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For other uses, see Dice (disambiguation).
Four traditional dice showing all six different sides.
Dice of different sizes being thrown in slow motion.

Dice (sg.: die or dice)[1] are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing games, and games of chance.

A traditional die is a cube 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 from one to six on its upper surface, with each value being equally likely. Dice may also have polyhedral 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 or entertainment.

History

Composite image of all sides of a 12 mm (1⁄2 in) Roman die, found in Leicestershire, England

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.[2] The Egyptian game of senet (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.[3] Perhaps the oldest known dice were excavated as part of a backgammon-like game set at the Burnt City, an archeological site in south-eastern Iran, estimated to be from between 2800 and 2500 BCE.[4][5] Bone dice from Skara Brae, Scotland have been dated to 3100–2400 BCE.[6] Excavations from graves at Mohenjo-daro, an Indus Valley civilization settlement, unearthed terracotta dice dating to 2500–1900 BCE,[7] including at least one die whose opposite sides all add up to seven, as in modern dice.[8]

Games involving dice are mentioned in the ancient Indian Rigveda,[9] Atharvaveda, Mahabharata and Buddhist games list.[10] There are several biblical references to "casting lots" (Hebrew: יפילו גורל yappîlū ḡōrāl), as in Psalm 22, indicating that dicing (or a related activity) was commonplace when the psalm was composed. Knucklebones was a game of skill played in ancient Greece; a derivative form had the four sides of bones receive different values like modern dice.[11]

Roman wall painting showing two dice-players, Pompeii, 1st century

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.[12] Twenty-sided dice date back to the 2nd century CE[13] and from Ptolemaic Egypt as early as the 2nd century BCE.[14]

Dominoes and playing cards originated in China as developments from dice.[15] The transition from dice to playing cards occurred in China around the Tang dynasty (618–907 CE), and coincides with the technological transition from rolls of manuscripts to block printed books.[16] In Japan, dice were used to play a popular game called sugoroku. There are two types of sugoroku. Ban-sugoroku is similar to backgammon and dates to the Heian period (794–1185 CE), while e-sugoroku is a racing game.[17]

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). The face (or corner, in cases such as tetrahedral dice, or edge, for odd-numbered long dice) 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 (although luck is often credited for the results of a roll). A die roll is made random by uncertainty in minor factors such as tiny movements in the thrower's hand; they are thus a crude form of hardware random number generator.

One typical contemporary dice game is craps, 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 games, where they are often used to decide the distance through which a piece will move along the board (as in backgammon and Monopoly).

Construction

Arrangement

Chirality of dice. Faces may be placed counterclockwise (top) or clockwise.

Common dice are small cubes, most often 1.6 cm (0.63 in) across, whose faces are numbered from one to six, usually by patterns of round dots called pips. (While the use of Arabic numerals 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.[18] 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.[19]

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.[19][20]

  • Typical facets showing the more compact pip arrangement of an Asian-style die (top) vs. a Western-style die (bottom)
    Typical facets showing the more compact pip arrangement of an Asian-style die (top) vs. a Western-style die (bottom)

Manufacturing

Western, Asian, and casino dice

Non-precision dice are manufactured via the plastic injection molding 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.[21]

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 process similar to rock polishing. The abrasive agent scrapes off all of the paint except for the indents of the numbering. A finer abrasive is then used to polish the die. This process also produces the smoother, rounded edges on the dice.

Precision casino dice may have a polished or sand finish, making them transparent or translucent respectively. Casino dice have their pips drilled, then filled flush with a paint of the same density as the material used for the dice, such that the center of gravity of the dice is as close to the geometric center as possible. This mitigates concerns that the pips will cause a small bias.[22] All such dice are stamped with a serial number to prevent potential cheaters from substituting a die. Precision backgammon dice are made the same way; they tend to be slightly smaller and have rounded corners and edges, to allow better movement inside the dice cup and stop forceful rolls from damaging the playing surface.

Etymology and terms

The word die comes from Old French dé; from Latin datum "something which is given or played".[23]

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";[24] the others are 2 to 6 in Old French.[25]

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 traces use of the term as far back as 1919.[26] 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 names in the game of craps.

