A die (: dice, sometimes also used as ) is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating
random values, commonly as part of
tabletop game
Tabletop games or tabletops are games that are normally played on a Table (furniture), table or other flat surface, such as board games, card games, dice games, miniature wargames, Tabletop role-playing game, tabletop role-playing games, or ti ...
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 su ...
,
board game
A board game is a type of tabletop game that involves small objects () that are placed and moved in particular ways on a specially designed patterned game board, potentially including other components, e.g. dice. The earliest known uses of the ...
s,
role-playing game
A role-playing game (sometimes spelled roleplaying game, or abbreviated as RPG) is a game in which players assume the roles of player character, characters in a fictional Setting (narrative), setting. Players take responsibility for acting out ...
s, and
games of chance
A game of chance is in contrast with a game of skill. It is a game whose outcome is strongly influenced by some randomizing device. Common devices used include dice, spinning tops, playing cards, roulette wheels, numbered balls, or in the case ...
.
A traditional die is a
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
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 is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
from one to six on its upper surface, with each value being equally likely. Dice may also have other
polyhedral
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surfa ...
or irregular shapes, may have faces marked with numerals or symbols instead of pips and may have their numbers carved out from the material of the dice instead of marked on it. Loaded dice are specifically designed or modified to favor some results over others, for
cheating
Cheating generally describes various actions designed to subvert or disobey rules in order to obtain unfair advantages without being noticed. This includes acts of bribery, cronyism and nepotism in any situation where individuals are given pr ...
or entertainment purposes.
History
Dice have been used since before recorded history, and their origin is uncertain. It is hypothesized that dice developed from the practice of fortune-telling with the
talus of hoofed animals, colloquially known as
knucklebones
Knucklebones, also known as scatter jacks, snobs, astragaloi (''singular'': astragalus), tali, dibs, fivestones, jacks, jackstones, or jinks, among many other names, is a game of Fine motor skill, dexterity played with a number of small objects ...
.
The
Ancient Egypt
Ancient Egypt () was a cradle of civilization concentrated along the lower reaches of the Nile River in Northeast Africa. It emerged from prehistoric Egypt around 3150BC (according to conventional Egyptian chronology), when Upper and Lower E ...
ian game of
senet
Senet or senat (; cf. Coptic language, Coptic , 'passing, afternoon') is a board game from ancient Egypt that consists of ten or more pawns on a 30-square playing board.Crist 2019 p. 107 The earliest representation of senet is dated to 2620 BC ...
(played before 3000
BCE and up to the 2nd century CE) 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.
Perhaps the oldest known dice were excavated as part of a
backgammon
Backgammon is a two-player board game played with counters and dice on tables boards. It is the most widespread Western member of the large family of tables games, whose ancestors date back at least 1,600 years. The earliest record of backgammo ...
-like game set at the
Burnt City
Shahr-e Sukhteh (, meaning "Burnt City"), c. 3550–2300 BC,Ascalone, E., and P. F. Fabbri, (2022)"Demographic considerations regarding the settlement and necropolis of Shahr i Sokhta" in: E. Ascalone and S.M.S. Sajjadi (eds.), Excavations and R ...
, an archeological site in south-eastern
Iran
Iran, officially the Islamic Republic of Iran (IRI) and also known as Persia, is a country in West Asia. It borders Iraq to the west, Turkey, Azerbaijan, and Armenia to the northwest, the Caspian Sea to the north, Turkmenistan to the nort ...
, estimated to be from between 2800 and 2500 BCE. Bone dice from
Skara Brae
Skara Brae is a stone-built Neolithic settlement, located on the Bay of Skaill in the parish of Sandwick, Orkney, Sandwick, on the west coast of Mainland, Orkney, Mainland, the largest island in the Orkney archipelago of Scotland. It consiste ...
, Scotland have been dated to 3100–2400 BCE. Excavations from graves at
Mohenjo-daro
Mohenjo-daro (; , ; ) is an archaeological site in Larkana District, Sindh, Pakistan. Built 2500 BCE, it was one of the largest settlements of the ancient Indus Valley Civilisation, and one of the world's earliest major city, cities, contemp ...
