For other uses, see either Die or Dice (disambiguation).
Two standard six-sided pipped dice with rounded corners.

Dice (the plural of Die, from Old French , from Latin datum "something given or played"[1]) are small polyhedral objects, usually cubic, used for generating random numbers or other symbols. Plural is a Grammatical number, typically referring to more than one of the Referent in the real world Old French was the Romance Dialect continuum spoken in territories which span roughly the northern half of modern France and parts of modern Belgium Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities sequences such as the results of an ideal die roll, or the digits The musical instrument is spelled Cymbal. A symbol is something --- such as an object, Picture, written word a sound a piece This makes dice suitable as gambling devices, especially for craps or sic bo, or for use in non-gambling tabletop games. Craps is a Dice game played against other players or a bank Craps developed from a simplification of the Old English game hazard. Sic bo (骰寶 also known as tai sai (大細 dai siu (大小 big and small or hi-lo, is an unequal game of chance played with Tabletop game is a general term used to refer to Board games Card games Dice games Miniatures wargames Tile-based games and other

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. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. All of these pips have the same appearance within a pair, or larger set of dice, and are sized for ease of recognizing the pattern 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. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

More generally, a variety of analogous devices are often described as dice, but necessarily in a context, or with a word or two preceding "die" or "dice", that avoids the assumption that traditional dice are intended. Such specialized dice may have cubical or other polyhedral shapes, with faces marked with various collections of symbols, and be used to produce other random results than one through six. There are also "loaded" or "crooked" dice (especially otherwise traditional ones), meant to produce skewed or even predictable results, for purposes of deception or amusement.

## Ordinary dice

European-style, Asian-style, and casino dice.

The common dice are small cubes 1 to 2 cm along an edge (16mm being the standard), whose faces are numbered from one to six (usually by patterns of dots called pips). A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. It is traditional to assign pairs of numbers that total seven to opposite faces (it has been since at least classical antiquity); this implies that at one vertex the faces 1, 2 and 3 intersect. Classical antiquity (also the classical era or classical period) is a broad term for a long period of cultural History centered on the Mediterranean In Geometry, a vertex (plural "vertices" is a special kind of point. 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. A clockwise motion is one that proceeds 'like the Clock 's hands' from the top to the right then down and then to the left and back to the top

The pips on traditional European-style dice are arranged in specific 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. For Sir Francis Galton 's machine for demonstrating the Normal distribution named "quincunx" see Bean machine. 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, and the "one" pip is larger than the others. A unique feature about Asian dice is the fact that the pips for both "one" and "four" are colored red. Why this is so is unknown. But 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 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. The Ming Dynasty ( or Empire of the Great Ming ( was the ruling dynasty of China from 1368 to 1644 following the collapse of the Mongol -led Background Li Xiǎn was born in 656 as the seventh son of his father Emperor Gaozong and the third son of his mother Emperor Gaozong's second wife Empress Wu refers to two different forms of Japanese Board game, one similar to western Backgammon and the other similar to western Snakes and Ladders. This story, however, is questionable at best, as it is also probable that "red fours" are also of common Indian origin. [2][3]

Typical facets of an Asian-style (top) and a European-style die (bottom). Note the compactedness of the pips in the Asian-style die compared to those of the European-style one.

Dice are thrown to provide random numbers for gambling and other games, and thus are a type of hardware random number generator. Randomness is a lack of order Purpose, cause, or predictability A game is a structured activity, usually undertaken for Enjoyment and sometimes also used as an Educational tool In Computing, a hardware random number generator is an apparatus that generates Random numbers from a physical process 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. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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. The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in Chaos theory. Also, because the numbers on typical dice are marked with small indentations, slightly more material is removed from the higher numbered faces. This results in a small bias, and they do not provide fair (uniform) random numbers. The bias is reduced somewhat in the Japanese die with its oversized single pip (pictured). Casino dice have markings that are flush with the surface and come very close to providing true uniformly distributed random numbers.

Dice are thrown, singly or in groups, from the hand or from a cup or box designed for the purpose, onto a flat surface. The face of each die that is uppermost when it comes to rest provides the value of the throw. A typical dice game today is craps, wherein two dice are thrown at a time, and wagers are made on the total value of up-facing pips on the two dice. Dice games are games that use or incorporate a die as their sole or central component usually as a Random device. Craps is a Dice game played against other players or a bank Craps developed from a simplification of the Old English game hazard. 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. A board game is a Game in which counters or pieces that are placed on removed from or moved across a "board" (a premarked surface usually specific to that game Ludo (from Latin ludo, "I play" is a simple Board game for two to four players in which the players race their four tokens from start to Backgammon is a Board game for two players in which the playing pieces are moved according to the roll of Dice.

