Editing Dice (section)

==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.