, an
Indus Valley civilization
The Indus Valley Civilisation (IVC), also known as the Indus Civilisation, was a Bronze Age civilisation in the northwestern regions of South Asia, lasting from 3300 BCE to 1300 BCE, and in its mature form from 2600 BCE ...
settlement, unearthed
terracotta
Terracotta, also known as terra cotta or terra-cotta (; ; ), is a clay-based non-vitreous ceramic OED, "Terracotta""Terracotta" MFA Boston, "Cameo" database fired at relatively low temperatures. It is therefore a term used for earthenware obj ...
dice dating to 2500–1900 BCE, including at least one die whose opposite sides all add up to seven, as in modern dice.
Games involving dice are mentioned in the
ancient Indian
The following Outline (list), outline is provided as an overview of and topical guide to ancient India:
Ancient India is the Indian subcontinent from prehistoric times to the start of Medieval India, which is typically dated (when the term is ...
''
Rigveda
The ''Rigveda'' or ''Rig Veda'' (, , from wikt:ऋच्, ऋच्, "praise" and wikt:वेद, वेद, "knowledge") is an ancient Indian Miscellany, collection of Vedic Sanskrit hymns (''sūktas''). It is one of the four sacred canoni ...
'', ''
Atharvaveda
The Atharvaveda or Atharva Veda (, , from ''wikt:अथर्वन्, अथर्वन्'', "priest" and ''wikt:वेद, वेद'', "knowledge") or is the "knowledge storehouse of ''wikt:अथर्वन्, atharvans'', the proced ...
,'' ''
Mahabharata
The ''Mahābhārata'' ( ; , , ) is one of the two major Sanskrit Indian epic poetry, epics of ancient India revered as Smriti texts in Hinduism, the other being the ''Ramayana, Rāmāyaṇa''. It narrates the events and aftermath of the Kuru ...
'' and the
Buddhist games list. Knucklebones was a
game of skill
A game of skill 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 randomizing device, such as dice, spinn ...
played in
ancient Greece
Ancient Greece () was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity (), that comprised a loose collection of culturally and linguistically r ...
; a derivative form had the four sides of bones receive different values like modern dice.
Although gambling was illegal, many
Romans 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 CE and from
Ptolemaic Egypt Ptolemaic is the adjective formed from the name Ptolemy, and may refer to:
Pertaining to the Ptolemaic dynasty
* Ptolemaic dynasty, the Macedonian Greek dynasty that ruled Egypt founded in 305 BC by Ptolemy I Soter
*Ptolemaic Kingdom
Pertaining ...
as early as the 2nd century BCE.
Dominoes
Dominoes is a family of tile-based games played with gaming pieces. Each domino is a rectangular tile, usually with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also called ''Pip (counting), pips ...
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 f ...
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 (, ; zh, c=唐朝), or the Tang Empire, was an Dynasties of China, imperial dynasty of China that ruled from 618 to 907, with an Wu Zhou, interregnum between 690 and 705. It was preceded by the Sui dynasty and followed ...
(618–907 CE), 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
(literally 'double six') refers to two different forms of a Japanese board game: ''ban-sugoroku'' (盤双六, 'board-sugoroku') which is similar to western tables games like backgammon, and ''e-sugoroku'' (絵双六, 'picture-sugoroku') which i ...
. There are two types of sugoroku. ''Ban-sugoroku'' is similar to
backgammon
Backgammon is a two-player board game played with counters and dice on tables boards. It is the most widespread Western member of the large family of tables games, whose ancestors date back at least 1,600 years. The earliest record of backgammo ...
and dates to the
Heian period
The is the last division of classical Japanese history, running from 794 to 1185. It followed the Nara period, beginning when the 50th emperor, Emperor Kammu, moved the capital of Japan to Heian-kyō (modern Kyoto). means in Japanese. It is a ...
(794–1185 CE), while ''e-sugoroku'' is a
racing game
Racing games are a video game genre in which the player participates in a motor racing, racing competition. They may be based on anything from real-world racing leagues to fantastical settings. They are distributed along a spectrum between more re ...
.
