**Dice** (singular **die** or **dice**^{[1]}) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random numbers, 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 and may have faces marked with numerals or symbols instead of pips. Loaded dice are designed to favor some results over others for cheating or entertainment.

## History

Dice have been used since before recorded history, and it is uncertain where they originated. It is theorized that dice developed from the practice of fortune-telling with the talus of hoofed animals, colloquially known as knucklebones.^{[2]} The Egyptian game of senet was played with flat two-sided throwsticks which indicated the number of squares a player could move, and thus functioned as a form of dice. Senet was played before 3000 BC and up to the 2nd century AD.^{[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 BC.^{[4]} Bone dice from Skara Brae have been dated to 3100–2400 BC.^{[5]} Excavations from graves at Mohenjo-daro, an Indus Valley civilization settlement, unearthed terracotta dice dating to 2500–1900 BC.^{[6]}

Games involving dice are mentioned in the ancient Indian *Rigveda*, *Atharvaveda,* *Mahabharata* and Buddhist games list.^{[7]} 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.^{[8]}

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.^{[10]} Twenty-sided dice date back to the 2nd century AD^{[11]} and from Ptolemaic Egypt as early as the 2nd century BC.^{[12]}

Dominoes and playing cards originated in China as developments from dice.^{[citation needed]} The transition from dice to playing cards occurred in China around the Tang dynasty, and coincides with the technological transition from rolls of manuscripts to block printed books.^{[13]} 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, while *e-sugoroku* is a racing game.^{[14]}

## Use

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

The result of a die roll is determined by the way it is thrown, according to the laws of classical mechanics. A die roll is 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

Common dice are small cubes, most often 1.6 cm (0.63 in) across,^{[15]} 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.^{[16]} 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.^{[17]}

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.^{[17]}^{[18]}

### Manufacturing

Non-precision dice are manufactured via the plastic injection molding process. 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.^{[19]}

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 creates the smoother, rounded edges on the dice.^{[20]}

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.^{[21]} 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".^{[22]}

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";^{[23]} the others are 2 to 6 in Old French.^{[24]}
The term "snake eyes" is the outcome of rolling the dice and getting only one pip on each die. The *Online Etymology Dictionary* traces use of the term as far back as 1919.^{[25]}
The term "boxcars", also known as "midnight", is the outcome of rolling the dice and getting a six on each die. The pair of six pips resembles a pair of boxcars on a freight train.

### Unicode representation

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

to `⚅`

.^{[26]}

## Loaded dice

A loaded, weighted, cheat, or crooked die is one that has been tampered with so that it will land with a specific side facing upwards more or less often than a fair die would. There are several methods for creating 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.^{[27]}

## Variants

### Polyhedral dice

Various shapes like two-sided or four-sided dice are documented in archaeological findings e.g. from Ancient Egypt or 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,^{[28]} 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

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.^{[29]}

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 same number printed near each vertex on all sides. In this case, the number around the vertex pointing up is used. Alternatively, the numbers on a tetrahedral die can be placed at the middles of the edges, in which case the numbers around the base are used.

Normally, the faces on a die will be placed so opposite faces will add up to one more than the number of faces. (This is not possible with 4-sided dice and dice with an odd-number of faces.) Some dice, such as those with 10 sides, are usually numbered sequentially beginning with 0, in which case the opposite faces will add to one less than the number of faces.

Some twenty-sided dice sport 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.^{[30]}

Faces/ Sides |
Shape | Notes | |
---|---|---|---|

4 | 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 | A common die. The sum of the numbers on opposite faces is 7. | |

8 | 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 | 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 | Each face is a regular pentagon. The sum of the numbers on opposite faces is usually 13. | |

20 | 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

"Uniform fair dice" are dice where all faces have equal probability of outcome due to the symmetry of the die as it is face-transitive. Theoretically, these include:

- Platonic solids, the five regular polyhedra: 4, 6, 8, 12, 20 sides
- Catalan solids, the duals of the 13 Archimedean solids: 12, 24, 30, 48, 60, 120 sides
- Bipyramids, the duals of the infinite set of prisms, with triangle faces: any multiple of 4 (so that a face will face up) above 8
- 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) above 6
- 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:

- prisms: the basis of long dice and teetotums
- antiprisms: the basis of barrel dice

Long dice and teetotums can in principle be made with any number of faces, including odd numbers.^{[31]} 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. | |

2 | Cylinder | 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 that is thin enough to land either on its "edge" or "face". When landing on an edge, the result is displayed by digits (2–4) close to the prism's top edge. The triangular faces are labeled with the digits 1 and 5. | ||

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 labelled 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.^{[32]}^{[33]} | |

12 | Rhombic dodecahedron | Each face is a rhombus. | |

12 | Hexagonal trapezohedron | Each face is a kite. | |

14 | Heptagonal trapezohedron | Each face is a kite. | |

16 | Octagonal bipyramid | Each face is an isosceles triangle. | |

16 | Octagonal trapezohedron | Each face is a kite. | |

18 | Rounded rhombicuboctahedron | 18 faces are squares. The 8 triangular faces are rounded and cannot be landed on. | |

20 | Decagonal trapezohedron | Each face is a kite. | |

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. | ||

Dodecagonal trapezohedron | Each face is a kite. | ||

30 | Rhombic triacontahedron | Each face is a rhombus. | |

34 | Heptadecagonal trapezohedron | Each face is a kite. | |

48 | Disdyakis dodecahedron | Each face is a scalene triangle. | |

50 | Icosikaipentagonal 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. 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

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.^{[34]}^{[35]} 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.^{[35]} Other numbered variations include Sicherman dice and intransitive dice.

