Editing Dice

{{Short description|Throwable objects with marked sides, used for generating random numbers}}
{{Other uses}}
{{Use dmy dates|date=October 2021}}
[[File:6sided dice (cropped).jpg|thumb|Four traditional dice showing all six different sides.]]
[[File:Dice-rolling-slowmotion-2018-9.webm|thumb|Dice of different sizes being thrown in slow motion.]]

A '''die''' ({{plural form}}: '''dice''', sometimes also used as {{singular}})<ref>{{Cite web |title=die, n.1 |url=https://www.oed.com/view/Entry/52390  |access-date=1 December 2023 |website=[[Oxford English Dictionary|OED Online]] |publisher=Oxford University Press}}</ref> is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating [[Statistical randomness|random values]], commonly as part of [[tabletop game]]s, including [[List of dice games|dice games]], [[board game]]s, [[role-playing game]]s, and [[Game of chance|games of chance]].

A traditional die is a [[cube]] with each of its six faces marked with a different number of dots ([[pip (counting)|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 other [[polyhedron|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==
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 bone|talus]] of hoofed animals, colloquially known as [[knucklebones]].<ref name="Laird">{{Cite book |last=Laird |first=Jay |title=Encyclopedia of Play in Today's Society |publisher=SAGE Publications |year=2009 |isbn=978-1-4522-6610-7 |pages=171–173}}</ref> The [[Ancient Egypt]]ian game of [[senet]] (played before 3000&nbsp;[[Common Era|BCE]] and up to the 2nd century CE) was played with flat two-sided [[Binary lot#Staves|throwsticks]] which indicated the number of squares a player could move, and thus functioned as a form of dice.<ref name="Aruz">{{Cite book |last=Finkel |first=Irving |chapter-url=https://books.google.com/books?id=gr5BgOwEJicC&pg=PA151 |title=Beyond Babylon: Art, Trade, and Diplomacy in the Second Millennium B.C. |publisher=Metropolitan Museum of Art |year=2008 |isbn=978-1-58839-295-4 |page=151 |chapter=Board Games}}</ref> 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&nbsp;BCE.<ref>{{Cite web |date=2017-11-29 |title=8 Oldest Board Games in the World |url=https://www.oldest.org/entertainment/board-games/ |access-date=2022-03-12 |website=Oldest.org |language=en-US}}</ref><ref>{{Cite journal |last=NASER MOGHADASI |first=Abdorreza |date=September 2015 |title=The Burnt City and the Evolution of the Concept of "Probability" In the Human Brain |journal=Iranian Journal of Public Health |volume=44 |issue=9 |pages=1306–1307 |issn=2251-6085 |pmc=4645795 |pmid=26587512}}</ref> Bone dice from [[Skara Brae]], Scotland have been dated to 3100–2400&nbsp;BCE.<ref>{{Cite web |title=National Museums of Scotland - Two bone dice |url=https://nms.scran.ac.uk/database/record.php?usi=000-100-040-457-C |access-date=9 May 2020 |website=nms.scran.ac.uk}}</ref> Excavations from graves at [[Mohenjo-daro]], an [[Indus Valley civilization]] settlement, unearthed [[terracotta]] dice dating to 2500–1900&nbsp;BCE,<ref>Possehl, Gregory. "Meluhha". In: J. Reade (ed.) ''The Indian Ocean in Antiquity''. London: Kegan Paul Intl. 1996a, 133–208</ref> including at least one die whose opposite sides all add up to seven, as in modern dice.<ref>{{Cite web |last=Ashmolean Museum |first=University of Oxford |title=Terracotta Dice |url=http://jameelcentre.ashmolean.org/collection/4/6739/6741/11231 |access-date=2023-03-19 |website=jameelcentre.ashmolean.org |language=en}}</ref>

Games involving dice are mentioned in the [[Outline of ancient India|ancient Indian]] ''[[Rigveda]]'',<ref>{{Cite web |last=www.wisdomlib.org |date=2021-08-27 |title=Rig Veda 10.34.1 [English translation] |url=https://www.wisdomlib.org/hinduism/book/rig-veda-english-translation/d/doc838881.html |access-date=2024-07-28 |website=www.wisdomlib.org |language=en}}</ref> ''[[Atharvaveda]],'' ''[[Mahabharata]]'' and the [[List of games that Buddha would not play|Buddhist games list]].<ref>2.3, 4.38, 6.118, 7.52, 7.109</ref> 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.<ref>{{Cite web |last=Good |first=Alexandra |title=Knucklebones |url=http://archaeologicalmuseum.jhu.edu/the-collection/object-stories/archaeology-of-daily-life/childhood/knucklebones/ |access-date=16 April 2015 |website=Johns Hopkins Archaeological Museum}}</ref>

