Dice as Random Number Generators
The main purpose of pipped and numbered dice is the generation of random numbers. In this section we describe various such number generators based mainly on isohedral shapes. “Regular” N-sided dice typically generate random numbers in the range 1..N or 0..N-1, but other ranges are possible, as shown in the sequel.
Let M denote the number of different numbers on an N-sided die. In most cases, N=M, however there are dice where the same number appears more than once, in which case M<N.
In this section we start with a short math section, mainly for mathematically interested alealogists. We then show dice with M=1 up to M=100. For each value of M, we list dice according to the characteristics of the random numbers they generate:
uniform distributions: rolling a die produces
numbers which are equally likely
0..M-1: the most common member of this
family is the D10 with numbers from 0 to 9
1..M: the most common member of this
family is the “normal” cube, a hexahedron with numbers 1..6
Consecutive numbers other than 0..M-1 or
1..M: These dice generate random numbers with an “offset”, e.g. a D6
numbered 7,8,9,10,11,12.
Non-consecutive numbers: The numbers of such a
die are not consecutive (i.e., there are some numbers “missing” in the
sequence), but they still are rolled equally likely. The most prominent member
of this family is the backgammon doubling D6 with numbers 2,4,8,16,32,64.
non-uniform distributions: some numbers are more
likely than others. These dice are specially designed to produce these numbers,
e.g. as cheaters or for special games.
Towards the end some special dice sets, such as Sicherman dice, Non-transitive dice, cheater dice or trick dice are shown and explained.
Math
The main purpose of dice, apart from being collected, is to generate random numbers, or, in mathematical terms, a discrete random variable. The statistical properties of a die can be described by its probability density function. The most common die, a cube (D6) numbered 1,2,3,4,5,6 generates a random variable with discrete values 1,2,3,4,5, and 6. If the die is fair, each of these numbers is rolled with equal probability of 1/6.
Let us assume that a die has N faces with M different numbers. If all the numbers are different, then N=M. If all numbers appear K times, then N=K·M. Such dice are “fair” in the sense that the probability density function is uniform. These dice allow to make equal-probability random selections between a number of choices.
On most dice each number is printed only once, in which case K=1, as for the “regular” cube mentioned above. Other examples include the “binary cube”, a D6 numbered 0,0,0,1,1,1, hence K=3 because each number appears three times. The die with the largest K in my collection is the D-Total by A. Simkin / L. Zocchi, a deltoidal icositetrahedron with N=24, showing, among others, 8 x 1..3 pips.
For many common fair dice, its M different numbers can be derived from the numbers 1..M with a linear function aX+b. For a=1, the resulting random variable assumes values of consecutive integers with “offset” b.
Examples:
a=1, b=0:
This generates the sequence 1,2,3,..,M. The most common die in this family is
the regular cube, numbered 1..6
a=1, b=-1:
This generates the sequence 0,1,2,..,M-1. The most common die in this family
is the D10, numbered 0..9
M=6, a=1, b=6:
This is used by an educational die with numbers 7,8,9,10,11,12.
M=6, a=10,
b=0: This yields die with numbers 10,20,30,40,50,60.
N=6, M=2, K=3,
a=4, b=-2: This yields a loaded die numbered 2,2,2,6,6,6
M=6, a=1/6,
b=0: This yields a fractional die with numbers 1/6, 1/3, 1/2, 2/3, 5/6,
1
Another fairly common die is the backgammon doubling. Its probability distribution function is uniform (i.e., it is fair), and its numbers can be generated from the sequence 1..M with the function 2^(X-b), power of 2.
Examples:
N=6, M=6, K=1, b=0: D6 backgammon doubling die numbered 2,4,8,16,32,64.
N=8, M=8, K=1, b=-1: D8 backgammon doubling die numbered 1,2,4,8,16,32,64,128.
There are fractional dice that also use a power law, X^(-1)
N=6, M=6, K=1: D6 fractional die numbered 1,1/2,1/3,1/4,1/5,1/6.
There are some dice where some numbers are printed more often than others. Even if the original die is a perfect isohedron (i.e., a fair shape), the resulting random variable is not fair. In mathematical terms, its probability density function is non-uniform. Examples are cheaters, e.g. a D6 numbered 6,6,5,4,3,2. The “6” appears twice, but there is no “1”. Other dice with non uniform distributions include non-transitive dice, Sicherman dice and dice specially designed for particular games. Some of them are described in more detail at the end of this section.
