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Ordinary dice The common dice are small cubes 1 to 2 cm along an edge (16mm being the standard), whose faces are numbered from one to six (usually by patterns of dots called pips). It is traditional to assign pairs of numbers that total seven to opposite faces (it has been since at least classical antiquity); this implies that at one vertex the faces 1, 2 and 3 intersect. It leaves one other abstract design choice: the faces representing 1, 2 and 3 respectively can be placed in either clockwise or counterclockwise order about this vertex. Dice are thrown to provide random numbers for gambling and other games and thus are a type of hardware random number generator. However, because the numbers on toy dice are marked with small indentations, slightly more material is removed from the higher numbered faces. This results in a small bias, and they do not provide fair (uniform) random numbers. The bias is reduced somewhat in the Japanese die with its oversized single pip (pictured). Casino dice have markings that are flush with the surface and come very close to providing true uniformly distributed random numbers. Dice are thrown, singly or in groups, from the hand or from a cup or box designed for the purpose, onto a flat surface. The face of each die that is uppermost when it comes to rest provides the value of the throw. A typical dice game today is craps, wherein two dice are thrown at a time, and wagers are made on the total value of up-facing pips on the two dice. They are also frequently used to randomize allowable moves in board games such as Backgammon. History
Materials Dice have been made from a wide variety of materials throughout history, including stone, wood, and animal bones, and more recently, bakelite and plastic. Precision dice Precision casino dice, used for the game of craps, are made from cellulose acetate. These dice may have a polished finish, making them transparent, or a sand finish, making them translucent. Casino dice have their pips drilled, and then filled flush with a paint of the same specific gravity as the acetate, such that the dice remain in perfect balance. In casino play, a stick of 5 dice are used, all stamped with a matching serial number to prevent a cheat from substituting a die. Precision backgammon dice are also made from acetate, or a similar material, with the pips filled in as is done with casino dice. While casino dice are noticeably larger than common dice, with sharp edges and corners, precision backgammon dice tend to be somewhat smaller. Their corners and edges are beveled to allow greater movement inside the dice cup and prevent chaotic rolls from damaging the playing surface. Polyhedral dice
Terms While the terms ace, deuce, trey, cater, cinque and sice are hardly common today having been replaced with the ordinary names of the numbers one to six, they are still used by some professional gamblers to describe the different sides of the dice. Ace is from the Latin as, meaning "a unit" *; the others are the numbers 2–6 in old French. Dice Notation Often the names of the dice appear in formulas for calculating game parameters: e.g., hit points. "6d8+10", for example, will yield a number between 16 (6×1+10) and 58 (6×8+10), as it means "Roll an eight-sided die six times and add ten to the total." Occasionally they may be written "1d6×10+20"; this means "Roll one six-sided die. Multiply it by ten and add twenty." "Crooked" dice "Crooked dice" refers to dice that have been altered in some way to change the distribution of the dice's outcome. Loaded dice A loaded or gaffed die is a die that has been tampered with to land with a selected side facing upwards more often than it would simply by chance. There are methods of creating loaded dice, including having some edges round and other sharp and slightly off square faces. If the dice are not transparent, weights can be added to one side or the other. They can be modified to produce winners ("passers") or losers ("miss-outs"). "Tappers" have a drop of mercury in a reservoir at the center of the cube, with a capillary tube leading to another mercury reservoir at the side of the cube. The load is activated by tapping the die on the table so that the mercury leaves the center and travels to the side. Often one can see the circle of the cut used to remove the face and bury the weight. In a professional die, the weight is inserted in manufacture; in the case of a wooden die, this can be done by carving the die around a heavy inclusion, like a pebble around which a tree has grown. A variable loaded die is hollow with a small weight and a semi-solid substance inside, usually wax, whose melting point is just lower than the temperature of the human body. This allows the cheater to change the loading of the die by breathing on it or holding it firmly in hand, causing the wax to melt and the weight to drift down, making the chosen opposite face more likely to land up. A less common type of variable die can be made by inserting a magnet into the die and embedding a coil of wire in the game table. Then, either leave the current off and let the die roll unchanged or run current through the coil to increase the likelihood that the north side or the south side will land on the bottom depending on the direction of the current. Plastic dice can be biased to roll a certain number by heating them (for example in an oven) with the desired face upward, so that the plastic will soften slightly and "pool" at the opposite (bottom) side of the die without showing much, if any, visible distortion. Transparent acetate dice, used in all reputable casinos, are harder to tamper with. Cheat dice Cheat dice (see below) are often sold as loaded dice but usually are not technically loaded. Shaved dice A die can be "shaved" on one side i.e. slightly shorter in one dimension, making it slightly rectangular and thus affecting its outcome. One countermeasure employed by casinos against shaved dice is to measure the dice with a micrometer. Dice with faces other than digit sequences As noted, the faces of most dice are labelled using an unbroken