### 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. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different An acetate, or ethanoate, is either a salt or Ester of Acetic acid. 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. Backgammon is a Board game for two players in which the playing pieces are moved according to the roll of Dice. While casino dice are noticeably larger than common dice, with sharp edges and corners, precision backgammon dice tend to be somewhat smaller. Their corners and edges are beveled to allow greater movement inside the dice cup and prevent chaotic rolls from damaging the playing surface.

## History

Bone die found at Cantonment Clinch (1823 - 1834), an American fort used in the Civil War by both Confederate and Union troops at separate times. The fort was also used in 1898 in the Spanish-American War.
A collection of historical dice from Asia

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. Soapstone (also known as steatite or soaprock) is a Metamorphic rock, a talc- Schist. See Talus for other meanings of the word The talus bone or astragalus of the Ankle joint connects the leg to the foot Oxen (singular ox) are Cattle trained as draft animals. Often they are adult castrated males Knucklebones also known as hucklebones, dibs, dibstones, jackstones, chuckstones or five-stones, is a game of very ancient A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. Modern Mongolians still use such bones, known as shagai, for games and fortunetelling. Mongolia (mɒŋˈɡoʊliə, literally Mongol country/nation,) is a Landlocked Country in East Shagai (Шагай is the Mongolian word for the Cuboid bone of the ankle of a Sheep. Fortune-telling is the practice of predicting the future usually of an individual through mystical or supernatural means and often for commercial gain In addition to bone, ivory, wood, metal, and stone materials have been commonly used. Bones are rigid organs that form part of the Endoskeleton of Vertebrates They function to move support and protect the various organs of the body produce Ivory is formed from Dentine and constitutes the bulk of the Teeth and Tusks of animals such as the Elephant, Hippopotamus, Wood is hard fibrous lignified structural tissue produced as secondary Xylem in the stems of Woody plants notably trees but also shrubs The M acro E xpansion T emplate A ttribute L anguage complements TAL, providing macros which allow the reuse of code across In Geology, rock is a naturally occurring aggregate of Minerals and/or Mineraloids The Earth's outer solid layer the ‘ Lithosphere Recently, the use of plastics, including cellulose acetate and bakelite, is nearly universal. Plastic is the general common term for a wide range of synthetic or semisynthetic organic solid materials suitable for the manufacture of industrial products Cellulose acetate, first prepared in 1865 is the Acetate Ester of Cellulose. Bakelite (ˈbеɪkɨlaɪt is a material based on the thermosetting Phenol formaldehyde resin polyoxybenzylmethylenglycolanhydride developed in 1907–1909 by 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.

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 archeological site in south-eastern Iran. Backgammon is a Board game for two players in which the playing pieces are moved according to the roll of Dice. Shahr-e Sukhte "Burnt City" (شهر سوخته is an archaeological site of a sizable Bronze Age urban settlement associated with the Jiroft culture. For a topic outline on this subject see List of basic Iran topics. Excavations from ancient tombs in the Harappan civilization,[4] seem to further indicate a South Asian origin. The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin Dicing is mentioned as an Indian game in the Rig Veda, Atharva Veda[5] and Buddha games list. This article is about the history of South Asia prior to the Partition of British India in 1947 The Rigveda ( Sanskrit sa ऋग्वेद ṛgveda, a compound of ṛc "praise verse" and veda "knowledge" The Atharvaveda ( Sanskrit: अथर्ववेद atharvaveda, a Tatpurusha compound of {{IAST|atharvan}}, an ancient Rishi The Buddha games list is a list of Games which it is reputed that Gautama Buddha said that he would not play It is also mentioned in the great Hindu epic, the Mahabharata, where Yudhisthira plays a game of dice against the Kauravas for the northern kingdom of Hastinapura. A Hindu ( Devanagari: हिन्दू is an adherent of the philosophies and scriptures of Hinduism, a set of religious, Philosophical In the great Hindu epic Mahabharata, Yudhisthira ( Sanskrit: युधिष्ठिर yudhiṣṭhira) the eldest son of King The term Kaurava ( Sanskrit:कौरव is a Sanskrit term that means a descendant of Kuru, a legendary king who is the ancestor of many of the characters WikipediaWikiProject Indian cities for details --> Hastinapur (हस्‍तिनापुर (Hastinapura in Sanskrit) is a town and a Nagar panchayat In its primitive form knucklebones was essentially a game of skill played by women and children. A game of skill is a Game where the outcome is determined mainly by mental and/or physical Skill, rather than by pure chance. 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. Greece (Ελλάδα transliterated: Elláda, historically, Ellás,) officially the Hellenic Republic (Ελληνική Δημοκρατία Symposium originally referred to a drinking party (the Greek verb sympotein means "to drink together" but has since come to refer to any Academic conference