M=1
These are dice that allow rolling a single number only, i.e. a degenerate case of a random number generator. They are mainly used as cheaters, one is a non-transitive die.
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6,6,6,6,6,6 K=6, a=6, b=0 Loaded die |
5,5,5,5,5,5 K=6, a=5, b=0 Loaded die |
4,4,4,4,4,4 K=6, a=4, b=0 Loaded die |
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3,3,3,3,3,3 K=6, a=3, b=0 Loaded die |
3,3,3,3,3,3 K=6, a=3, b=0 Non-transitive die Grand Illusions |
0,0 Lord of the Rings Free Peoples Games Workshop |
M=2
Binary dice.
Range 0..1
Not yet found: D2 (coin), D4 2x(0-1); D10 5x(0-1), etc
Range 1..2
Fair binary dice, generating numbers 1 and 2. All shapes with an even number of faces are possible. Shapes in my collection: coin, tetrahedron, and hexahedron.
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“Coin” 1-2, pipped Bear Cub Machine |
“Coin” 1-2, numbered fascom / Shapeways |
Tetrahedron 2 x 1-2 K=2 Formula Dé |
Hexahedron 3 x 1-2 K=3 |
Not yet found: D8, 4x(1-2); etc.
Others
Dice with only two numbers, cheaters and non-transitive dice and special dice designed for games.
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150,150,100,100,100,100 Non-uniform P(150)=1/3, P(100)=2/3 Monopoly Dice Game |
6,6,6,2,2,2 3x(6,2) K=3, a=4, b=-2 Loaded die |
6,6,2,2,2,2 Non-uniform P(6)=1/3, P(2)=2/3 Non-transitive die Grand Illusions |
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6,3,3,3,3,3 Non-uniform P(6)=1/6, P(3)=5/6 Non-transitive die Grand Illusions |
5,5,5,2,2,2 3x(5,2) K=3, a=3, b=-1 Non-transitive die Grand Illusions |
5,5,5,1,1,1 3x(5,1) K=3, a=4, b=-3 Non-transitive die Grand Illusions |
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4,4,4,4,4,1 Non-uniform P(4)=5/6, P(1)=1/6 Non-transitive die Grand Illusions |
4,4,4,4,0,0 Non-uniform P(4)=2/3, P(0)=1/3 Non-transitive die Grand Illusions |
M=3
Dice with three different numbers.
Range 0..2
There is no fair die in my collection that generates the numbers 0..2. However, there are some dice with non-uniform distributions.
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2,2,1,1,1,1,0,0 Non-uniform P(2)=1/4, P(1)=1/2, P(0)=1/4 2D2 (sum of 2 binary dice) Exile Game Studio (Ubiquity Dice) |
2,1,1,0,0,0 Non-uniform P(2)=1/6, P(1)=1/3, P(0)=1/2 |
Range 1..3
Fair ternary dice, generating numbers 1..3. All shapes with a number of faces which is divisible by 3 are possible. Shapes in my collection: triangular prism, hexahedron, and deltoidal icositetrahedron.
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Elongated triangular prism HABA |
Elongated triangular prism Abraham Neddermann |
Rounded-off triangular prism GameScience |
Rounded-off triangular prism Crystal Caste |
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Hexahedron 2 x 1-3 K=2 |
Hexahedron 2 x 1-3 K=2 |
Deltoidal Icositetrahedron D24, 8 x 1-3 1..3 pips K=8 D-Total by A. Simkin / GameScience |
Not yet found: D12, 4x(1-3)
This is an unfair die:
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3,3,2,2,2,1 Non-uniform, Dragonland Game? |
Others
Dice with three numbers other than 1..3 are used as cheaters or in special games.