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

## Applications

Polyhedral dice are commonly used in role-playing games. The fantasy role-playing game *Dungeons & Dragons* (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.

Many board games use dice to randomize how far pieces move or to settle conflicts. Typically, this has meant that rolling higher numbers is better. Some games, such as *Axis & Allies*, have inverted this system by making the lower values more potent. In the modern age, a few games and game designers have approached dice in a different way by making each side of the die similarly valuable. In *Castles of Burgundy*, players spend their dice to take actions based on the die's value. In this game, a six is not better than a one, or vice versa. In *Quarriors* (and its descendant, Dicemasters), different sides of the dice can offer completely different abilities. Several sides often give resources while others grant the player useful actions.^{[36]}

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.^{[37]}

Astrological dice are a specialized set of three 12-sided dice for divination; the first die represents planets, the Sun, the Moon, and the nodes of the Moon, the second die represents the 12 zodiac signs, and the third represents the 12 houses. 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.^{[38]} 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.^{[39]}

### Notation

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

denotes a four-sided die. If several dice of the same type are to be rolled, this is indicated by a leading number specifying the number of dice. Hence, `6d8`

means the player should roll six eight-sided dice and add the results. Modifiers to a die roll can also be indicated as desired. For example, `3d6+4`

instructs the player to roll three six-sided dice, calculate the total, and add four to it.

## See also

## References

**^**"die, n.1".*OED Online*. Oxford University Press. Retrieved 10 April 2020.**^**Laird, Jay (2009).*Encyclopedia of Play in Today's Society*. SAGE Publications. pp. 171–173. ISBN 978-1-4522-6610-7.**^**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.**^**"presstv.ir". presstv.ir. 11 April 2007. Archived from the original on 20 June 2007. Retrieved 18 June 2012.**^**"National Museums of Scotland - Two bone dice".*nms.scran.ac.uk*. Retrieved 9 May 2020.**^**Possehl, Gregory. "Meluhha". In: J. Reade (ed.)*The Indian Ocean in Antiquity*. London: Kegan Paul Intl. 1996a, 133–208**^**2.3, 4.38, 6.118, 7.52, 7.109**^**Good, Alexandra. "Knucklebones".*Johns Hopkins Archaeological Museum*. Retrieved 16 April 2015.**^**"Twenty-sided die (icosahedron) with faces inscribed with Greek letters". metmuseum.org. Retrieved 28 March 2015.**^**Matz, David (2002).*Daily Life of the Ancient Romans*. Greenwood Publishing Group. pp. 94–95. ISBN 978-0-313-30326-5.**^**"christies.com". christies.com. Retrieved 18 June 2012.**^**"Twenty-sided die (icosahedron) with faces inscribed with Greek letters". metmuseum.org. Retrieved 28 March 2015.**^**Ronan, Colin; Needham, Joseph (1986).*The Shorter Science and Civilisation in China*. Cambridge University Press. p. 55. ISBN 978-0-521-31560-9.**^**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.**^**"Dice Sizes Explained".*Dice Game Depot*. 2018.**^**Cf.*Greek Anthology*Book 14, §8: "The Opposite Pairs of Numbers on a Die. The numbers on a die run so: six one, five two, three four."- ^
^{a}^{b}"Standard Dice". Archived from the original on 30 July 2013. **^**"Chinese Dice". Elliott Avedon Museum & Archive of Games. Archived from the original on 1 May 2008.**^**"7 Surprising Facts About How Dice Are Made (With Pictures)".*gmdice.com*. Retrieved 20 August 2019.**^**How Dice Are Made from Awesome Dice**^**"Casino Dice".*www.dice-play.com*. Retrieved 1 July 2018.**^**"die". Oxford Dictionaries. Retrieved 14 February 2015.**^**"ace". AskOxford. Retrieved 18 June 2012.**^**Conant, Levi Leonard (1896).*The Number Concept: Its Origin and Development*. Macmillan. p. 124.**^**Online Etymology Dictionary**^**"Dice faces in block Miscellaneous Symbols" (PDF).*The Unicode standard*.**^**Horridge, Kevin (27 November 2017). "6 Ways Casinos Stop Dice Cheats".*Casino.org*. Retrieved 11 October 2019.**^**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.**^**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.**^**Girdwood, Andrew (30 March 2019). "What's a spindown dice and are standard d20s any fairer?". Retrieved 9 July 2020.**^**Kybos, Alea. "Properties of Dice" (PDF). Archived from the original (PDF) on 28 May 2012. Retrieved 7 October 2012.**^**"games.rengeekcentral.com". games.rengeekcentral.com. Retrieved 18 June 2012.**^**"wwmat.mat.fc.ul.pt" (PDF). Retrieved 18 June 2012.**^**"Specialist D6".*em4miniatures*. Retrieved 18 August 2017.- ^
^{a}^{b}"[TMP] 'What are Average Dice' Topic".*The Miniatures Page*. 20 November 2009. Retrieved 18 August 2017. **^**"Making better use of dice in games". 2 October 2014.**^**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.**^**"EFF Dice-Generated Passphrases".*Electronic Frontier Foundation*. 12 May 2016. Retrieved 16 November 2018.**^**"EFF's New Wordlists for Random Passphrases".*Electronic Frontier Foundation*. 19 July 2016. Retrieved 4 December 2016.

- 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, ISBN 0-7160-2112-9

## External links

Look up in Wiktionary, the free dictionary.dice |

Wikimedia Commons has media related to Dice. |

- Weisstein, Eric W. "Dice".
*MathWorld*. – analysis of dice probabilities, also featuring Uspenski's work on rolling multiple dice

This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Dice". *Encyclopædia Britannica*. **8** (11th ed.). Cambridge University Press. p. 176–177.