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. Endless such activities were also conducted in other cities under ancient Rome The Roman Empire was the post-Republican phase of the ancient Roman civilization, characterised by an autocratic form of government and large territorial Saturnalia is the feast with which the Romans commemorated the dedication of the temple of the god Saturn 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. Quintus Horatius Flaccus, ( Venosa, December 8, 65 BC - Rome, November 27, 8 BC known in the English-speaking world as Horace Youth is the period from infancy or childhood to maturity Usage Around the world the terms "youth" " Adolescent " "teenager" and The horse ( Equus caballus) is a hoofed ( Ungulate) Mammal, one of eight living species of the family Equidae. Throwing dice for money was the cause of many special laws in Rome. Law is a system of rules enforced through a set of Institutions used as an instrument to underpin civil obedience politics economics and society Rome ( Roma ˈroma Roma is the capital city of Italy and Lazio, and is Italy's largest and most populous city with more than 2 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. In law a lawsuit is a civil action brought before a Court in which the party commencing the action the Plaintiff, seeks a legal or equitable remedy Professional gamblers were common, and some of their loaded dice are preserved in museums. A museum is a "permanent institution in the service of society and of its development open to the public which acquires conserves researches communicates and exhibits the The common public-houses were the resorts of gamblers, and a fresco is extant showing two quarrelling dicers being ejected by the indignant host. Fresco (plural either frescos or frescoes) is any of several related Painting types done on Plaster on walls or

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. Publius (or Gaius) Cornelius Tacitus (ca 56 &ndash ca 117 was a senator and a Historian of the Roman Empire. The Germanic peoples are a historical group of Indo-European -speaking peoples originating in Northern Europe and identified by their use of the Germanic Centuries later, during the Middle Ages, dicing became the favorite pastime of the knights, and both dicing schools and guilds of dicers existed. Knight is the English term for a social position originating in the Middle Ages. After the downfall of feudalism the famous German mercenaries called landsknechts established a reputation as the most notorious dicing gamblers of their time. Feudalism, a term first used in the early modern period (17th century in its most classic sense refers to a Medieval Europe Political system composed A mercenary is a person who takes part in an armed conflict who is not a national or a party to the conflict and is "motivated to take part in the hostilities essentially by Landsknechts (singular Landsknecht, German plural Landsknechte, sometimes also in English publications were European most often German, Mercenary 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 article is about the country For a topic outline on this subject see List of basic France topics. 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. China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country For a topic outline on this subject see List of basic Japan topics. Korea is a geographic area composed of two sovereign countries a civilization and a former state situated on the Korean Peninsula in East Asia. The markings on Chinese dominoes evolved from the markings on dice, taken two at a time. Chinese Dominoes are used in several Tile-based games namely Tien Gow, Pai Gow, Gwat Pai, Che Deng, Tiu U,

## 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" [1]; the others are the numbers 2–6 in old French. The word " ace " comes from the Old French word 'as' (from Latin 'as' meaning 'a unit' from the name of a small Roman coin (The dice game marketed as Kismet uses ace, deuce, and trey. Kismet is the trademarked name of a commercial Dice game introduced in 1964. )

#### Notation

Main article: Dice 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. Dice notation (also known as dice algebra, common dice notation, RPG dice notation, and several other titles is a system to represent different combinations 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 die is a die that has been tampered with to land with a selected side facing upwards more often than it otherwise would simply by chance. Probability is the likelihood or chance that something is the case or will happen 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. Mercury (ˈmɜrkjʊri also called quicksilver or hydrargyrum, is a Chemical element with the symbol Hg ( Latinized hydrargyrum Capillary action, capillarity, capillary motion, or wicking is the ability of a substance to draw another substance into it 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 pebble is a clast of rock with a Particle size of 4 to 64 Millimeters based on the Krumbein phi scale of Sedimentology