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30,30,20,20,10,10 2x(30,20,10) K=2, a=10, b=0 |
8,8,6,6,4,4 K=2, a=2, b=2 |
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6,6,5,5,1,1 2x(6,5,1) K=2, non-linear Loaded die Koplow |
6,6,4,4,2,2 2x(6,4,2) K=2, a=2, b=0 Loaded die Koplow |
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6,6,3,3,2,2 2x(6,3,2) K=2, non-linear Loaded die Koplow |
5,5,4,4,3,3 2x(5,4,3) K=2, a=1, b=2 Consecutive numbers Loaded die Koplow |
5,4,2 K=1, non-linear HABA Schleckermaul |
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5,5,4,4,1,1 2x(5,4,1) K=2, non-linear Loaded die Koplow |
5,5,3,3,1,1 2x(5,3,1) K=2 a=2, b=-1 Loaded die Koplow |
4,3,2 K=1, a=1, b=1 Consecutive numbers HABA Schleckermaul |
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4,4,4,3,3,2 Non-uniform, Formula Dé Jeux Descartes Also used in Dragonland game? |
4,3,1 K=1, non-linear HABA Schleckermaul |
M=4
The tetrahedron is the most common die in this family.
Range 0..3
There is no fair die in my collection that generates the numbers 0..3. However, there are some dice with non-uniform distributions.
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3,3,2,1,1,0 Non-uniform P(3)=1/3, P(2)=1/6 P(1)=1/3, P(0)=1/6
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3,2,2,1,1,0 Non-uniform P(3)=1/6, P(2)=1/3 P(1)=1/3, P(0)=1/6 |
3,2,1,1,1,0 Non-uniform P(3)=1/6, P(2)=1/6 P(1)=1/2, P(0)=1/6 Babylon 5 Defense Grid Die |
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3,2,2,2,1,1,1,0 Non-uniform P(3)=1/8, P(2)=3/8 P(1)=3/8, P(0)=1/8 3D2 (sum of 3 binary dice) Exile Game Studio (Ubiquity Dice) |
Range 1..4
Fair quaternary dice, generating numbers 1..4. All shapes with a number of faces which is divisible by 4 are possible. Shapes in my collection: tetrahedron, square prism, octahedron, dodecahedron, and deltoidal icositetrahedron.
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Regular Tetrahedron |
Isosceles Tetrahedron |
Scalene Tetrahedron friz / Shapeways |
Modified square prism Crystal Caste (left) Bear Cub Machine (right) |
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Octahedron D8, 2 x 1-4 K=2 |
Dodecahedron D12, 3 x 1-4 K=3 Koplow |
Deltoidal Icositetrahedron D24, 6 x 1-4 K=6 Numbers in triangle D-Total by A. Simkin / GameScience |
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Spherical D4 This die is not numbered, but it could be used to generate numbers 1..4 |
Spinner D4 |
Not yet found: D20, 5x(1-4)
Unfair dice generating 1..4 (Sicherman):
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4,3,3,2,2,1 Non-uniform P(4)=1/6, P(3)=1/3 P(2)=1/3, P(1)=1/6 Sicherman die GameStation |
4,3,3,2,2,1 Non-uniform P(4)=1/6, P(3)=1/3 P(2)=1/3, P(1)=1/6 Sicherman die Grand Illusions |
Others
Dice with four numbers other than 1..4 are used as cheaters or in special games.
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500,200,200,100,50,50 Non-uniform, P(100)=1/6, P(50)=1/3 Monopoly Dice Game |
400,300,250,200,200,200 Non-uniform, P(250)=1/6, P(200)=1/2 Monopoly Dice Game |
9,9,8,7,6,6 Non-uniform P(7)=1/6, P(6)=1/3
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8,7,7,6,6,5 Non-uniform P(8)=1/6, P(7)=1/3 P(6)=1/3, P(5)=1/6 |
6,5,4,4,3,3 Non-uniform P(6)=1/6, P(5)=1/6 P(4)=1/3, P(3)=1/3 |
6,5,4,3 K=1, a=1, b=3 |
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5,5,4,4,3,2 Non-uniform P(5)=1/3, P(4)=1/3 P(3)=1/6, P(2)=1/6 |
5,4,4,3,3,2 Non-uniform Averaging die This die has the same average as a “regular” D6, namely 3.5 |
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5,4,3,3,2,2 Non-uniform P(5)=1/6, P(4)=1/6 P(3)=1/3, P(2)=1/3 |
M=5
Range 0..4
Fair dice generating numbers 0..4. All shapes with a number of faces which is divisible by 5 are possible. Shapes in my collection: pentagonal trapezohedron.
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Pentagonal Trapezohedron D10, 2 x 0-4 K=2 a=1, b=-1 |
Range 1..5
Fair dice generating numbers 1..5. All shapes with a number of faces which is divisible by 5 are possible. Shapes in my collection: pentagonal prism, pentagonal trapezohedron.
