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.
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.
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 |
Binary dice.
Not yet found: D2 (coin), D4 2x(0-1); D10 5x(0-1), etc
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 Bear Cub Machine |
Tetrahedron 2 x 1-2 K=2 Formula Dé |
Hexahedron 3 x 1-2 K=3 |
Not yet
found: D8, 4x(1-2); etc.
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 |
Dice with
three different numbers.
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 |
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? |
Dice with three numbers other than 1..3 are used as cheaters or in
special games.
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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 |
6,6,3,3,2,2 2x(6,3,2) K=2,
non-linear Loaded die Koplow |
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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 |
5,5,4,4,1,1 2x(5,4,1) K=2,
non-linear Loaded die Koplow |
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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 |
4,4,4,3,3,2 Non-uniform, Formula Dé Jeux Descartes Also used in Dragonland game? |
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4,3,1 K=1,
non-linear HABA Schleckermaul |
The tetrahedron is the most common die in this family.
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 |
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 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) |
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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Tetrahedron |
Isosceles Tetrahedron |
Modified
square
prism Crystal Caste (left) Bear
Cub Machine (right) |
Octahedron D8, 2 x
1-4 K=2 |
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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 |
Spherical
D4 This
die is not numbered,
but it could
be used to generate
numbers 1..4 |
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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 |
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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8,8,7,7,6,6,5,5 2x(8…5) K=2, a=1, b=3 Formula Dé |
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 |
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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 |
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 |
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 |
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 |
There are some hexahedra with one
number printed twice, yielding an unfair die.
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20,15,10,5,5,0 Non-uniform P(5)=1/3, 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 |
The most common shape of this family is the cube (hexahedron).
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Hexahedron numbered |
Hexahedron pipped |
Not yet
found: D12, 2 x 0-5
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 |
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Holes |
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 |
954,855,756,657,558,459 Trick die |
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840,741,642,543,345,147 Trick die |
780,681,483,384,285,186 Trick die |
300,280,260,240,220,200 K=1 a=20, b=180 |
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260,240,220,200,180,160 K=1 a=20, b=140 |
250,225,200,175,150,125 K=1 a=25, b=100 |
200,180,160,140,120,100 K=1 a=20, b=80 |
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64,32,16,8,4,2 K=1, 2^X Backgammon |
60,50,40,30,20,10 K=1, a=10, b=0 |
35,32,25,18,11,4 Lotto die? |
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35,28,21,17,14,7 Lotto die? |
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? |
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 |