Although gambling was illegal, many [[Culture of ancient Rome|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.<ref>{{Cite book |last=Matz |first=David |url=https://archive.org/details/dailylifeofancie00matz/page/94 |title=Daily Life of the Ancient Romans |publisher=Greenwood Publishing Group |year=2002 |isbn=978-0-313-30326-5 |pages=[https://archive.org/details/dailylifeofancie00matz/page/94 94–95] |url-access=registration}}</ref> Twenty-sided dice date back to the 2nd&nbsp;century CE<ref>{{Cite web |title=christies.com |url=http://www.christies.com/Lotfinder/lot_details.aspx?intObjectID=4205385 |access-date=18 June 2012 |publisher=christies.com}}</ref> and from [[Ptolemaic Kingdom|Ptolemaic Egypt]] as early as the 2nd century&nbsp;BCE.<ref name="auto"/>

[[Dominoes]] and [[playing card]]s originated in China as developments from dice.<ref>{{Cite web |title=domino {{!}} game piece {{!}} Britannica |url=https://www.britannica.com/topic/domino-game-piece |access-date=2022-11-20 |website=www.britannica.com |language=en}}</ref> 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 [[woodblock printing|block printed]] books.<ref>{{Cite book |last1=Ronan |first1=Colin |url=https://books.google.com/books?id=CjRAiqGSJ50C&pg=PA55 |title=The Shorter Science and Civilisation in China |last2=Needham |first2=Joseph |publisher=Cambridge University Press |year=1986 |isbn=978-0-521-31560-9 |page=55}}</ref> 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 [[Race game|racing game]].<ref>{{Cite book |last=Salter |first=Rebecca |title=Japanese Popular Prints: From Votive Slips to Playing Cards |publisher=University of Hawaii Press |year=2006 |isbn=978-0-8248-3083-0 |page=164 |chapter=Board Games}}</ref>

<gallery mode=packed heights=150px>
File:Knuck dice Steatite 37x27x21 mm.JPG|[[Knucklebones]] die, made of [[soapstone]]
File:Twenty-sided die (icosahedron) with faces inscribed with Greek letters MET 10.130.1158 001.jpg|Twenty-sided [[serpentinite]] die from [[Ptolemaic Kingdom|Ptolemaic Egypt]]<ref name="auto">{{Cite web |title=Twenty-sided die (icosahedron) with faces inscribed with Greek letters |url=http://www.metmuseum.org/collection/the-collection-online/search/551072 |access-date=28 March 2015 |publisher=metmuseum.org}}</ref>
File:Roman dice IMG 4367.JPG|Roman die
File:9BFE00 -roman lead die (FindID 103936).jpg|Composite image of all sides of a {{cvt|12|mm|in|frac=4}} [[Culture of ancient Rome|Roman]] die, found in [[Leicestershire]], 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]], 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]] Museum, China.
</gallery>

==Etymology and terms==
The word die comes from [[Old French]] ''[[wikt:dé#Old French|dé]]''; from [[Latin]] ''[[wikt:datum#Latin|datum]]'' "something which is given or played".<ref>{{Cite web |title=die |url=http://www.oxforddictionaries.com/definition/english/die?searchDictCode=all#die-2 |archive-url=https://web.archive.org/web/20150214101611/http://www.oxforddictionaries.com/definition/english/die?searchDictCode=all#die-2 |url-status=dead |archive-date=14 February 2015 |access-date=14 February 2015 |publisher=Oxford Dictionaries}}</ref>

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";<ref>{{Cite web |title=ace |url=http://www.askoxford.com/concise_oed/ace |archive-url=https://web.archive.org/web/20070929095518/http://www.askoxford.com/concise_oed/ace |url-status=dead |archive-date=29 September 2007 |access-date=18 June 2012 |publisher=AskOxford}}</ref> the others are 2 to 6 in [[Old French]].<ref name="Conant">{{Cite book |last=Conant |first=Levi Leonard |url=http://www.gutenberg.org/files/16449/16449-h/16449-h.htm |title=The Number Concept: Its Origin and Development |publisher=Macmillan |year=1896 |page=124}}</ref>

When rolling two dice, certain combinations have slang names. The term ''snake eyes'' is a roll of one [[pip (counting)|pip]] on each die. The ''[[Online Etymology Dictionary]]'' traces use of the term as far back as 1919.<ref name="ref1">{{Cite web|url=https://www.etymonline.com/search?q=s&page=28|title=s &#124; Search Online Etymology Dictionary|website=www.etymonline.com}}</ref>

The US term ''boxcars'', also known as ''midnight'', is a roll of six pips on each die. The pair of six pips resembles a pair of [[boxcar]]s 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 [[Dice tower|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 [[Randomness|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]]. 