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Elongated pentagonal prism Abraham Neddermann |
Pentagonal Trapezohedron D10, 2 x 1-5 K=2 a=1, b=0 |
Not yet found: D20, 4x1-5
This die is not fair:
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Triangular Prism. Not fair. GameScience |
Others
There are some D10 with all numbers printed twice. These dice are fair.
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50,50,40,40,30,30,20,20,10,10 K=2, a=10, b=0 Sumator GameScience |
40,40,30,30,20,20,10,10,00,00 K=2, a=10, b=-10 GameScience |
This is an octahedron with 5 different numbers:
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8,8,7,7,6,6,5,4 Non-uniform P(8)=1/3, P(7)=1/3 P(6)=1/3, P(5)=1/6, P(4)=1/6 Formula Dé |
There are some hexahedra with one number printed twice, yielding an unfair die.
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40,32,24,16,8,8 Non-uniform P(16)=1/6, P(8)=1/3 |
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20,15,10,5,5,0 Non-uniform P(5)=1/3, P(0)=1/6 |
12,11,10,10,9,2 Non-uniform P(9)=1/6, P(0)=1/6 |
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6,6,5,4,3,2 Non-uniform P(3)=1/6, P(2)=1/6 Cheater, 1 replaced by 6 |
6,5,4,2,0,0 (blank,blank) Non-uniform P(6)=1/3, P(5)=1/6, P(4)=1/6, P(3)=1/6, P(2)=1/6 |
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5,4,3,2,1,1 Non-uniform P(5)=1/6, P(4)=1/6, P(3)=1/6, P(2)=1/6, P(1)=1/3 Cheater, 6 replaced by 1 |
5,3,2,1,1,0 Non-uniform P(5)=1/6, P(3)=1/6, P(2)=1/6, P(1)=1/3, P(0)=1/6 Babylon 5 |
M=6
The most common shape of this family is the cube (hexahedron).
Range 0..5
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Hexahedron numbered |
Hexahedron pipped |
Not yet found: D12, 2 x 0-5
Range 1..6
Fair dice generating numbers 1..6. All shapes with a number of faces which is divisible by 6 are possible. Shapes in my collection: hexahedron, 6-sided antiprism, 6-sided prism, dodecahedron, deltoidal icositetrahedron
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Hexahedron pipped |
Hexahedron numbered |
Modified 6-sided Antiprism Crystal Caste (left) Hasbro / Monopoly (right) |
Football 6-sided prism Hasbro (Monopoly) |
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Dodecahedron D12, 2x1-6 K=2 Hasbro (Monopoly) |
Deltoidal Icositetrahedron D24, 4 x 1-6 K=4 Numbers in square D-Total by A. Simkin / GameScience |
Spinner D6 |
Other shapes:
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Sphere |
Flattened Sphere |
Concave
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Crooked Not fair |
Jumping Not fair |
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Elliptical Not fair |
Tactile |
Crystal pips* |
Holes |
Others
There is a large number of D6 with numbers other than 1..6. These include trick dice, backgammon dice, educational dice, mathematical dice (non-transitive and Sicherman) and special dice designed for games. Most of these dice are hexahedra, and there are also a few dodecahedra.