File:Knuck dice Steatite 37x27x21 mm.JPG, Knucklebones
Knucklebones, also known as scatter jacks, snobs, astragaloi (''singular'': astragalus), tali, dibs, fivestones, jacks, jackstones, or jinks, among many other names, is a game of Fine motor skill, dexterity played with a number of small objects ...
die, made of soapstone
Soapstone (also known as steatite or soaprock) is a talc-schist, which is a type of metamorphic rock. It is composed largely of the magnesium-rich mineral talc. It is produced by dynamothermal metamorphism and metasomatism, which occur in sub ...
File:Twenty-sided die (icosahedron) with faces inscribed with Greek letters MET 10.130.1158 001.jpg, Twenty-sided serpentinite
Serpentinite is a metamorphic rock composed predominantly of serpentine group minerals formed by serpentinization of mafic or ultramafic rocks. The ancient origin of the name is uncertain; it may be from the similarity of its texture or color ...
die from Ptolemaic Egypt Ptolemaic is the adjective formed from the name Ptolemy, and may refer to:
Pertaining to the Ptolemaic dynasty
* Ptolemaic dynasty, the Macedonian Greek dynasty that ruled Egypt founded in 305 BC by Ptolemy I Soter
*Ptolemaic Kingdom
Pertaining ...
File:Roman dice IMG 4367.JPG, Roman die
File:9BFE00 -roman lead die (FindID 103936).jpg, Composite image of all sides of a Roman
Roman or Romans most often refers to:
*Rome, the capital city of Italy
*Ancient Rome, Roman civilization from 8th century BC to 5th century AD
*Roman people, the people of Roman civilization
*Epistle to the Romans, shortened to Romans, a letter w ...
die, found in Leicestershire
Leicestershire ( ) is a Ceremonial counties of England, ceremonial county in the East Midlands of England. It is bordered by Derbyshire, Nottinghamshire and Lincolnshire to the north, Rutland to the east, Northamptonshire to the south-east, Warw ...
, England
File:Wall painting - scenes around the pub - Pompeii (VI 14 35-36) - Napoli MAN 111482 - 04.jpg, Roman wall painting showing two dice-players, Pompeii
Pompeii ( ; ) was a city in what is now the municipality of Pompei, near Naples, in the Campania region of Italy. Along with Herculaneum, Stabiae, and Villa Boscoreale, many surrounding villas, the city was buried under of volcanic ash and p ...
, 1st century
File:historical dice.jpg, A collection of historical dice from various regions of Asia
File:Chinese dice from Late Yuan Dynasty to early Qing Dynasty.jpg, Chinese dice from Late Yuan Dynasty to early Qing Dynasty in Jiangyin
Jiangyin (, Jiangyin dialect: ) is a county-level city on the southern bank of the Yangtze River. It is administered by the Wuxi, Jiangsu province. Jiangyin is an important transport hub on the Yangtze River and one of the most developed counties ...
Museum, China.
Etymology and terms
The word die comes from
Old French
Old French (, , ; ) was the language spoken in most of the northern half of France approximately between the late 8th ''
dé''; from
wikt:dé#Old French">dé''; from Latin ''wikt:datum#Latin">datum
Data ( , ) are a collection of discrete or continuous value (semiotics), values that convey information, describing the quantity, qualitative property, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols t ...
'' "something which is given or played".
While the terms ''ace'', ''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 (, , ; ) was the language spoken in most of the northern half of France approximately between the late 8th .
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">pip (counting)">pip on each die. The ''Online Etymology Dictionary'' 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 boxcars on a freight train. Many rolls have Craps#Names of rolls, names in the game of craps.
Use
Dice are thrown onto a surface either from the hand or from a container designed for this (such as a cup, tray, or
tower
A tower is a tall Nonbuilding structure, structure, taller than it is wide, often by a significant factor. Towers are distinguished from guyed mast, masts by their lack of guy-wires and are therefore, along with tall buildings, self-supporting ...
). The face (or corner, in cases such as tetrahedral dice, or edge, for odd-numbered
long dice
Long dice (sometimes oblongFinkel 2004, p 39. or stick dice) are dice, often roughly right Prism (geometry), prisms or (in the case of barrel dice) antiprisms, designed to land on any of several marked lateral faces, but neither end. Landing on e ...