==Construction==
===Arrangement===

[[File:Chiralität von Würfeln V.2.svg|thumb|[[Chirality]] of dice. Faces may be placed [[clockwise and counterclockwise|counterclockwise (top) ''or'' clockwise]].]]

Common dice are small [[cube]]s, most often {{convert|1.6|cm|in|abbr=on}} across, whose faces are numbered from one to six, usually by patterns of round dots called [[pip (counting)|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 (geometry)|vertex]].<ref>Cf. {{Cite book|url=http://archive.org/details/greekanthology05newyuoft|title=The Greek anthology|date=1918|location=New York |publisher=G.P. Putnam's and Sons|translator-first=W.R.|translator-last=Paton|at = Book 14, §8|quote=The Opposite Pairs of Numbers on a Die. The numbers on a die run so: six one, five two, three four.}}</ref> The faces of a die may be placed [[clockwise|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.<ref name="diceplay">{{Cite web |title=Standard Dice |url=http://homepage.ntlworld.com/dice-play/DiceStandard.htm |archive-url=https://web.archive.org/web/20130730085649/http://homepage.ntlworld.com/dice-play/DiceStandard.htm |archive-date=30 July 2013}}</ref>

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.<ref name="diceplay" /><ref>{{Cite web |title=Chinese Dice |url=http://www.gamesmuseum.uwaterloo.ca/Archives/Culin/Dice1893/dice.html |archive-url=https://web.archive.org/web/20080501172614/http://www.gamesmuseum.uwaterloo.ca/Archives/Culin/Dice1893/dice.html |archive-date=1 May 2008 |publisher=Elliott Avedon Museum & Archive of Games}}</ref>

<gallery mode=packed>
File:Die Faces.svg|Typical facets showing the more compact pip arrangement of an Asian-style die (top) vs. a Western-style die (bottom)
</gallery>


===Manufacturing===
[[File:Snake eyes dice.jpg|thumb|right|Typical western ''non-precision'' dice use opaque plastic and have recessed pips.]]
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.<ref name=Scarne74/>{{rp|23–24}}

Non-precision dice are manufactured via the plastic [[injection molding]] process, often made of [[Poly(methyl_methacrylate) |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.<ref>{{Cite web |title=7 Surprising Facts About How Dice Are Made (With Pictures) |url=https://www.gmdice.com/blogs/dnd/how-are-dice-made |access-date=20 August 2019 |website=gmdice.com|date=19 August 2019 }}</ref>

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 [[Abrasion (mechanical)|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.

[[File:The Game (33285019256).jpg|thumb|left|These canceled precision dice from [[Tropicana Atlantic City]] are made from translucent red cellulose acetate and flush pips.]]
Precision dice are generally made from bars of extruded [[cellulose acetate]], sawed to the proper length to ensure that each face is as square as practical, generally with edges {{cvt|3/4|±|1/5000|in}} in length, with pips drilled {{cvt|17|±|4|mil}} deep and filled with opaque paint or epoxy which matches the [[density]] of cellulose, ensuring the dice remain balanced.<ref name=Scarne74/>{{rp|23}}<ref>{{Cite web |title=Casino Dice |url=http://www.dice-play.com/DiceCasino.htm |access-date=1 July 2018 |website=www.dice-play.com}}</ref> 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.<ref name=Scarne74/>{{rp|240}} Precision casino dice are transparent or translucent. This makes it difficult to incorporate internal weights which could bias the roll.<ref name=Scarne74/>{{rp|213–214;253}} To discourage cheating by dice substitution, each die carries a [[serial number]] and the casino's logo or name.<ref name=Scarne74>{{cite book |title=Scarne on Dice |first=John |last=Scarne |date=1974 |url=https://archive.org/details/scarneondice0000scar/mode/2up |publisher=Stackpole Books |location=Harrisburg, Pennsylvania |isbn=0-8117-1516-7 |url-access=registration |author-link=John Scarne}}</ref>{{rp|23–24}} 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 {{cvt|0.775|in}} on a side, except for the dice used in [[pai gow]], which range from {{cvt|0.637–0.643|in}} on a side.<ref>{{cite web |url=https://www.nj.gov/oag/ge/docs/Regulations/CHAPTER69E.pdf |title=New Jersey Administrative Code, Title 13, Chapter 69E: Gaming Equipment |publisher=State of New Jersey |access-date=24 December 2024}}</ref>{{rp|13:69E-1.15}}