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. Wax has traditionally referred to a substance that is secreted by Bees ( Beeswax) and used by them in constructing their The melting point of a solid is the temperature range at which it changes state from solid to Liquid. 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. A magnet (from Greek grc μαγνήτης λίθος " Magnesian stone" is a material or object that produces a Magnetic field. 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. An acetate, or ethanoate, is either a salt or Ester of Acetic acid. A casino is in the modern sense of the word a facility that houses and accommodates certain types of Gambling activities

#### 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.

## 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. Poker dice are Dice which instead of having number pips have representations of Playing cards upon them A playing card is a piece of specially prepared heavy paper thin card or thin plastic figured with distinguishing motifs and used as one of a set for playing Card games Several varieties exist, but the most common contain the following pattern: 9♣, 10♦, Jack (blue), Queen (green), King (red), A♠
• 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. Boggle is a Word game designed by Allan Turoff and trademarked by Parker Brothers and Hasbro. Irregular military refers to any non-standard military Being defined by exclusion there is a lot of variance in what comes under the term 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:
• one face each with two through five, and two with sixes, or
• for craps, a pair of dice in which one die has five on each face, and its mate has a mixture of twos and sixes, guaranteeing rolls of seven or 11. Craps is a Dice game played against other players or a bank Craps developed from a simplification of the Old English game hazard.
• dice with a single sequence of markings repeated multiple times, for example:
• a cubical die numbered twice from 1 to 3, or thrice from 1 to 2.
• icosahedral dice numbered twice from 1 to 10 (commonly used in Dungeons & Dragons before the popularization of ten-sided dice). Dungeons & Dragons (abbreviated as D&D or DnD) is a Fantasy Role-playing game (RPG originally designed by
• Fudge dice, numbered twice from −1 to 1, represented as −, blank, +. Fudge is a Generic role-playing game system for use in Freeform role-playing games The name " FUDGE " was once an acronym for Freeform
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. A wargame is a Game that simulates or represents a Military operation. Warhammer The Game of Fantasy Battles, formerly Warhammer Fantasy Battle and often abbreviated to Warhammer WFB or WHFB 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. Backgammon is a Board game for two players in which the playing pieces are moved according to the roll of Dice. 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. A board game is a Game in which counters or pieces that are placed on removed from or moved across a "board" (a premarked surface usually specific to that game
• 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 pair of 6-sided dice bearing only positive integers other than a pair of normal dice which have the same probability distribution for the sum 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. The I Ching ( Wade-Giles) or “Yì Jīng” ( Pinyin) also called “Classic of Changes” or “Book of Changes” is one of the oldest of the 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. An overhead projector (OHP is a variant of Slide projector that is used to display images to an audience

### Non-cubical dice

Some dice are polyhedral other than cubical in shape. A barrel die is an uncommon type of polyhedral die used most often by players of Role playing games and Wargames. What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions They became popular around the early 1950s among players of wargames and have since been used extensively in trading card games, German-style board games, and role-playing games. A wargame is a Game that simulates or represents a Military operation. Collectible card games (CCGs also called trading card games (TCGs or customizable card games (CCGs are played using specially designed sets of cards German-style board games are a broad class of games produced in recent decades in Germany and other countries A role-playing game ( RPG; often roleplaying game) is a Game in which the participants assume the roles of Fictional characters. Although polyhedral dice are a relative novelty during modern times, some ancient cultures appear to have used them in games (as evidenced by the presence of two icosahedral dice dating from the days of ancient Rome on display in the British Museum). Ancient Rome was a Civilization that grew out of a small agricultural community founded on the Italian Peninsula as early as the 10th century BC The British Museum is a Museum of human history and culture in London. 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. In Geometry, a Platonic solid is a convex Regular polyhedron. 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. Zocchihedron is the Trademark of the most common 100-sided die, which was invented by Lou Zocchi, and debuted in 1985 ) The 4 sided platonic solid is difficult to roll, and a few games like Daldøs use a 4 sided rolling pin instead. Daldøs ]s] is a Running-fight Board game only known from a few coastal locations in Scandinavia, where its history can be traced

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. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one ---> In Probability Summing multiple dice approximates a normal distribution (a "bell curve"), while eliminating high or low throws can be used to skew the distribution in various ways. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields Using these techniques, games can closely approximate the real probability distributions of the events they simulate.

