In many gaming contexts, especially tabletop role-playing games, shorthand notations representing different dice rolls are used. A "d" or "D" is used to indicate a die with a specific number of sides; for example,`d4`

denotes a four-sided die. 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 and add the results. 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.
See also

* Crown and Anchor * d20 System * Dreidel * Fuzzy dice * ''Musikalisches Würfelspiel'' * Dice throw (review) * Quincunx * Senary * Teetotum * Urim and ThummimReferences

* Persi Diaconis, Diaconis, Persi; and Keller, Joseph B."Fair Dice"

''The American Mathematical Monthly'', 96(4):337–339, 1989 (Discussion of dice that are fair "by symmetry" and "by continuity".) * Iverson, G. R.; Longcour, W. H.; et al.; ''Bias and Runs in Dice Throwing and Recording: A Few Million Throws'', Psychometrika, vol. 36, no. 1, March 1971 * Reiner Knizia, Knizia, Reiner, ''Dice Games Properly Explained'', Elliot Right Way Books, 1999,

External links

* analysis of dice probabilities, also featuring Uspenski's work on rolling multiple dice {{Authority control Dice, Gaming devices