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971,872,773,377,278,179 Trick die |
960,762,663,564,366,168 Trick die |
960,663,564,366,267,168 Trick die |
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954,855,756,657,558,459 Trick die |
913,814,616,517,418,319 Trick die |
902,803,704,605,506,209 Trick die |
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840,741,642,543,345,147 Trick die |
780,681,483,384,285,186 Trick die |
690,591,492,393,294,195 Trick die |
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300,280,260,240,220,200 K=1 a=20, b=180 |
260,240,220,200,180,160 K=1 a=20, b=140 |
250,225,200,175,150,125 K=1 a=25, b=100 |
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200,180,160,140,120,100 K=1 a=20, b=80 |
81,72,54,36,18,9 |
64,32,16,8,4,2 K=1, 2^X Backgammon |
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60,50,40,30,20,10 K=1, a=10, b=0 |
40,30,25,20,15,10 |
36,30,24,18,12,6 K=1, a=6, b=0 MEGADICE |
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35,32,25,18,11,4 Lotto die? |
35,29,23,17,11,5 K=1, a=6, b=-1 MEGADICE |
35,28,21,17,14,7 Lotto die? |
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34,28,22,16,10,4 K=1, a=6, b=-2 MEGADICE |
34,27,26,20,13,6 Lotto die? |
33,31,24,17,10,3 Lotto die? |
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33,28,26,19,12,5 Lotto die? |
33,27,21,15,9,3 K=1, a=6, b=-3 MEGADICE |
32,26,20,14,8,2 K=1, a=6, b=-4 MEGADICE |
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31,25,19,13,7,1 K=1, a=6, b=-5 MEGADICE |
30,29,28,27,26,25 K=1, a=1, b=24 |
30,23,18,16,9,2 Lotto die? |
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29,22,15,8,5,1 Lotto die? |
24,23,22,21,20,19 K=1, a=1, b=18 |
24,19,16,15,14,11 D6, Edged Cigar Shape Football Fever |
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18,17,16,15,14,13 K=1, a=1, b=12 |
16,15,14,13,12,11 K=1, a=1, b=10 Nin Gonost |
15,14,13,12,11,10 K=1, a=1, b=9 Nin Gonost |
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14,8,5,4,2,1 |
13,12,11,10,9,8 K=1, a=1, b=7 Nin Gonost |
13,12,9,8,4,3 Rummy Die |
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12,11,10,9,8,7 K=1, a=1, b=6 |
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12,12,11,11,10,10, 9,9,8,8,7,7 2x(12..7) K=2, a=1, b=6 Formula Dé |
12,11,8,7,3,2 Rummy Die |
12,11,7,6,3,2 Rummy Die |
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12,9,8,7,3,1 |
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11,10,9,8,7,6 K=1, a=1, b=5 Nin Gonost |
11,10,7,6,2,1 Rummy Die |
11,10,6,5,2,1 Rummy Die |
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11,8,7,5,3,0 |
10,9,8,7,6,5 K=1, a=1, b=4 |
10,9,8,7,6,0 |
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10,9,7,6,5,3 |
10,8,8,8,7,7,6,6,5,5,5,3 D12, non-uniform P(10)=1/12, P(8)=1/4, P(7)=1/6, P(6)=1/6, P(5)=1/4, P(3)=1/12 Golo Golf Par 5 Die |
9,8,7,6,5,4 K=1, a=1, b=3 |
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9,8,8,8,7,7,6,6,5,5,5,4 D12, non-uniform P(9)=1/12, P(8)=1/4, P(7)=1/6, P(6)=1/6, P(5)=1/4, P(4)=1/12 Golo Golf Par 5 Die |
9,8,7,3,2,1 9.UR8L3U2UL7L1U |
9,8,7,2,1,0 Everlasting calendar, matches with 5,4,3,2,1,0. 9 can also be read as 6. Also in Tumble Numble and Foggle |
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9,8,5,4,3,2 |
9,8,5,4,3,1 Non-transitive die Miwin |
9,7,6,5,2,1 Non-transitive die Miwin |
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9,4,3,2,1,0 |
8,7,6,5,4,3 K=1, a=1, b=2 |
8,7,7,7,6,6,5,5,4,4,4,3 D12, non-uniform P(8)=1/12, P(7)=1/4, P(6)=1/6, P(5)=1/6, P(4)=1/4, P(3)=1/12 Golo Golf Par 4 Die |
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8,7,7,6,6,6,5,5,4,3 D10, non-uniform P(8)=1/10, P(7)=1/5, P(6)=3/10, P(5)=1/5, P(4)=1/10, P(3)=1/10 |
8,7,6,4,3,2 Non-transitive die Miwin |
8,7,6,2,1,0 Everlasting calendar, matches with 5,4,3,2,1,0. 6 can also be read as 9. |
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8,7,4,3,2,1 8UR4L1L7LR2R3R |
8,7,4,2,1,0 8UL7R2U4LL1R0R |
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8,6,5,4,3,1 Sicherman die Grand Illusions |
8,6,5,4,3,1 Sicherman die GameStation |
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8,6,6,6,5,5,4,4,3,3,3,1 Same numbers as Sicherman die D12, non-uniform P(8)=1/12, P(6)=1/4, P(5)=1/6, P(4)=1/6, P(3)=1/4, P(1)=1/12 1 in a star, 3 in a square Golo Golf Par 3 Die |
7,6,5,4,3,2 K=1, a=1, b=1 |
7,6,6,6,5,5,4,4,3,3,3,2 D12, non-uniform P(7)=1/12, P(6)=1/4, P(5)=1/6, P(4)=1/6, P(3)=1/4, P(2)=1/12 2 in a circle, 3 in a square Golo Golf Par 3 Die |
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7,5,4,3,2,1 |
6,5,4,3,2,0 (blank) |
5,5,4,4,3,2,1,0 D8, non-uniform P(5)=1/4, P(4)=1/4, P(3)=1/8, P(2)=1/8, P(1)=1/8, P(0)=1/8 |
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5,4,3,2,1,0 K=1, a=1, b=-1 |
+5,+4,+3,+2-1,-2 |
6,4,2,-1,-3,-5 Odd Negative |
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5,3,1,-2,-4,-6 Even Negative |
3,2,1,0,-1,-2 K=1, a=1, b=-3 |
-6.,-5,-4,-3,-2,-1 K=1, a=1, b=-7 (or a=-1, b=0) |
M=7
There are only very few dice with 7 different numbers.