) 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
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
(although
luck
Luck is the phenomenon and belief that defines the experience of improbable events, especially improbably positive or negative ones. The Naturalism (philosophy), naturalistic interpretation is that positive and negative events may happen at a ...
is often credited for the results of a roll). A die roll is made
random
In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. ...
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, a hardware random number generator (HRNG), true random number generator (TRNG), non-deterministic random bit generator (NRBG), or physical random number generator is a device that generates random numbers from a physical process c ...
.
Construction
Arrangement
Common dice are small
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
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
The ten Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) are the most commonly used symbols for writing numbers. The term often also implies a positional notation number with a decimal base, in particular when contrasted with Roman numera ...
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. 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.
File:Die Faces.svg, Typical facets showing the more compact pip arrangement of an Asian-style die (top) vs. a Western-style die (bottom)
Manufacturing

In general, dice are classified as either ''precision'' or ''non-precision''. Precision dice (also known as ''perfect'' or ''gambling house'' dice) are used in casinos, while non-precision dice (also known as ''drugstore'' or ''candystore'' dice) are sold with social and board games. These types are easily distinguished with visual and tactile differences; precision dice generally are larger, translucent, and have flush markings, sharp corners and edges, while non-precision dice generally are smaller, opaque, and have recessed markings, rounded corners and edges.
[
Non-precision dice are manufactured via the plastic ]injection molding
Injection moulding (U.S. spelling: injection molding) is a manufacturing process for producing parts by injecting molten material into a mould, or mold. Injection moulding can be performed with a host of materials mainly including metals (for ...
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, a similar process called barreling, or barrel finishing,Degarmo, p. 781. wo ...
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 dice are generally made from bars of extruded cellulose acetate
In biochemistry, cellulose acetate refers to any acetate ester of cellulose, usually cellulose diacetate. It was first prepared in 1865. A bioplastic, cellulose acetate is used as a film base in photography, as a component in some coatings, and ...
, sawed to the proper length to ensure that each face is as square as practical, generally with edges in length, with pips drilled deep and filled with opaque paint or epoxy which matches the density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
of cellulose, ensuring the dice remain balanced.[ The dice are buffed and polished to a gloss or sand finish after the pips are set, and the edges usually are left sharp, also called square or razor edge, although beveled or rounded edges, if performed evenly and consistently for each edge, are acceptable.][ Precision casino dice are transparent or translucent. This makes it difficult to incorporate internal weights which could bias the roll.][ To discourage cheating by dice substitution, each die carries a ]serial number
A serial number (SN) is a unique identifier used to ''uniquely'' identify an item, and is usually assigned incrementally or sequentially.
Despite being called serial "numbers", they do not need to be strictly numerical and may contain letters ...
and the casino's logo or name. Local regulations and the intended game may affect the allowable dimensions and tolerances; for example, New Jersey specifies the maximum size of a die is on a side, except for the dice used in pai gow, which range from on a side.
Precision backgammon dice are made the same way and also feature pips flush with the surface of each face; 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.
Unicode representation
Using Unicode
Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Char ...
characters, the faces can be shown in text using the range U+2680 to U+2685 or using decimal ⚀
to ⚅
, and the emoji using U+1F3B2 or 🎲
from the Miscellaneous Symbols and Pictographs
Miscellaneous Symbols and Pictographs is a Unicode block containing meteorological and astronomical symbols, emoji characters largely for compatibility with Japanese telephone carriers' implementations of Shift JIS, and characters originally from ...
block.
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 often 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 transparent cellulose acetate
In biochemistry, cellulose acetate refers to any acetate ester of cellulose, usually cellulose diacetate. It was first prepared in 1865. A bioplastic, cellulose acetate is used as a film base in photography, as a component in some coatings, and ...
dice, as tampering is easier to detect than with opaque dice.
Variants
Polyhedral dice
Various shapes such as 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 of wargame
A normal wargame is a strategy game in which two or more players command opposing armed forces in a simulation of an armed conflict. Wargaming may be played for Recreational wargaming, recreation, to train military officers in the art of milit ...
s, and since have been employed extensively in role-playing game
A role-playing game (sometimes spelled roleplaying game, or abbreviated as RPG) is a game in which players assume the roles of player character, characters in a fictional Setting (narrative), setting. Players take responsibility for acting out ...
s and trading card games. Dice using both the numerals 6 and 9, which are reciprocally symmetric through rotation, typically distinguish them with a dot or underline.