There is some controversy over whether manufacturing processes create genuinely "fair" dice (dice that roll with even distributions over their number span). Casino dice are legally required to be fair; those used by others hold no such requirement. A casino is in the modern sense of the word a facility that houses and accommodates certain types of Gambling activities

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 obviously cause the dice to keep rolling.

Cowry shells or coins may be used as a kind of two-sided dice. Cowry, also sometimes spelled cowrie, plural always cowries, is the Common name for a group of small to large sea Snails marine main - title Coin keywords numismatics coin review (Because of their shape, cowry shells probably do not yield a uniform distribution. WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one ---> In Probability )

#### Standard variations

A matched Platonic-solids set of five dice, (from left) tetrahedron (4 sides), cube (6), octahedron (8), dodecahedron (12), and icosahedron (20).

The most common non-cubical dice — often sold in sets of five or six that are each differently shaped but with the same pair of background and marking colors — include one each of the five Platonic solids, which are highly symmetrical. In Geometry, a Platonic solid is a convex Regular polyhedron. The six-die versions add the pentagonal trapezohedron, in which the faces (identical to one another as to angles and edge lengths) each have two different lengths of side, and three different sizes of angle; the corners at which multiple faces meet are also of two different kinds. Forms Trigonal trapezohedron - 6 (rhombic faces - dual Octahedron * A Cube is a special case trigonal trapezohedron

SidesShapeNotes
4tetrahedronEach face has three numbers: they are arranged such that the upright number (which counts) is the same on all three visible faces. Four-sided dice are often used in Role-playing games such as Dungeons & Dragons, to get small numbers for things such as damage or character statistic increases A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. 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. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. The sum of the numbers on opposite faces is seven.
8octahedronEach face is triangular; looks something like two Egyptian pyramids attached at the base. An octahedron (plural octahedra is a Polyhedron with eight faces A pyramid is a Building where the upper surfaces are triangular and converge on one point Usually, the sum of the opposite faces is 9.
10pentagonal trapezohedronEach face is kite-shaped; five of them meet at the same sharp corner (as at the top of the diagram in this row), and five at another equally sharp one; about halfway between them, a different group of three faces converges at each of ten blunter corners. As a die Some Role-playing games and Miniature wargames use ten-sided dice typically pentagonal trapezohedra to get random decimal numbers such as percentages As a die Some Role-playing games and Miniature wargames use ten-sided dice typically pentagonal trapezohedra to get random decimal numbers such as percentages In Geometry a kite, or deltoid, is a Quadrilateral with two disjoint pairs of Congruent Adjacent sides in contrast The ten faces usually bear numbers from zero to nine, rather than one to ten (zero being read as "ten" in many applications), and often all odd numbered faces converge at the same sharp corner, and the even ones at the other. In Mathematics, the parity of an object states whether it is even or odd In Mathematics, the parity of an object states whether it is even or odd
12dodecahedronEach face is a regular pentagon. A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal
20icosahedronFaces are equilateral triangles. In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ Properties The area of an equilateral triangle with sides of length a\\! Typically, opposite faces add to twenty-one. A 2nd century CE Roman icosahedron die is in the collection of the British Museum, though the game it was used for is not known. The British Museum is a Museum of human history and culture in London. [6]