Unicode representation

Symbol ⚀ ⚁ ⚂ ⚃ ⚄ ⚅ 🎲
Unicode U+2680 U+2681 U+2682 U+2683 U+2684 U+2685 U+1F3B2
Decimal ⚀ ⚁ ⚂ ⚃ ⚄ ⚅ 🎲

Using Unicode characters, the faces can be shown in text using the range U+2680 to U+2685 or using decimal ⚀ to ⚅,[27] and the emoji using U+1F3B2 or 🎲 from the Miscellaneous Symbols and Pictographs 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 dice, as tampering is easier to detect than with opaque dice.[28]

Variants

Polyhedral dice

A typical set of polyhedral dice in various colors. They consist of the five Platonic solids, along with a ten-sided die that is also used for generating percentages.

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 wargames,[29] and since have been employed extensively in role-playing games 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.

Common variations

Further information: Isohedron

Dice are often sold in sets, matching in color, of six different shapes. Five of the dice are shaped like the Platonic solids, whose faces are regular polygons. 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 die, whose faces are ten kites, each with two different edge lengths, three different angles, and two different kinds of vertices. Such sets frequently include a second 10-sided die either of contrasting color or numbered by tens, allowing the pair of 10-sided dice to be combined to generate numbers between 1 and 100.

Using these dice in various ways, games can closely approximate a variety of probability distributions. For instance, 10-sided dice can be rolled in pairs to produce a uniform distribution of random percentages, and summing the values of multiple dice will produce approximations to normal distributions.[30]

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.

Some twenty-sided dice have a different arrangement used for the purpose of keeping track of an integer that counts down, such as health points. These spindown dice are arranged such that adjacent integers appear on adjacent faces, allowing the user to easily find the next lower number. They are commonly used with collectible card games.[31]

Faces/sides Shape Notes
4 Tetrahedron Tetrahedron Each face has three numbers, arranged such that the upright number, placed either near the vertex or near the opposite edge, is the same on all three visible faces. The upright numbers represent the value of the roll. This die does not roll well and thus it is usually thrown into the air instead.
6 Cube Cube A common die. The sum of the numbers on opposite faces is 7.
8 Octahedron Octahedron Each face is triangular and the die resembles two square pyramids attached base-to-base. Usually, the sum of the opposite faces is 9.
10 Pentagonal trapezohedron Pentagonal trapezohedron Each face is a kite. The die has two sharp corners, where five kites meet, and ten blunter corners, where three kites meet. The ten faces usually bear numbers from zero to nine rather than one to ten (zero being read as "ten" in many applications). Often, all odd numbered faces converge at one sharp corner, and the even ones at the other. The sum of the numbers on opposite faces is usually 9 (if numbered 0–9) or 11 (if numbered 1–10).
12 Dodecahedron Dodecahedron Each face is a regular pentagon. The sum of the numbers on opposite faces is usually 13.
20 Icosahedron Icosahedron Faces are equilateral triangles. Icosahedra have been found dating to Roman/Ptolemaic times, but it is not known if they were used as gaming dice. Modern dice with 20 sides are sometimes numbered 0–9 twice as an alternative to 10-sided dice. The sum of the numbers on opposite faces is 21 if numbered 1–20.

Rarer variations

Dice collection: D2–D22, D24, D26, D28, D30, D36, D48, D50, D60 and D100.

"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 addition to the Platonic solids, these theoretically include:

  • Catalan solids, the duals of the 13 Archimedean solids: 12, 24, 30, 48, 60, 120 sides
  • Trapezohedra, the duals of the infinite set of antiprisms, with kite faces: any even number not divisible by 4 (so that a face will face up), starting from 6
  • Bipyramids, 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
  • Disphenoids, 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 tetartoids 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:

Long dice and teetotums can, in principle, be made with any number of faces, including odd numbers.[32] 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 and daldøs.