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 game
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. The genre was introduced with ''Magic: The G ...
s.
Common variations
Dice are often sold in sets, matching in color, of six different shapes. Five of the dice are shaped like the Platonic solid
In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s, whose faces are regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
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
In geometry, a pentagonal trapezohedron is the third in the infinite family of trapezohedra, face-transitive polyhedra. Its dual polyhedron is the pentagonal antiprism. As a decahedron it has ten faces which are congruent kites.
It can be dec ...
die, whose faces are ten kites
A kite is a tethered heavier than air flight, heavier-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. Kites often have ...
, each with two different edge lengths, three different angles, and two different kinds of vertices.
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 middle 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.
Using these dice in various ways, games can closely approximate a variety of probability distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
s. The percentile dice system is used to produce a uniform distribution of random percentages, and summing the values of multiple dice will produce approximations to normal distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
f(x) = \frac ...
s.
Rarer variations
"Uniform fair dice" are dice where all faces have an equal probability of outcome due to the symmetry of the die as it is face-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its Face (geometry), faces are the same. More specifically, all faces must be not ...
. In addition to the Platonic solids, these theoretically include:
* Catalan solid
The Catalan solids are the dual polyhedron, dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to ...
s, the duals
''Duals'' is a compilation album by the Irish 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 Bowl, ...
of the 13 Archimedean solid
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...
s: 12, 24, 30, 48, 60, 120 sides
* Trapezohedra
In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron Remarks: the faces of a deltohedron are deltoids; a (non-twisted) kite or deltoid can be dissected into two isosceles triangles or "deltas" (Δ), ba ...
, the duals of the infinite set of antiprism
In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
s, with kite faces: any even number not divisible by 4 (so that a face will face up), starting from 6
* Bipyramid
In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise ...
s, the duals of the infinite set of prisms, with triangle faces: any multiple of 4 (so that a face will face up), starting from 8
* Disphenoid
In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same ...
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, pyritohedra and tetartoid
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular s ...
s are less symmetrical but still face-transitive dodecahedra: 12 sides.
Two other types of polyhedra are technically not face-transitive but are still fair dice due to symmetry:
* antiprism
In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
s: the basis of barrel dice
* prism
PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...
s: the basis of long dice and teetotums
Long dice
Long dice (sometimes oblongFinkel 2004, p 39. or stick dice) are dice, often roughly right Prism (geometry), prisms or (in the case of barrel dice) antiprisms, designed to land on any of several marked lateral faces, but neither end. Landing on e ...
and teetotum
A teetotum (or T-totum) is a form of spinning top most commonly used for gambling games. It has a polygonal body marked with letters or numbers, which indicate the result of each spin. Usage goes back to (at least) ancient Greeks and Romans, wi ...
s can, in principle, be made with any number of faces, including odd numbers. Long dice are based on the infinite set of prisms. 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 Tamils, Tamil dice game played by 2 or 4 people (or multiples) by forming teams. It originated in Tamil Nadu (a southern state of India) and is comparable to another dice game from the country called Pachisi. Dayakatta ...
and daldøs
Daldøs running-fight board game">ø">al'døs">ø.html" ;"title="al'dø">al'døsis a Running-fight game">running-fight board game only known from a few coastal locations in southern Scandinavia, where its history can be traced back to around 18 ...
.
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 for ''Boggle
''Boggle'' is a word game introduced in 1972 and in which players try to find as many words as they can from a grid of lettered dice, within a set time limit. It was invented by Allan Turoff and originally distributed by Parker Brothers.
Rules
...
'', directions for ''Warhammer
Warhammer may refer to:
* War hammer, a medieval weapon
''Warhammer'' franchise
*''Warhammer'', a series of games and related media:
** ''Warhammer'' (game), a table-top fantasy miniature wargame, and origin of the franchise
** ''Warhammer Fanta ...
'', ''Fudge
''Fudge'' is a generic role-playing game system for use in freeform role-playing games. The name "''FUDGE''" was once an acronym for ''Freeform Universal Donated'' (later, ''Do-it-yourself'') ''Gaming Engine'' and, though the acronym has since b ...
