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Terminology

Alphabetic String. An alphabetic string (or just “string”) is a representation of a dice set that is more convenient than the traditional “array listing” and takes advantage of the fact that it is assumed that no numeric values are repeated across the dice. In an alphabetic string of n dice, the first n letters of the alphabet (traditionally lowercase) are concatenated into a single string so that a letter's position in the string indicates that that particular numeric value is assigned to that die. For example, the alphabetic string
```
cbabccccbabc
```
is equivalent to the d2+d4+d6 set:
```
Die 1: 3, 10
Die 2: 2, 4, 9, 11
Die 3: 1, 5, 6, 7, 8, 12
```
Column Grouped. (see “Symmetry”)
Column Paired. (see “Symmetry”)
Fairness. There are different levels of fairness that a set of dice can exhibit. The original “go first dice” question dealt with only the weakest type of fairness, but the problem has now morphed into a search for dice with the permutation fairness (the strongest type). The three levels of fairness, from weakest to strongest are listed below (and there is a separate page about concepts of fairness with a bit more explanation).
- Go-First Fairness. Each player has an equal chance of rolling the highest number, and that person would go first.
- Place fairness. Each player has an equal chance of not only rolling highest, but also ending up in any ranking/position. In other words, each player also has an equal chance of rolling the second highest number, the third highest number (if three or more players rolling), and so on.
- Permutation Fairness. Not only does each player have an equal chance of ending up in any position in the ordering, but every possible ordering (permutation) of players has an equal chance of occurring.
Go-First Fairness. (see “Fairness”)
Homogeneous. A set of dice is homogeneous if all of the dice have the same number of faces. A set that does not have this property is said to be “inhomogeneous” or “heterogeneous”.
Inhomogeneous (aka Heterogeneous). Not homogeneous.
Mirrored/Palindromic. (see “Symmetry”)
Nice Dice. A “nice” set comprises dice that are all isohedral shapes that are not lenses or rolling logs (“nice” is definitely subjective, since 2n-lens-shaped dice with small n values are definitely functional and pleasing enough for some folks' tastes). Also, a d2 (a coin) is not considered “nice”, but that is purely the opinion of this author (Eric). “Nice” side counts, therefore, are the following: 4, 6, 8, 12, 20, 24, 30, 48, 60, and 120.
Permutation Fairness. (see “Fairness”)
Place Fairness. (see “Fairness”)
Symmetry. A set of dice may exhibit certain types of symmetry. Restricting dice sets with such symmetries can reduce the size of a proposed search space significantly. The main types of symmetry considered are as follows:
- Column Grouped symmetry occurs when, on a set of n dice, the first (lowest) n values are evenly distributed across the dice in the set (that is, one value per die), the next group of n values are evenly distributed, and so on, It is called “column grouped”, because, when the dice configuration is written in traditional format, each column lists a group of consecutive n values. Note that column grouped symmetry demands that a dice set be homogeneous. Here is an example of a dice set with column group symmetry:
```
Die 1: 1, 6, 8, 12, 13, 17
Die 2: 2, 4, 9, 11, 15, 16
Die 3: 3, 5, 7, 10, 14, 18
```
- Column Paired symmetry occurs when a set of dice not only has column grouped symmetry (see above), but also the property that the first (lowest) pair of numbers on the dice all add up to the same number, the second pair of numbers all add up to the same number, and so on. The following example exhibits column paired symmetry (note that the numbers in the first two columns of each die add up to 9, the numbers in the next two columns all added to 25, and so on…):
```
Die 1: 1, 8, 11, 14, 19, 22, 27, 30, 35, 38, 41, 48
Die 2: 2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, 47
Die 3: 3, 6, 12, 13, 17, 24, 25, 32, 36, 37, 43, 46
Die 4: 4, 5, 9,  16, 20, 21, 28, 29, 33, 40, 44, 45
```
- Mirror Symmetry occurs when a dice set's alphabetic string is palindromic (reads the same forward as it does backward). This type of symmetry is often called “Palindromic”, for obvious reasons. Here is an example:
```
ccbbbaaaaaabbbccccccccbbbaaaaaabbbcc
```
  Physical dice within a set that has mirror symmetry have the property that all pairs of opposite faces (of all of the dice) add up to the same number.

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