[[File:Backgammon PrecisionDice.jpg|thumb|right|Precision backgammon dice also share the flush-pip construction, but tend to be smaller with heavily rounded corners.]]
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.<ref>{{cite web |url=https://katgammon.com/about-precision-dice/ |title=About Precision Dice |date=November 21, 2020 |first=Mizban |last=Host |website=Katgammon (blog) |access-date=25 December 2024}}</ref>

===Unicode representation===
{| class=wikitable
|-
! Symbol
| style="font-size:250%" | ⚀
| style="font-size:250%" | ⚁
| style="font-size:250%" | ⚂
| style="font-size:250%" | ⚃
| style="font-size:250%" | ⚄
| style="font-size:250%" | ⚅ 
| style="font-size:250%" | 🎲
|-
! Unicode
| U+2680 
| U+2681
| U+2682
| U+2683
| U+2684
| U+2685
| U+1F3B2
|-
! Decimal
| <nowiki>&amp;#9856;</nowiki>
| <nowiki>&amp;#9857;</nowiki>
| <nowiki>&amp;#9858;</nowiki>
| <nowiki>&amp;#9859;</nowiki>
| <nowiki>&amp;#9860;</nowiki>
| <nowiki>&amp;#9861;</nowiki>
| <nowiki>&amp;#127922;</nowiki>
|}

Using [[Unicode]] characters, the faces can be shown in text using the range U+2680 to U+2685 or using decimal <code>&amp;#9856;</code> to <code>&amp;#9861;</code>,<ref>{{Cite web |title=Dice faces in block Miscellaneous Symbols |url=https://www.unicode.org/charts/PDF/U2600.pdf |archive-url=https://web.archive.org/web/20010603010228/http://www.unicode.org/charts/PDF/U2600.pdf |archive-date=2001-06-03 |url-status=live |website=The Unicode standard}}</ref> and the emoji using U+1F3B2 or <code>&amp;#127922;</code> 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.<ref>{{Cite web |last=Horridge |first=Kevin |date=27 November 2017 |title=6 Ways Casinos Stop Dice Cheats |url=https://www.casino.org/blog/6-ways-casinos-stop-dice-cheats/ |access-date=11 October 2019 |website=Casino.org}}</ref>

==Variants== 
===Polyhedral dice===
[[File:6dice(cropped).jpg|thumb|right|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 [[wargame]]s,<ref name="Peterson2012">{{Cite book |last=Peterson |first=Jon |title=Playing at the World: A History of Simulating Wars, People and Fantastic Adventures, from Chess to Role-Playing Games |date=July 2012 |publisher=Unreason Press |isbn=978-0-615-64204-8 |pages=315–318}}</ref> and since have been employed extensively in [[role-playing game]]s and [[collectible card game|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]]s.<ref>{{Cite web |last=Girdwood |first=Andrew |date=30 March 2019 |title=What's a spindown dice and are standard d20s any fairer? |url=https://www.geeknative.com/65027/whats-a-spindown-dice-and-are-standard-d20s-any-fairer/ |access-date=9 July 2020}}</ref>

====Common variations====
{{Further information|Isohedral figure{{!}}Isohedron}}
Dice are often sold in sets, matching in color, of six different shapes. Five of the dice are shaped like the [[Platonic solid]]s, whose faces are [[regular polygon]]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]] die, whose faces are ten [[kite (geometry)|kites]], each with two different edge lengths, three different angles, and two different kinds of vertices. 

Unlike other common dice, a [[four-sided die|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]]s. The percentile dice system is used to produce a [[Discrete uniform distribution|uniform distribution]] of random percentages, and summing the values of multiple dice will produce approximations to [[normal distribution]]s.<ref>{{Cite web |last1=Paret |first1=Michelle |last2=Martz |first2=Eston |year=2009 |title=Tumbling Dice & Birthdays: Understanding the Central Limit Theorem |url=http://www.minitab.com/uploadedFiles/Shared_Resources/Documents/Articles/CentralLimitTheorem.pdf |url-status=dead |archive-url=https://web.archive.org/web/20131101141751/http://www.minitab.com/uploadedFiles/Shared_Resources/Documents/Articles/CentralLimitTheorem.pdf |archive-date=1 November 2013 |access-date=29 September 2013 |publisher=Minitab}}</ref> 