'' dice, 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 and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
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 wargame
A normal wargame is a strategy game in which two or more players command opposing armed forces in a simulation of an armed conflict. Wargaming may be played for Recreational wargaming, recreation, to train military officers in the art of milit ...
s, where a narrower range of numbers is required. Other numbered variations include Sicherman dice and intransitive dice
A set of dice is intransitive (or nontransitive) if it contains X>2 dice, ''X1'', ''X2'', and ''X3''... with the property that ''X1'' rolls higher than ''X2'' more than half the time, and ''X2'' rolls higher than ''X3'' etc... more than half the ...
.
Spherical dice
A die can be constructed in the shape of a sphere, with the addition of an internal cavity in the shape of the dual polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...
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
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 su ...
is craps
Craps is a dice game in which players gambling, bet on the outcomes of the roll of a pair of dice. Players can wager money against each other (playing "street craps") or against a bank ("casino craps"). Because it requires little equipment, " ...
, 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
A board game is a type of tabletop game that involves small objects () that are placed and moved in particular ways on a specially designed patterned game board, potentially including other components, e.g. dice. The earliest known uses of the ...
s, where they are often used to decide the distance through which a piece will move along the board (as in backgammon
Backgammon is a two-player board game played with counters and dice on tables boards. It is the most widespread Western member of the large family of tables games, whose ancestors date back at least 1,600 years. The earliest record of backgammo ...
and ''Monopoly
A monopoly (from Greek language, Greek and ) is a market in which one person or company is the only supplier of a particular good or service. A monopoly is characterized by a lack of economic Competition (economics), competition to produce ...
'').
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
''Axis & Allies'' is a series of World War II strategy game, strategy board games. The first version was initially published in 1981 and a second edition known colloquially as ''Axis & Allies: Classic'' was published in 1984. Played on a board de ...
'', 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, ''Dice Masters Mike Elliott (sometimes credited as Michael Elliott) is a Seattle-based board game, card game and mobile game designer whose titles include ''Magic: The Gathering'', ''Thunderstone (card game), Thunderstone'', and ''Battle Spirits Trading Card Game' ...
''), 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
Divination () is the attempt to gain insight into a question or situation by way of an occultic ritual or practice. Using various methods throughout history, diviners ascertain their interpretations of how a should proceed by reading signs, ...
and using dice for such a purpose is called cleromancy
Cleromancy is a form of sortition (casting of lots) in which an outcome is determined by means that normally would be considered random, such as the rolling of dice ( astragalomancy), but that are sometimes believed to reveal the will of a deit ...
. A pair of common dice is usual, though other forms of polyhedra can be used. Tibetan Buddhists sometimes use this method of divination. It is highly likely that the Pythagoreans
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek co ...
used the Platonic solids
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edge ...
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 nodes of the Moon, the second die represents the 12 zodiac signs
In Western astrology, astrological signs are the twelve 30-degree sectors that make up Earth's 360-degree orbit around the Sun. The signs enumerate from the first day of spring, known as the First Point of Aries, which is the vernal equinox. T ...
, and the third represents the 12 houses. A specialized icosahedron die provides the answers of the Magic 8 Ball
The Magic 8 Ball is a plastic sphere, made to look like an oversized , that is used for fortune-telling or seeking advice. It was invented in 1946 by Albert C. Carter and Abe Bookman and is manufactured by Mattel. The user asks a yes–no questi ...
, conventionally used to provide answers to yes-or-no questions.
Dice can be used to generate random numbers for use in password
A password, sometimes called a passcode, is secret data, typically a string of characters, usually used to confirm a user's identity. Traditionally, passwords were expected to be memorized, but the large number of password-protected services t ...
s and cryptography
Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), ...
applications. The Electronic Frontier Foundation
The Electronic Frontier Foundation (EFF) is an American international non-profit digital rights group based in San Francisco, California. It was founded in 1990 to promote Internet civil liberties.
