Dice
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.
It is hypothesized that dice developed from the practice of fortune-telling with the talus of hoofed animals, colloquially known as knucklebones.
[3] Perhaps the oldest known dice were excavated as part of a backgammon-like game set at the Burnt City, an archeological site in south-eastern Iran, estimated to be from between 2800 and 2500 BCE.
[10] 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.
Ban-sugoroku is similar to backgammon and dates to the Heian period (794–1185 CE), while e-sugoroku is a racing game.
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.
These distributions can model real-world scenarios or mathematical constructs, making dice a practical tool for teaching and exploring concepts in probability theory.
Common dice are small cubes, most often 1.6 cm (0.63 in) across, whose faces are numbered from one to six, usually by patterns of round dots called pips.
Opposite sides of a modern die traditionally add up to seven, requiring the 1, 2, and 3 faces to share a vertex.
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.
[21]: 23–24 Non-precision dice are manufactured via the plastic injection molding process, often made of polymethyl methacrylate (PMMA).
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 3⁄4 ± 1⁄5000 in (19.0500 ± 0.0051 mm) in length, with pips drilled 17 ± 4 mils (0.43 ± 0.10 mm) deep and filled with opaque paint or epoxy which matches the density of cellulose, ensuring the dice remain balanced.
[21]: 213–214, 253 To discourage cheating by dice substitution, each die carries a serial number and the casino's logo or name.
[21]: 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 0.775 in (19.7 mm) on a side, except for the dice used in pai gow, which range from 0.637–0.643 in (16.2–16.3 mm) on a side.
[24]: 13:69E-1.15 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.
[26] 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.
Using Unicode characters, the faces can be shown in text using the range U+2680 to U+2685 or using decimal ⚀ to ⚅,[30] and the emoji using U+1F3B2 or 🎲 from the Miscellaneous Symbols and Pictographs block.
[31] 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.
Aside from the cube, the other four Platonic solids have 4, 8, 12, and 20 faces, allowing for those number ranges to be generated.
[33] 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.
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.
In addition to the Platonic solids, these theoretically include: Two other types of polyhedra are technically not face-transitive but are still fair dice due to symmetry: Long dice and teetotums can, in principle, be made with any number of faces, including odd numbers.
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.
The faces of most dice are labelled using sequences of whole numbers, usually starting at one, expressed with either pips or digits.
[42] 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.
For example, d4 denotes one four-sided die; 6d8 means the player should roll six eight-sided dice and sum the results.
When an amount is added, the notation is nds+c or nDs+c; for example, 3d6+4 instructs the player to roll three six-sided dice, calculate the total, and add four to it.