series of whole numbers, starting at one (or zero), expressed with either pips or digits. Common exceptions include: Non-cubical dice Some dice are polyhedra other than cubes in shape. They were once almost exclusively used by fortune-tellers and in other occult practices, but they have become popular lately (at least since the early 1950s) among players of wargames, trading card games, German-style board games, and role-playing games. Although polyhedral dice are a relative novelty during modern times, some ancient cultures appear to have used them in games (as evidenced by the presence of two icosahedral dice dating from the days of ancient Rome on display in the British Museum). Such dice are typically plastic, and have faces bearing numerals rather than patterns of dots. Reciprocally symmetric numerals are distinguished with a dot in the lower right corner (6. vs 9.) or by being underlined (6 vs 9). The platonic solids are commonly used to make dice of 4, 6, 8, 12, and 20 faces. Other shapes can be found to make dice with 2, 5, 7, 10, 16, 24, 30, 34, 50, or 100 sides, but other than the 10 sided, they are rarely used. (See Zocchihedron.) The 4 sided platonic solid is difficult to roll, and a few games like Daldøs use a 4 sided rolling pin instead. A large number of different probability distributions can be obtained using these dice in various ways; for example, 10-sided dice (or 20-sided dice labeled with single digits) are often used in pairs to produce a linearly-distributed random percentage. Summing multiple dice approximates a normal distribution (a "bell curve"), while eliminating high or low throws can be used to skew the distribution in various ways. Using these techniques, games can closely approximate the real probability distributions of the events they simulate. There is some controversy over whether manufacturing processes create genuinely "fair" dice (dice that roll with even distributions over their number span). Casino dice are legally required to be fair; those used by all others hold no such requirement. Spherical dice also exist; these function like the plain cubic dice, but have an octahedral internal cavity in which a weight moves which causes them to settle in one of six orientations when rolled. However, these dice are somewhat awkward in use because they require a flat and level surface to roll properly — an uneven surface often causes them to stop partway between two numbers, while a sloped surface will obviously cause the dice to keep rolling. Cowry shells or coins may be used as a kind of two-sided dice. (Because of their shape, cowry shells probably do not yield a uniform distribution.) Standard variations
Rarer variations The full geometric set of "uniform fair dice" (with all congruent sides) are: Probability For a single roll of an -sided die, the probability of rolling each value, 1 through , is exactly 1/s. This is an example of a discrete uniform distribution. For a double roll, however, the total of both rolls is not evenly distributed, but is distributed in a triangular curve. For a six-sided die, for example, the probability distribution is as follows: For three or more die rolls, the curve becomes more bell-shaped with each additional die (according to the central limit theorem). The exact probability distribution for any number of dice can be calculated as the repeated convolution of the single-die probability distribution with itself. For example, in the triangular curve described above, Equivalently, one can calculate the probability that an -sided die, rolled times, will yield some sum using combinations: The probability of rolling any exact sequence of numbers is simply . For example, the chance of rolling 1, 2, and 3 in that order with three rolls of a six-sided die is , or . Rolling any single number times in a row, regardless of which number, is times more likely, at a chance. Application in role-playing games The fantasy role-playing game Dungeons & Dragons introduced the use of polyhedral dice during modern times and paved the way for their use in other role-playing games, using 20-, 12-, 10-, 8- and 4-sided dice in addition to the traditional 6 sided die. Such dice are often sold in sets. Types of polyhedral dice are distinguished by prefixing a "d" to the number of faces; for example, a ten-sided die is a d10. Players use polyhedral dice together in a number of ways. For example, a d10 can be used in conjunction with a d6 instead of using a d20. If the d6 displays a 1, 2 or 3, the number on the d10 is resolved as 1–10. If the d6 displays a 4, 5 or 6, the number shown on the d10 is resolved to 11–20 ("1" is 11, "2" is 12, etc.). In cases like this, almost any sided die can be used as a "resolver". Two d10 are often used to generate a number between 1 and 100. When tossing these dice, the player indicates which die is "high" (representing the tens position). This may be done by color, or using a custom die marked with multiples of ten. Similar methods can be used for additional digits. Use of dice for divination Some people believe that dice can be used for divination. Using dice for such a purpose is called cleromancy. A pair of standard 6-sided dice is generally used. Astrological dice are a specialized set of three 12-sided dice for divination, using the concepts of astrology and containing astrological symbols for the planets, the zodiac signs and the astrological houses. The first die represents planets, the Sun, the Moon, and two nodes (North Node and South Node). The second die represents the 12 zodiac signs, and the third represents the 12 houses. In simplified terms, the planets, etc. could represent the 'actor'; the zodiac signs could represent the 'role' being played by the actor; and the house could represent the 'scene' in which the actor plays. Rune dice are a specialized set of dice for divination (runecasting), using the symbols of the runes printed on the dice. An icosahedron is used to provide the answers of a Magic 8-Ball, which is conventionally used to provide advice on yes-or-no questions. See also | |||||||||||||||
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