It provides funds for legal defense in court, ...
describes a method by which dice can be used to generate passphrase
A passphrase is a sequence of words or other text used to control access to a computer system, program or data. It is similar to a password in usage, but a passphrase is generally longer for added security. Passphrases are often used to control ...
s. Diceware
Diceware is a method for creating passphrases, passwords, and other cryptographic variables using ordinary dice as a hardware random number generator. For each word in the passphrase, five rolls of a six-sided die are required. The numbers f ...
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.
Thrown or simulated dice are sometimes used to generate specific probability distributions, which are fundamental to probability theory
Probability theory or probability calculus 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 ...
. For example, rolling a single six-sided die yields a uniform distribution, where each number from 1 to 6 has an equal chance of appearing. However, when rolling two dice and summing the results, the probability distribution shifts, as some sums (like 7) become more likely than others (like 2 or 12). These distributions can model real-world scenarios or mathematical constructs, making dice a practical tool for teaching and exploring concepts in probability theory.
Notation
In many gaming contexts, especially tabletop role-playing games, shorthand notations are used to differentiate between different types of dice. The most commonly-used notation, considered the standard Standard may refer to:
Symbols
* Colours, standards and guidons, kinds of military signs
* Standard (emblem), a type of a large symbol or emblem used for identification
Norms, conventions or requirements
* Standard (metrology), an object ...
, is written as . In this expression, is the number of sides on the dice and is the number of rolls; if there is only one roll, the is omitted. As an illustration, the d20 (twenty-sided dice) is to ''Dungeons & Dragons
''Dungeons & Dragons'' (commonly abbreviated as ''D&D'' or ''DnD'') is a fantasy tabletop role-playing game (TTRPG) originally created and designed by Gary Gygax and Dave Arneson. The game was first published in 1974 by TSR (company)#Tactical ...
'' what the d6 (six-sided dice) is to many board games. ''Monopoly
A monopoly (from Greek language, Greek and ) is a market in which one person or company is the only supplier of a particular good or service. A monopoly is characterized by a lack of economic Competition (economics), competition to produce ...
'' uses 2d6 rolls (the total value of two six-sided dice) to determine player movement.
The notation also allows for adding or subtracting a constant amount ''c'' to the roll. When an amount is added, the notation is . For example, "3d6+4" instructs the player to roll three six-sided dice, calculate the total, and add four to it. When an amount is to be subtracted, the notation is . Thus, "3d6−4" instructs the player to subtract four from the total value of three six-sided dice. The notation is also sometimes used, with the modifier "L" (or less commonly "H") representing the lowest amount (or highest amount) of each roll combined. For instance, 4d6−L instructs the player to sum up the total of four six-sided dice and subtract the lowest value. If the result of a modified dice roll is negative, it is often taken to be zero or one; for instance, when the dice roll determines the amount of damage to a creature.
Role-playing dice sets and percentile dice
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 (TTRPG) originally created and designed by Gary Gygax and Dave Arneson. The game was first published in 1974 by TSR (company)#Tactical ...
'' (''D&D'') is largely credited with popularizing dice in such games. Some games use only one type, like ''Exalted
''Exalted'' is a high fantasy tabletop role-playing game originally published by White Wolf Publishing in July 2001. The game is currently in its third edition. It was originally created by Robert Hatch, Justin Achilli and Stephan Wieck, a ...
'' 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 notation
Dice 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 suc ...
is frequently used.
A role-playing dice set typically comprises seven dice: one each of d4, d6, d8, d12 and d20, and two d10s, one labeled from 0 to 9 and the other labeled in tens from 00 to 90. These last two, taken together, are called ''percentile dice''. In standard dice notation, a roll of percentile dice can be expressed as "d100", although "d%" is also seen. A d100 roll is typically performed like a 2d10 roll, using both decahedral dice together. Adding the numbers shown gives a random number from 0 to 99 – although 0+00 is typically read as 100, for consistency with other dice notation.
The zocchihedron was invented as an alternative to percentile dice. Unlike percentile dice, the zocchihedron is a true d100 die, though no single die of 100 sides can be as consistently fair.[https://rpggeek.com/blog/1051/blogpost/38414/interview-with-lou-zocchi-the-first-dice-idea-i-wa]
See also
*
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*
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*
*
References
Bibliography
* 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
* 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
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