Faces/sides Shape Image Notes
1 Möbius strip or sphere Most commonly a joke die, this is either a sphere with a 1 marked on it or shaped like a Möbius strip. It entirely defies the aforementioned use of a die.
2 Flat Cylinder or Flat Prism A coin flip. Some coins with 1 marked on one side and 2 on the other are available, but most simply use a common coin.
3 Rounded-off triangular prism A long die intended to be rolled lengthwise. 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.
4 Capped 4-sided long die A long die intended to be rolled lengthwise. It cannot stand on end as the ends are capped.
5
Triangular prism A prism thin enough to land either on its "edge" or "face". When landing on an edge, the result is displayed by digits (2–4) close to the prism's top edge. The triangular faces are labeled with the digits 1 and 5.
Capped 5-sided long die Five-faced long die for the Korean Game of Dignitaries; notches indicating values are cut into the edges, since in an odd-faced long die these land uppermost.
6 Capped 6-sided long die Two six-faced long dice are used to simulate the activity of scoring runs and taking wickets in the game of cricket. Originally played with labeled six-sided pencils, and often referred to as pencil cricket.
7 Pentagonal prism Similar in constitution to the 5-sided die. Seven-sided dice are used in a seven-player variant of backgammon. Seven-sided dice are described in the 13th century Libro de los juegos as having been invented by Alfonso X in order to speed up play in chess variants.[33][34]
Truncated sphere A truncated sphere with seven landing positions.
9 Truncated sphere A truncated sphere with nine landing positions.
10 Decahedron A ten-sided die made by truncating two opposite vertices of an octahedron.
11 Truncated sphere A truncated sphere with eleven landing positions.
12 Rhombic dodecahedron Each face is a rhombus.
13 Truncated sphere A truncated sphere with thirteen landing positions.
14 Heptagonal trapezohedron Each face is a kite.
Truncated octahedron A truncated octahedron. Each face is either a square or a hexagon.
Truncated sphere A truncated sphere with fourteen landing positions. The design is based on the cuboctahedron.
15 Truncated sphere A truncated sphere with fifteen landing positions.
16 Octagonal bipyramid Each face is an isosceles triangle.
17 Truncated sphere A truncated sphere with seventeen landing positions.
18 Rounded rhombicuboctahedron Eighteen faces are squares. The eight triangular faces are rounded and cannot be landed on.
19 Truncated sphere A truncated sphere with nineteen landing positions.
21 Truncated sphere A truncated sphere with twenty-one landing positions.
22 Truncated sphere A truncated sphere with twenty-two landing positions.
24 Triakis octahedron Each face is an isosceles triangle.
Tetrakis hexahedron Each face is an isosceles triangle.
Deltoidal icositetrahedron Each face is a kite.
Pseudo-deltoidal icositetrahedron Each face is a kite.
Pentagonal icositetrahedron Each face is an irregular pentagon.
26 Truncated sphere A truncated sphere with twenty-six landing positions.
28 Truncated sphere A truncated sphere with twenty-eight landing positions.
30 Rhombic triacontahedron Each face is a rhombus.
32 Truncated sphere A truncated sphere with thirty-two landing positions. The design is similar to that of a truncated icosahedron.
34 Heptadecagonal trapezohedron Each face is a kite.
36 Truncated sphere A truncated sphere with thirty-six landing positions. Rows of spots are present above and below each number 1 through 36 so that this die can be used to roll two six-sided dice simultaneously.
48 Disdyakis dodecahedron Each face is a scalene triangle.
50 Icosipentagonal trapezohedron Each face is a kite.
60 Deltoidal hexecontahedron Each face is a kite.
Pentakis dodecahedron Each face is an isosceles triangle.
Pentagonal hexecontahedron Each face is an irregular pentagon.
Triakis icosahedron Each face is an isosceles triangle.
100 Zocchihedron A sphere containing another sphere with 100 facets flattened into it. Note that this design is not isohedral; it does not function as a uniform fair die as some results are more likely than others.
120 Disdyakis triacontahedron Each face is a scalene triangle.

Non-numeric dice

A set of Fudge 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, directions for Warhammer Fantasy Battle, Fudge 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.[35][36] These are six-sided dice with sides numbered 2, 3, 3, 4, 4, 5, which have the same arithmetic mean as a standard die (3.5 for a single die, 7 for a pair of dice), but have a narrower range of possible values (2 through 5 for one, 4 through 10 for a pair). They are used in some table-top wargames, where a narrower range of numbers is required.[36] Other numbered variations include Sicherman dice and intransitive dice.