Range 0..6
There is no fair die numbered 0..6 in my collection. However, there is a special die generating numbers 0..6 as sum of two quaternary dice. It consists of two halves, each with 0,1,2, or 3 dots. When the die is rolled, these halves can move (more or less) independently.
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0..6 Not fair P(0)=1/16, P(1)=1/8, P(2)=3/8, P(3)=1/2, P(4)=3/8, P(5)=1/8, P(6)=1/8 Y2K Die |
Range 1..7
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Elongated heptagonal prism Abraham Neddermann |
Pentagonal Prism. Not fair |
Not yet found: D14, 2x(1-7)
Others
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11,10,9,9,8,8,7,7,6,5 Non uniform P(11)=1/10, P(10)=1/10, P(9)=1/5, P(8)=1/5, P(7)=1/5, P(6)=1/10, P(5)=1/10 |
M=8
There are many D8 on the market, mostly octahedral numbered 1..8. Besides that there are very few other dice.
Range 1..8
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Octahedron |
Modified 8-sided Antiprism Crystal Caste |
Deltoidal Icositetrahedron D24, 3 x 1-8 K=3 Numbers in diamond D-Total by A. Simkin / GameScience |
Wanted: D8 pipped
Not yet found: D16, 2x(1-8)
Others
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128,64,32,16,8,4,2,1 D8, non-uniform 2^(X-1) Doubling die |
9,8,7,6,5,4,3,2 K=1, a=1, b=1 |
12,11,10,9,9,8,8,8,7,7,6,5 D12, non-uniform P(12)=1/12, P(11)=1/12, P(10)=1/12, P(9)=1/6, P(8)=1/4, P(7)=1/6, P(6)=1/12, P(5)=1/12 |
M=9
There are only a few dice with nine different numbers in my collection. There is also a heptagonal prism (missing in my collection).
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45,44,35,34,30, 29,25,25,21,20 D10, non-uniform P(45)=1/10, P(44)=1/10, P(35)=1/10, P(34)=1/10, P(30)=1/10, P(29)=1/10, P(25)=1/5, P(21)=1/10, P(20)=1/10 25 appears twice Bomb Die Football Fever |
16,15,14,13,13,12,12,11,11,10,9,8 D12, non-uniform P(16)=1/12, P(15)=1/12, P(14)=1/12, P(13)=1/6, P(12)=1/6, P(11)=1/6, P(10)=1/12, P(9)=1/12, P(9)=1/12 |
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9,8,7,6,5,4,3,2,1 K=1, a=1, b=0 Elongated nonagonal prism Abraham Neddermann |
9,8,7,6,5,4,3,2,0,0 D10, Non-uniform P(9)=P(8)=…=P(2)=1/10 P(0)=1/5 Cheater, 1 replaced by 0 |
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Missing in my collection |
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Heptagonal prism Not fair |
M=10
Range 0..9
In addition to the “classical” pentagonal trapezohedron, there are also a decagonal
edged cigar, a 10-sided antiprism, an icosahedron and a rhombic triacontahedron with numbers 0..9. There are also two types of flattened octahedra, which are not fair.