{| class="wikitable"
|-
! Faces/sides
! colspan="2" | Shape
! Notes
|-
| [[Four-sided dice|4]] || [[Tetrahedron]] || [[File:Tetrahedron.png|48px|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 is usually thrown into the air instead {{Citation needed|date=April 2025}}.
|-
| 6 || [[Cube]] || [[File:Hexahedron.png|48px|Cube]]
| The most common variation of die. The sum of the numbers on opposite faces is 7.
|-
| 8 || [[Octahedron]] || [[File:Octahedron.png|48px|Octahedron]]
| Each face is triangular and the die resembles two [[square pyramid]]s attached base-to-base. Usually, the sum of the opposite faces is 9.
|-
| [[Ten-sided die|10]] || [[Pentagonal trapezohedron]] || [[File:Trapezohedron5.jpg|40px|Pentagonal trapezohedron]]
| Each face is a [[Kite (geometry)|kite]]. The die has two sharp corners, where five kites meet, and ten blunter corners, where three kites meet. Often, all [[Parity (mathematics)|odd numbered]] faces converge at one sharp corner, and the [[Parity (mathematics)|even]] ones at the other. The 10-sided die is usually numbered 0–9, though the 0 can also be read as a 10.
|-
| 12 || [[regular dodecahedron|Dodecahedron]] || [[File:Dodecahedron.png|48px|Dodecahedron]]
| Each face is a regular pentagon. The sum of the numbers on opposite faces is usually 13.
|-
| 20 || [[regular icosahedron|Icosahedron]] || [[File:Icosahedron.png|48px|Icosahedron]]
| Faces are [[equilateral triangle]]s. 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====
[[Image:Dices collection.png|upright=1.9|thumb|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 [[isohedral figure|face-transitive]]. In addition to the Platonic solids, these theoretically include:

* [[Catalan solid]]s, the [[dual polyhedron|duals]] of the 13 [[Archimedean solid]]s: 12, 24, 30, 48, 60, 120 sides
* [[Trapezohedron|Trapezohedra]], the duals of the infinite set of [[antiprism]]s, with kite faces: any even number not divisible by 4 (so that a face will face up), starting from 6
* [[Bipyramid]]s, the duals of the infinite set of [[Prism (geometry)|prisms]], with triangle faces: any multiple of 4 (so that a face will face up), starting from 8
* [[Disphenoid]]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, [[Pyritohedron|pyritohedra]] and [[tetartoid]]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]]s: the basis of [[barrel dice]]
* [[prism (geometry)|prism]]s: the basis of long dice and teetotums

[[Long dice]] and [[teetotum]]s can, in principle, be made with any number of faces, including odd numbers.<ref>{{Cite web |last=Kybos |first=Alea |title=Properties of Dice |url=http://www.aleakybos.ch/Properties%20of%20Dice.pdf |url-status=dead |archive-url=https://web.archive.org/web/20120528013233/http://www.aleakybos.ch/Properties%20of%20Dice.pdf |archive-date=28 May 2012 |access-date=7 October 2012}}</ref> Long dice are based on the infinite set of [[prism (geometry)|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]].