#### Rarer variations

SidesShapeNotes
1sphereMost commonly a joke die, this is just a sphere with a 1 marked on it. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe There exist spherical dice constructed of a spherical weight glued between two hemispherical pieces, which have been hollowed to create a certain number places where the weight can rest. The weight effectively pins one side of the sphere down, revealing the number directly opposite.
2cylinderThis is nothing more than a coin shape with 1 marked on one side and 2 on the other. A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis While some tasks in roleplaying require flipping a coin, it is usually referred to as such, and not as rolling 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.
3Rounded-off triangular prismThis is a rounded-off triangular prism, intended to be rolled like a rolling-pin style die. In Geometry, a triangular prism or three-sided prism is a type of prism; it is a Polyhedron made of a triangular base a translated 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. American football, known in the United States and Canada simply as football, is a competitive Team sport known for mixing strategy with A football is a Ball used to play one of the various sports known as Football.
5Triangular prismThis is a prism that is thin enough to land either on its "edge" or "face". In Geometry, a triangular prism or three-sided prism is a type of prism; it is a Polyhedron made of a triangular base a translated 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.
7Pentagonal prismSimilar in constitution to the 5-sided die. In Geometry, the pentagonal prism is a prism with a Pentagonal base 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. The Libro de los Juegos, ("Book of games" or Libro de acedrex dados e tablas, ("Book of chess dice and tables" was commissioned Backgammon is a Board game for two players in which the playing pieces are moved according to the roll of Dice. Some variants have heptagonal ends and rectangular faces. Construction A regular heptagon is not constructible with Compass and straightedge but is constructible with a marked Ruler and compass
12rhombic dodecahedronEach face is in the shape of a rhombus. The rhombic dodecahedron is a convex Polyhedron with 12 rhombic faces In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses
14heptagonal trapezohedronEach face is in the shape of a kite. Construction A regular heptagon is not constructible with Compass and straightedge but is constructible with a marked Ruler and compass Forms Trigonal trapezohedron - 6 (rhombic faces - dual Octahedron * A Cube is a special case trigonal trapezohedron In Geometry a kite, or deltoid, is a Quadrilateral with two disjoint pairs of Congruent Adjacent sides in contrast
16octagonal dipyramidEach face is in the shape of an isosceles triangle. Regular octagons A regular octagon is an octagon whose sides are all the same length and whose internal angles are all the same size Equilateral triangle bipyramids Only three kinds of bipyramids can have all edges of the same length (which implies that all faces are Equilateral triangles: the A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line
24tetrakis hexahedronEach face is in the shape of an isosceles triangle. A tetrakis hexahedron is an Archimedean dual solid or a Catalan solid. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line
24deltoidal icositetrahedronEach face is in the shape of a kite. Related polyhedra The deltoidal icositetrahedron is topologically identical to a Cube which has all of its edges bisected In Geometry a kite, or deltoid, is a Quadrilateral with two disjoint pairs of Congruent Adjacent sides in contrast
30rhombic triacontahedronEach face is in the shape of a rhombus (diamond-shaped). In Geometry, the rhombic triacontahedron is a convex Polyhedron with 30 rhombic faces In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses A thirty sided die is the largest multi-faced die that is a symmetrical polyhedron. It is considered to have the highest rate of generating random numbers due to its large array of numbers and natural spherical shape. Although it is not included in most dice kits, 30 sided die can be found in most hobby and game stores.
34heptadecagonal trapezohedronEach face is in the shape of a kite. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit Forms Trigonal trapezohedron - 6 (rhombic faces - dual Octahedron * A Cube is a special case trigonal trapezohedron In Geometry a kite, or deltoid, is a Quadrilateral with two disjoint pairs of Congruent Adjacent sides in contrast
50icosakaipentagonal trapezohedronSimilar to the 14- and 16-sided dice, the faces of the 50-sided die are kites, although very narrow. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit Forms Trigonal trapezohedron - 6 (rhombic faces - dual Octahedron * A Cube is a special case trigonal trapezohedron In Geometry a kite, or deltoid, is a Quadrilateral with two disjoint pairs of Congruent Adjacent sides in contrast
100ZocchihedronDice of this sort are rare. Zocchihedron is the Trademark of the most common 100-sided die, which was invented by Lou Zocchi, and debuted in 1985 However, they can be found at most hobby and game stores. They are not, however, a true polyhedron. A 100-sided die is made by taking a sphere and placing 100 equally spaced flat planes on the surface. The name Zocchihedron is coined from its creator, Lou Zocchi.