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Pentagonal Trapezohedron numbered |
Pentagonal Trapezohedron pipped |
Decagonal Edged Cigar Phi Sports |
Modified 10-sided Antiprism Crystal Caste |
Icosahedron D20, 2 x 0-9 K=2 |
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Rhombic Triacontahedron D30, 3 x 0-9 +9..+0,9..0,-9..-0 K=3 |
Truncated Octahedron Not fair. |
Range 1..10
Dice numbered 1..10 are less popular than those numbered 0..9 because with two of the latter dice numbers 1..100 can be generated (00=100).
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Pentagonal Trapezohedron |
Modified 10-sided Prism Bear Cub Machine |
10 sided spinner |
Not yet found: D20, 2x(1-10)
Others
There are surprisingly many dice with 10 numbers other than 0..9 or 1..10. Most of them are so called place value dice, but there are also 10 sided antiprisms, icosahedra and rhombic triacontahedra.
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900000,…,100000,000000 K=1, a=100000, b=-100000 Koplow |
90000,…,10000,00000 K=1, a=10000, b=-10000 Koplow |
9000,…,1000,0000 K=1, a=1000, b=-1000 Koplow |
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900,…,100,000 K=1, a=100, b=-100 Koplow |
90,80,…,10,00 K=1, a=10, b=-10 |
90,80,…,10,00 K=1, a=10, b=-10 Crystal Caste |
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90,80,…,10,00 K=1, a=10, b=-10 GRIDIRON MASTER / PHI SPORTS |
30,30,30,29,29,29,…, 21,21,21 D30 3x(30…21) K=3, a=1, b=20 Formula Dé |
20,20,19,19,...,11,11 2x(20…11) K=2, a=1, b=10 Formula Dé, Truant |
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19,18,18,17,17,16,16,15,15,15,14,14,14,13,13,13,12,12,11,10 D20, non-uniform P(19)=1/20, P(18)=1/10, P(17)=1/10, P(16)=1/10, P(15)=3/20, P(14)=3/20, P(13)=3/20, P(12)=1/10, P(11)=1/20, P(10)=1/20 |
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15,14,14,13,13,12,12,11,11,11,10,10,10,9,9,9,8,8,7,6 D20, non-uniform P(15)=1/20, P(14)=1/10, P(13)=1/10, P(12)=1/10, P(11)=3/20, P(10)=3/20, P(9)=3/20, P(8)=1/10, P(7)=1/20, P(6)=1/20 |
M=11
Dice with 11 different numbers are very rare. There do not even seem to be cheaters (e.g. a D12 with the 12 replaced by a 1 or vice versa).
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25,24,23,22,21,21,21,20,20,20,20,19,19,19,19,18,18,17,16,15 D20, non-uniform P(25)=1/20, P(24)=1/20, P(23)=1/20, P(22)=1/20, P(21)=3/20, P(20)=1/5, P(19)=1/5, P(18)=1/10, P(17)=1/20, P(16)=1/20, P(15)=1/20 |
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11,10,9,8,7,6,5,4,3,2,1 K=1, a=1, b=0 Elongated hendecagonal prism Abraham Neddermann |
M=12
Range 1..12
There are many different shapes for dice numbered 1..12: pentagonal dodecahedron, rhombic dodecahedron, 12-sided antiprism, and deltoidal icositetrahedron (with each number printed twice).
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Pentagonal Dodecahedron |
Rhombic Dodecahedron AskAstro |
Modified 12-sided Antiprism Crystal Caste |
Deltoidal Icositetrahedron D24, 2 x 1-12 K=2 Numbers in pentagon D-Total by A. Simkin / GameScience |
Others
This die can be used to roll minutes from 0 to 55 in steps of 5:
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55,50,…5,0 K=1, a=5, b=-5 |
M=13
Dice with 13 different numbers are very rare. This one is numbered 1..13.
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Elongated tridecagonal prism Abraham Neddermann |
M=14
This heptagonal trapezohedron is numbered 1..14.
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Heptagonal Trapezohedron |
There is also a 14 sided die based on the cuboctahedron. However, this is not a numbered die, but one with poker symbols (Card Dice)
M=15
Dice with 15 different numbers are very rare. This one is numbered 1..15.
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Elongated pentadecagonal prism Abraham Neddermann |
M=16
The octagonal bipyramid with 16 faces exists both with “regular” and with hexadecimal numbers (0..F).