{| class="wikitable"
|-
! Faces/sides
! Shape
! Image
! Notes
|-
| 1
| [[Möbius strip]] or [[sphere]]
| [[File:D1 dice.JPG|48px]]
| Most commonly a [[practical joke|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 (geometry)|Cylinder]] or Flat [[Prism_(geometry)|Prism]]
|[[File:D02.JPG|48x48px]]
| 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. (See also [[Binary lot]].)
|-
| 3
| Rounded-off [[triangular prism]]
| [[File:D03 wood.jpg|48px]]
| 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]]
| [[File:Daldøs die.jpg|48px]]
| A long die intended to be rolled lengthwise. It cannot stand on end as the ends are capped.
|-
| rowspan=3 | 5
|-
| [[Triangular prism]]
| [[File:D05.jpg|48px]]
| 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 - similar to a 4-sided die. The triangular faces are labeled with the digits 1 and 5. 
|-
| Capped 5-sided [[long die]]
| [[File:Game of Dignitaries long die Culin 1898 fig 136.png|48px]]
| 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]]
| [[File:Owzthat Dice Game.jpg|48px]]
| 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]].
|-
| rowspan=2 | 7
| [[Pentagonal prism]]
| [[File:D07.jpg|48px]]
| Similar in constitution to the 5-sided die. Seven-sided dice are used in a [[Seven-sided backgammon|seven-player variant]] of [[backgammon]]. Seven-sided dice are described in the 13th century {{lang|es|[[Libro de los juegos]]}} as having been invented by [[Alfonso X of Castile|Alfonso X]] in order to speed up play in [[chess variants]].<ref>{{Cite web |title=games.rengeekcentral.com |url=http://games.rengeekcentral.com/tc4.html |access-date=18 June 2012 |publisher=games.rengeekcentral.com}}</ref><ref>{{Cite web |title=wwmat.mat.fc.ul.pt |url=http://wwmat.mat.fc.ul.pt/~jnsilva/HJT2k9/AlfonsoX.pdf |archive-url=https://web.archive.org/web/20110302034551/http://wwmat.mat.fc.ul.pt/~jnsilva/HJT2k9/AlfonsoX.pdf |archive-date=2011-03-02 |url-status=live |access-date=18 June 2012 }}</ref> 
|-
| [[Spherical_cap|Truncated sphere]]
| [[File:D7_dice.JPG|48px]]
| A truncated sphere with seven landing positions. 
|-
| 9
| Truncated sphere
| [[File:D9-dice-impact.png|48px]]
| A truncated sphere with nine landing positions. 
|-
| 10
| Decahedron
| [[File:D10_truncated.jpg|48px]]
| A ten-sided die made by truncating two opposite vertices of an octahedron. 
|-
| 11
| Truncated sphere
| [[File:D11_dice.JPG|48px]]
| A truncated sphere with eleven landing positions. 
|-
| 12
| [[Rhombic dodecahedron]]
| [[File:D12_rhombic_dodecahedron.JPG|48px]]
| Each face is a [[rhombus]].
|-
| 13
| Truncated sphere
| [[File:D13_dice.JPG|48px]]
| A truncated sphere with thirteen landing positions. 
|-
| rowspan=3 | 14
| [[Heptagonal trapezohedron]]
| [[File:14面体ダイス.jpg|48px]]
| Each face is a [[kite (geometry)|kite]].
|-
| [[Truncated octahedron]]
| [[File:Korean14dice2.JPG|48px]]
| A truncated octahedron. Each face is either a square or a hexagon. 
|-
| Truncated sphere
| [[File:D14_truncated_octahedron.jpg|48px]]
| A truncated sphere with fourteen landing positions. The design is based on the [[cuboctahedron]]. 
|-
| 15
| Truncated sphere
| [[File:D15_dice.JPG|48px]]
| A truncated sphere with fifteen landing positions. 
|-
| 16
| [[Octagonal bipyramid]]
| [[File:D16_dice.JPG|48px]]
| Each face is an isosceles triangle.
|-
| 17
| Truncated sphere
| [[File:D17_dice_2.JPG|48px]]
| A truncated sphere with seventeen landing positions. 
|-
| 18
| Rounded [[rhombicuboctahedron]]
| [[File:D18_rhombicuboctahedron.JPG|48px]]
| Eighteen faces are squares. The eight triangular faces are rounded and cannot be landed on. 
|-
| 19
| Truncated sphere
| [[File:D19_dice.JPG|48px]]
| A truncated sphere with nineteen landing positions. 
|-
| 21
| Truncated sphere
| [[File:D21_dice.webp|48px]]
| A truncated sphere with twenty-one landing positions. 
|-
| 22
| Truncated sphere
| [[File:D22_dice.JPG|48px]]
| A truncated sphere with twenty-two landing positions. 
|-
| rowspan=5 | 24
| [[Triakis octahedron]]
| [[File:D24_triakis_octahedron_dice.JPG|48px]]
| Each face is an isosceles triangle.
|-
| [[Tetrakis hexahedron]]
| [[File:D24_tetrakis_hexahedron.JPG|48px]]
| Each face is an isosceles triangle.
|-
| [[Deltoidal icositetrahedron]]
| [[File:D24_deltoidal_icositetrahedron.JPG|48px]]
| Each face is a kite.
|-
| Pseudo-deltoidal icositetrahedron
| [[File:D24 pseudo uniform polyhedrondice.jpg|48px]]
| Each face is a kite.
|-
| [[Pentagonal icositetrahedron]]
| [[File:D24_pentagonal_icositetrahedron_dice.JPG|48px]]
| Each face is an irregular pentagon.
|-
| 26
| Truncated sphere
| [[File:D26_dice.webp|48px]]
| A truncated sphere with twenty-six landing positions.  
|-
| 28
| Truncated sphere
| [[File:D28_dice.webp|48px]]
| A truncated sphere with twenty-eight landing positions. 
|-
| 30
| [[Rhombic triacontahedron]]
| [[File:D30.jpg|48px]]
| Each face is a rhombus.
|-
| 32
| Truncated sphere
| [[File:D32_dice.JPG|48px]]
| A truncated sphere with thirty-two landing positions. The design is similar to that of a [[truncated icosahedron]]. 
|-
| 34
| [[Heptadecagon]]al trapezohedron
| [[File:D34.jpg|48px]]
| Each face is a kite.
|-
| 36
| Truncated sphere
| [[File:D36_dice.webp|48px]]
| 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]]
| [[File:D48_dice.JPG|48px]]
| Each face is a [[scalene triangle]].
|-
| 50
| Icosipentagonal trapezohedron
| [[File:D50 trapezohedron dice.JPG|48px]]
| Each face is a kite.
|-
| rowspan=4 | 60
| [[Deltoidal hexecontahedron]]
| [[File:D60_60men-saikoro.JPG|48px]]
| Each face is a kite.
|-
| [[Pentakis dodecahedron]]
| [[File:D60_pentakis_dodecahedron_dice.JPG|48px]]
| Each face is an isosceles triangle.
|-
| [[Pentagonal hexecontahedron]]
| [[File:D60_pentagonal_hexecontahedron_dice.JPG|48px]]
| Each face is an irregular pentagon.
|-
| [[Triakis icosahedron]]
| [[File:D60_triakis_icosahedron_dice.JPG|48px]]
| Each face is an isosceles triangle.
|-
| 100
| [[Zocchihedron]]
| [[File:Zocchihedron2.jpg|48px]]
| 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]]
| [[File:D120.jpg|48px]]
| Each face is a scalene triangle.
|}