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

• Platonic solids: 5 regular polyhedra: (4, 6, 8, 12, 20 sides)
• Catalan solids: 13 Archimedean duals: (12, 24, 30, 48, 60, 120 sides)
• Bipyramids: infinite set of prism duals, triangle faces: (6, 8, 10, 12, . In Geometry, a Platonic solid is a convex Regular polyhedron. In Mathematics, a Catalan solid, or Archimedean dual, is a Dual polyhedron to an Archimedean solid. In Geometry an Archimedean solid is a highly symmetric semi-regular convex Polyhedron composed of two or more types of Regular polygons meeting Equilateral triangle bipyramids Only three kinds of bipyramids can have all edges of the same length (which implies that all faces are Equilateral triangles: the General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces . . sides)
• Trapezohedrons: infinite set of antiprism duals, kite faces: (6, 8, 10, 12, . Forms Trigonal trapezohedron - 6 (rhombic faces - dual Octahedron * A Cube is a special case trigonal trapezohedron Cartesian coordinates Cartesian coordinates for the vertices of a right antiprism with n -gonal bases and isosceles triangles are ( \cos(k\pi/n . . sides)
• Disphenoids: infinite set of tetrahedra made from congruent non-regular triangles (4 sides)
• "Rolling-pin style dice" (also called "rolling logs" [2]) are the only way to make dice with an odd number of faces[3]. A disphenoid is a Polyhedron whose four faces are identical isosceles or scalene triangles They are based on an infinite set of prisms. General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces 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. WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one ---> In Probability 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:

Probability distribution for the sum of two six-sided dice
Sum
2
3
4
5
6
7
8
9
10
11
12
Probability
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
Probability (simplified)
1/36
1/18
1/12
1/9
5/36
1/6
5/36
1/9
1/12
1/18
1/36

For three or more die rolls, the curve becomes more bell-shaped with each additional die (according to the central limit theorem). The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields The central limit theorem (CLT states that the sum of a sufficiently large number of identically distributed independent Random variables each with finite The exact probability distribution Fs,i of a sum of i s-sided dice can be calculated as the repeated convolution of the single-die probability distribution with itself. In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and

$F_{s,i}(k) = \sum_n {F_{s,1}(n) F_{s,i-1}(k - n)} \,$

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 $F_{s,x+y}(k) = \sum_n {F_{s,x}(n) F_{s,y}(k - n)} \,$. Exponentiating by squaring is an Algorithm used for the fast computation of large Integer powers of a Number.

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:

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

The probability of rolling any exact sequence of numbers is simply $\frac{1}{s^i}$. In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x   k term in the Polynomial 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. An illustrative example of how to compute with the probabilities associated to Dice, as well as the Analysis of algorithms, is the following problem

## Application in role-playing games

Full set of matching dice used in roleplaying: 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. Fantasy is a Genre that uses magic and other Supernatural forms as a primary element of plot, theme, and/or setting A role-playing game ( RPG; often roleplaying game) is a Game in which the participants assume the roles of Fictional characters. Dungeons & Dragons (abbreviated as D&D or DnD) is a Fantasy Role-playing game (RPG originally designed by 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. Exalted is a Role-playing game published by White Wolf Publishing. Dungeons & Dragons (abbreviated as D&D or DnD) is a Fantasy Role-playing game (RPG originally designed by 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. This is generally considered fairer than decision by game master fiat, since success and failure are decided randomly based on a flat probability. A Gamemaster or Game Master (often abbreviated as GM) is a player in a multiplayer Game who acts as organizer arbitrator and officiant in rules situations 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.

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 many types of dice, a dice notation is used. Dice notation (also known as dice algebra, common dice notation, RPG dice notation, and several other titles is a system to represent different combinations For example, a six-sided die is referred to as a d6, while two such dice rolled with the results totaled would be called 2d6. A bonus or penalty applied to the final result is referred to like 2d6+4 or 2d6-2. 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 generally 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, sets of percentile dice exist where one is marked in tens (00, 10, 20. . . up to 90) and the other from 0 to 9.

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

Some people believe that dice can be used for divination. Divination (from Latin divinare "to be inspired by a god" related to Divine, Diva and Deus) is the attempt of ascertaining Using dice for such a purpose is called cleromancy. Cleromancy is a form of Divination using Sortition, casting of lots or casting bones in which an outcome is determined by means that normally would be considered 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. In Geometry, a Platonic solid is a convex Regular polyhedron. 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. Astrology (from Greek grc ἄστρον astron, "constellation star" and grc -λογία -logia) is a group of Systems A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is Zodiac denotes an annual cycle of twelve stations along the Ecliptic, the apparent path of the sun across the heavens through the Constellations that divide the ecliptic Most horoscopic traditions of astrology systems divide the Horoscope into a number (usually twelve of houses whose positions depend on time and location rather The first die represents planets, the Sun, the Moon, and two nodes (North Node and South Node). The Sun (Sol is the Star at the center of the Solar System. 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. The Magic 8-Ball, manufactured by Mattel, is a Toy used for Fortune-telling.