For puzzles involving any form of dice.
Dice are small throwable objects used for generating random numbers.
- A traditional die is a rounded cube, with each of its six faces showing a different number of dots (pips) from 1 to 6. When thrown or rolled, the die comes to rest showing on its upper surface a random integer from 1 to 6, each value being equally likely.
- A variety of similar items are also described as dice; such specialized dice may be tetrahedrons, dodecahedrons, or irregular solids, and may have faces marked with symbols instead of numbers. Different dice are often referred to as 'd#' where the '#' references the highest value on the dice (i.e. d6 for 6-sided, d20 for 20-sided, etc.).
Non-standard numberings include the Sicherman dice ($1, 2, 2, 3, 3, 4$ and $1, 3, 4, 5, 6, 8$ - throw them together and the distribution is the same as with two normal dice), and nontransitive dice, where A beats B over $50\%$ of the time, B beats C over $50\%$ of the time, but A doesn't beat C over $50\%$ of the time, for example $[2, 2, 4, 4, 9, 9]$, $[1, 1, 6, 6, 8, 8]$ and $[3, 3, 5, 5, 7, 7]$.
Typically, dice faces are arranged such that opposite faces always add up to the sum of the highest and lowest values on the dice. For example, with a d6 (six-sided die) the highest value is 6 and the lowest is 1. Therefore the numbers are typically arranged so that opposite faces add up to 7 (1 and 6, 2 and 5, 3 and 4).
A loaded dice is one that doesn't have a uniform distribution for the probability of landing on each face.