Spherical dice

A spherical die

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

Applications

Playing Dice by Master Jean de Mauléon (c. 1520)

Many board games use dice to randomize how far pieces move or to settle conflicts. Typically, this has meant that rolling higher numbers is better. Some games, such as Axis & Allies, have inverted this system by making the lower values more potent. In the modern age,[when?] a few games and game designers have approached dice in a different way by making each side of the die similarly valuable. In Castles of Burgundy, players spend their dice to take actions based on the die's value. In this game, a six is not better than a one, or vice versa. In Quarriors (and its descendant, Dicemasters), different sides of the dice can offer completely different abilities. Several sides often give resources while others grant the player useful actions.[37]

Dice can be used for divination and using dice for such a purpose is called cleromancy. A pair of common dice is usual, though other forms of polyhedra can be used. Tibetan Buddhists sometimes use this method of divination. It is highly likely that the Pythagoreans used the Platonic solids as dice. They referred to such dice as "the dice of the gods" and they sought to understand the universe through an understanding of geometry in polyhedra.[38]

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.

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

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, and the third represents the 12 houses. A specialized icosahedron die provides the answers of the Magic 8 Ball, conventionally used to provide answers to yes-or-no questions.

Dice can be used to generate random numbers for use in passwords and cryptography applications. The Electronic Frontier Foundation describes a method by which dice can be used to generate passphrases.[39] Diceware is a method recommended for generating secure but memorable passphrases, by repeatedly rolling five dice and picking the corresponding word from a pre-generated list.[40]

Notation

Main article: Dice notation

In many gaming contexts, especially tabletop role-playing games, shorthand notations representing different dice rolls are used. A very common notation, considered a standard, expresses a dice roll as nds or nDs, where n is the number of dice rolled and s is the number of sides on each die; if only one die is rolled, n is normally not shown. For example, d4 denotes one four-sided die; 6d8 means the player should roll six eight-sided dice and sum the results.

The notation also allows for adding or subtracting a constant amount c to the roll. When an amount is added, the notation is nds+c or nDs+c; 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 nds-c or nDs-c; so 3d6-4 instructs the player to subtract four from the result of rolling 3d6. 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.[41][42]