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Octagonal Bipyramid |
Octagonal Bipyramid Hexi Die |
M=19
There are two icosahedral cheater dice:
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20,20,19,18,…3,2 Non-uniform P(20)=1/10, P(19)=…=P(2)=1/20 average 239/20=11.95 Cheater, 1 replaced by 20 Chessex/Koplow/Truant |
19,18,…3,2,1,1 Non-uniform P(1)=1/10, P(19)=…=P(2)=1/20 average 201/20=10.05 Cheater, 1 replaced by 20 Truant |
M=20
The most popular die numbered 1..20 is the icosahedron, but there is also a 20-sided antiprism.
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Icosahedron |
Modified 20-sided Antiprism Crystal Caste |
There is only one die in my collection with 20 different numbers other than 1..20:
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57,51,50,46,45,43,38,35,32,29,24,20,19,16,13,11,07,04,02,00 |
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49,47,46,44,39,37,36,32,28,27,24,19,18,16,15,14,13,9,5,1 Pic-6 Lotto Dice |
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49,46,43,40,37,34,31,28,25,22,21,19,16,13,10,7,4,3,2,1 Pic-6 Lotto Dice |
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48,47,46,43,42,41,39,38,35,34,33,31,30,29,26,20,17,8,4,2 Pic-6 Lotto Dice |
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48,44,42,38,37,36,34,33,32,31,29,28,27,26,24,23,22,12,6,3 Pic-6 Lotto Dice |
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47,45,44,41,38,35,32,29,26,23,22,21,20,17,14,11,8,7,6,5 Pic-6 Lotto Dice |
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45,43,42,41,40,39,36,33,30,27,25,24,23,21,18,15,12,11,10,9 Pic-6 Lotto Dice |
M=24
There are two shapes of dice with 24 different numbers, tetrakishexahedron and deltoidal icositetrahedron.
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Tetrakishexahedron |
Deltoidal Icositetrahedron Franck Dutrain |
Deltoidal Icositetrahedron Numbers in center of face D-Total by A. Simkin / GameScience |
M=30
Rhombic triacontahedron, numbered 1..30:
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Rhombic Triacontahedron |
M=34
Dekaeptagonal Trapezohedron, numbered 1..34:
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Dekaeptagonal Chessex |
M=50
Two different shapes for dice numbered 0..49, the eikosipendegonal
trapezohedron and a flattened sphere:
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Eikosipendegonal Trapezohedron GameScience Big 50 Topper |
Flattened Sphere Not fair Alan Davies
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M=100
The Zocchihedron, a spherical die numbered 1..100:
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Zocchihedron Not fair GameScience |
Fractions
The numbers on the dice presented so far were all integers. There are a few dice with fractions, mainly D6 but also some D8 and D10:
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1,5/6,2/3,1/2,1/3,1/6 K=1, a=1/6, b=0 |
1/1,1/2,1/3,1/4,1/5,1/6 X^(-1) |
11/12,7/8,5/6,3/4,2/3,1/2 |
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3/4,2/3,2/4,1/2,1/3,1/4 |
1/2,1/2,1/3,1/3,1/4,1/6 Opposite faces are identical |
1/2,1/3,1/4,1/4,1/6,1/6 Non-uniform |
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1/2,1/3,1/4,1/5,1/6,1/8 |
1/2,1/3,1/4,1/6,1/8,1/12 |
1/2,1/4,1/4,1/8,1/8,1/8 Non-uniform |
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1/3,1/6,1/6,1/12,1/12,1/12 Non-uniform |
1/6,1/6,1/8,1/8,1/9,1/9 |
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1,.75,.67,.50.,33,.25 |
1.00,0.75,0.67,0.50,0.33, 0.25 |
1.00,0.50,0.25,0.10,0.05, 0.01 |
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1,7/8,3/4,5/8,1/2,3/8,1/4, 1/8 K=1, a=1/8, b=0 Fractional D8 |
10/10,9/10,…,1/10 K=1, a=1/10, b=0 Fractional D10 |
0.9,0.8,…,0.1,0.0 K=1, a=0.1, b=-0.1 Koplow |
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0.09,0.08,…,0.01,0.00 K=1, a=0.01, b=-0.01 Koplow |
0.009,0.008,…,0.001,0.000 K=1, a=0.001, b=-0.001 Koplow |