===Non-numeric dice===
[[File:White Fudge Dice.jpg|right|thumb|A set of [[Fudge (role-playing game system)|''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 (game)|Warhammer]]'', ''[[Fudge (role-playing game system)#Fudge dice|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.<ref name="em4">{{Cite web |title=Specialist D6 |url=http://www.em4miniatures.com/acatalog/SPECIALIST_D6.html |access-date=18 August 2017 |website=em4miniatures}}</ref><ref name="tmp">{{Cite web |date=20 November 2009 |title=[TMP] 'What are Average Dice' Topic |url=http://theminiaturespage.com/boards/msg.mv?id=184673 |access-date=18 August 2017 |website=The Miniatures Page |language=en}}</ref> These are six-sided dice with sides numbered <code>2, 3, 3, 4, 4, 5</code>, 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 [[wargame]]s, where a narrower range of numbers is required.<ref name="tmp" /> Other numbered variations include [[Sicherman dice]] and [[intransitive dice]].

===Spherical dice===
[[File:Black and red round 6-sided die (cropped).jpg|thumb|upright=.5|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==
{{see also|List of dice games}}
[[File:Master Jean de Mauléon - Playing Dice - Walters W4492V (cropped).jpg|thumb|left|''Playing Dice'' by Master Jean de Mauléon (c. 1520)]]

One typical contemporary [[List of dice games|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 game]]s, where they are often used to decide the distance through which a piece will move along the board (as in [[backgammon]] and ''[[Monopoly (game)|Monopoly]]'').

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|date=August 2021}}<!-- State since what year, decade or century this is referring to --> 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]]''), different sides of the dice can offer completely different abilities. Several sides often give resources while others grant the player useful actions.<ref>{{Cite web |date=2 October 2014 |title=Making better use of dice in games |url=http://boingboing.net/2014/10/02/making-better-use-of-dice-in-g.html}}</ref>

Dice can be used for [[divination]] and using dice for such a purpose is called [[cleromancy]]. A pair of common dice is usual, though other forms of polyhedra can be used. Tibetan Buddhists sometimes use this [[Mo (divination)|method of divination]]. It is highly likely that the [[Pythagoreans]] used the [[Platonic solids]] as dice. They referred to such dice as "the dice of the gods" and they sought to understand the universe through an understanding of geometry in polyhedra.<ref>{{Cite book |last=Guthrie |first=Kenneth |title=The Pythagorean sourcebook and library: an anthology of ancient writings which relate to Pythagoras and Pythagorean philosophy |publisher=Phanes Press |year=1988 |isbn=978-0-933999-50-3 |location=Grand Rapids, MI |oclc=255212063}}</ref>

Astrological dice are a specialized set of three 12-sided dice for divination; the first die represents the planets, the Sun, the Moon, and the [[orbital node|nodes]] of the Moon, the second die represents the 12 [[Astrological sign|zodiac signs]], and the third represents the [[House (astrology)|12 houses]]. A specialized icosahedron die provides the answers of the [[Magic 8 Ball]], conventionally used to provide answers to yes-or-no questions.