See also

References

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  2. ^ Laird, Jay (2009). Encyclopedia of Play in Today's Society. SAGE Publications. pp. 171–173. ISBN 978-1-4522-6610-7.
  3. ^ Finkel, Irving (2008). "Board Games". Beyond Babylon: Art, Trade, and Diplomacy in the Second Millennium B.C. Metropolitan Museum of Art. p. 151. ISBN 978-1-58839-295-4.
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  5. ^ NASER MOGHADASI, Abdorreza (September 2015). "The Burnt City and the Evolution of the Concept of "Probability" In the Human Brain". Iranian Journal of Public Health. 44 (9): 1306–1307. ISSN 2251-6085. PMC 4645795. PMID 26587512.
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  7. ^ Possehl, Gregory. "Meluhha". In: J. Reade (ed.) The Indian Ocean in Antiquity. London: Kegan Paul Intl. 1996a, 133–208
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  9. ^ www.wisdomlib.org (27 August 2021). "Rig Veda 10.34.1 [English translation]". www.wisdomlib.org. Retrieved 28 July 2024.
  10. ^ 2.3, 4.38, 6.118, 7.52, 7.109
  11. ^ Good, Alexandra. "Knucklebones". Johns Hopkins Archaeological Museum. Retrieved 16 April 2015.
  12. ^ Matz, David (2002). Daily Life of the Ancient Romans. Greenwood Publishing Group. pp. 94–95. ISBN 978-0-313-30326-5.
  13. ^ "christies.com". christies.com. Retrieved 18 June 2012.
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  15. ^ "domino | game piece | Britannica". www.britannica.com. Retrieved 20 November 2022.
  16. ^ Ronan, Colin; Needham, Joseph (1986). The Shorter Science and Civilisation in China. Cambridge University Press. p. 55. ISBN 978-0-521-31560-9.
  17. ^ Salter, Rebecca (2006). "Board Games". Japanese Popular Prints: From Votive Slips to Playing Cards. University of Hawaii Press. p. 164. ISBN 978-0-8248-3083-0.
  18. ^ Cf. The Greek anthology. Translated by Paton, W.R. New York: G.P. Putnam's and Sons. 1918. Book 14, §8. The Opposite Pairs of Numbers on a Die. The numbers on a die run so: six one, five two, three four.
  19. ^ a b "Standard Dice". Archived from the original on 30 July 2013.
  20. ^ "Chinese Dice". Elliott Avedon Museum & Archive of Games. Archived from the original on 1 May 2008.
  21. ^ "7 Surprising Facts About How Dice Are Made (With Pictures)". gmdice.com. 19 August 2019. Retrieved 20 August 2019.
  22. ^ "Casino Dice". www.dice-play.com. Retrieved 1 July 2018.
  23. ^ "die". Oxford Dictionaries. Archived from the original on 14 February 2015. Retrieved 14 February 2015.
  24. ^ "ace". AskOxford. Archived from the original on 29 September 2007. Retrieved 18 June 2012.
  25. ^ Conant, Levi Leonard (1896). The Number Concept: Its Origin and Development. Macmillan. p. 124.
  26. ^ "s | Search Online Etymology Dictionary". www.etymonline.com.
  27. ^ "Dice faces in block Miscellaneous Symbols" (PDF). The Unicode standard. Archived (PDF) from the original on 3 June 2001.
  28. ^ Horridge, Kevin (27 November 2017). "6 Ways Casinos Stop Dice Cheats". Casino.org. Retrieved 11 October 2019.
  29. ^ Peterson, Jon (July 2012). Playing at the World: A History of Simulating Wars, People and Fantastic Adventures, from Chess to Role-Playing Games. Unreason Press. pp. 315–318. ISBN 978-0-615-64204-8.
  30. ^ Paret, Michelle; Martz, Eston (2009). "Tumbling Dice & Birthdays: Understanding the Central Limit Theorem" (PDF). Minitab. Archived from the original (PDF) on 1 November 2013. Retrieved 29 September 2013.
  31. ^ Girdwood, Andrew (30 March 2019). "What's a spindown dice and are standard d20s any fairer?". Retrieved 9 July 2020.
  32. ^ Kybos, Alea. "Properties of Dice" (PDF). Archived from the original (PDF) on 28 May 2012. Retrieved 7 October 2012.
  33. ^ "games.rengeekcentral.com". games.rengeekcentral.com. Retrieved 18 June 2012.
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  35. ^ "Specialist D6". em4miniatures. Retrieved 18 August 2017.
  36. ^ a b "[TMP] 'What are Average Dice' Topic". The Miniatures Page. 20 November 2009. Retrieved 18 August 2017.
  37. ^ "Making better use of dice in games". 2 October 2014.
  38. ^ Guthrie, Kenneth (1988). The Pythagorean sourcebook and library: an anthology of ancient writings which relate to Pythagoras and Pythagorean philosophy. Grand Rapids, MI: Phanes Press. ISBN 978-0-933999-50-3. OCLC 255212063.
  39. ^ "EFF Dice-Generated Passphrases". Electronic Frontier Foundation. 12 May 2016. Retrieved 16 November 2018.
  40. ^ "EFF's New Wordlists for Random Passphrases". Electronic Frontier Foundation. 19 July 2016. Retrieved 4 December 2016.
  41. ^ Sim, Chris (2007). Dungeons & Dragons Rules Compendium. Renton, WA USA. p. 17. ISBN 9780786947256.{{cite book}}: CS1 maint: location missing publisher (link)
  42. ^ Mearls, Mike; et al. (2016). "System Reference Document 5.1" (PDF). p. 96. Retrieved 12 September 2024.

Bibliography

  • Diaconis, Persi; and Keller, Joseph B.; "Fair Dice", Archived 21 February 2008 at the Wayback Machine, 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, ISBN 0-7160-2112-9
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 This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Dice". Encyclopædia Britannica. Vol. 8 (11th ed.). Cambridge University Press. p. 176–177.

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