Dice can be used to generate random numbers for use in [[password]]s and [[cryptography]] applications. The [[Electronic Frontier Foundation]] describes a method by which dice can be used to generate [[passphrase]]s.<ref>{{Cite news |date=12 May 2016 |title=EFF Dice-Generated Passphrases |language=en |work=Electronic Frontier Foundation |url=https://www.eff.org/dice |access-date=16 November 2018}}</ref> [[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.<ref>{{Cite web |date=19 July 2016 |title=EFF's New Wordlists for Random Passphrases |url=https://www.eff.org/deeplinks/2016/07/new-wordlists-random-passphrases |access-date=4 December 2016 |website=Electronic Frontier Foundation}}</ref>

Thrown or simulated dice are sometimes used to generate specific probability distributions, which are fundamental to [[probability theory]]. 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===
{{Main|Dice 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 [[Dice notation|standard]], is written as {{math|''n''d''s''}}. In this expression, {{math|''s''}} is the number of sides on the dice and {{math|''n''}} is the number of rolls; if there is only one roll, the {{math|''n''}} is omitted. As an illustration, the d20 (twenty-sided dice) is to ''[[Dungeons & Dragons]]'' what the d6 (six-sided dice) is to many board games. ''[[Monopoly (game)|Monopoly]]'' 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 {{math|''n''d''s''+''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 {{math|''n''d''s''−''c''}}. Thus, "3d6−4" instructs the player to subtract four from the total value of three six-sided dice. The notation {{math|''n''d''s''−L}} 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.<ref>{{Cite book |last=Sim |first=Chris |title=Dungeons & Dragons Rules Compendium |collaboration= |year=2007 |isbn=9780786947256 |location=Renton, WA USA |pages=17}}</ref><ref>{{Cite web |last=Mearls |first=Mike |display-authors=etal |date=2016 |title=System Reference Document 5.1 |url=https://media.wizards.com/2016/downloads/DND/SRD-OGL_V5.1.pdf |access-date=September 12, 2024 |page=96}}</ref>

===Role-playing dice sets and percentile dice===
[[File:Wuerfel5.jpg|thumb|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.

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,<ref>https://www.dicedragons.co.uk/blogs/dice-advice/how-to-roll-a-d100</ref> 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.<ref>https://rpggeek.com/blog/1051/blogpost/38414/interview-with-lou-zocchi-the-first-dice-idea-i-wa</ref>

== See also ==
{{div col}}
* {{annotated link|Astragalomancy}}
* {{annotated link|Crown and Anchor}}
* {{annotated link|d20 System}}
* {{annotated link|Dice pool}}
* {{annotated link|Dreidel}}
* {{annotated link|Fuzzy dice}}
* {{annotated link|Musikalisches Würfelspiel|''Musikalisches Würfelspiel''}}
* {{annotated link|Die throw (review)}}
* {{annotated link|Intransitive dice}}
* {{annotated link|Quincunx}}
* {{annotated link|Random number generation}}
* {{annotated link|Teetotum}}
* {{annotated link|Urim and Thummim}}
{{div col end}}

==References==
{{reflist}}

===Bibliography===
* [[Persi Diaconis|Diaconis, Persi]]; and Keller, Joseph B.; [http://www-stat.stanford.edu/~cgates/PERSI/papers/fairdice.pdf "Fair Dice"], {{webarchive |url=https://web.archive.org/web/20080221165308/http://www-stat.stanford.edu/~cgates/PERSI/papers/fairdice.pdf |date=21 February 2008 }}, ''The American Mathematical Monthly'', 96(4):337–339, 1989 (Discussion of dice that are fair "by symmetry" and "by continuity".)
* Iverson, G. R.; Longcour, W. H.; et al.; ''Bias and Runs in Dice Throwing and Recording: A Few Million Throws'', Psychometrika, vol. 36, no. 1, March 1971
* [[Reiner Knizia|Knizia, Reiner]], ''Dice Games Properly Explained'', Elliot Right Way Books, 1999, {{ISBN|0-7160-2112-9}}

==External links==
{{wiktionary|dice}}
{{Commons category|Dice}}
* {{mathworld |urlname=Dice |title=Dice}}{{snd}} analysis of dice probabilities, also featuring Uspenski's work on rolling multiple dice

{{EB1911|wstitle=Dice|volume=8|page=176–177}}
{{Craps}}
{{Authority control}} 
[[Category:Articles containing video clips]]
[[Category:Dice| ]]
[[Category:Gaming devices]]