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” Nsided dice typically generate random numbers in the range 1..N or 0..N1, but other ranges are possible, as shown in the sequel.
Let M denote the number of different numbers on an Nsided 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..M1: 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..M1 or 1..M: These dice generate random numbers with an “offset”, e.g. a D6 numbered 7,8,9,10,11,12.
Nonconsecutive 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.
nonuniform 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, Nontransitive 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 equalprobability 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 DTotal 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,..,M1. 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^(Xb), 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 nonuniform. 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 nontransitive 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 nontransitive die.



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 



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 Nontransitive die Grand Illusions 
0,0 Lord of the Rings Free Peoples Games Workshop 
Binary dice.
Not yet found: D2 (coin), D4 2x(01); D10 5x(01), 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.




“Coin” 12, pipped Bear Cub Machine 
“Coin” 12, numbered fascom / Shapeways 
Tetrahedron 2 x 12 K=2 Formula Dé 
Hexahedron 3 x 12 K=3 
Not yet found: D8, 4x(12); etc.
Dice with only two numbers, cheaters and nontransitive dice and special dice designed for games.



150,150,100,100,100,100 Nonuniform 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 Nonuniform P(6)=1/3, P(2)=2/3 Nontransitive die Grand Illusions 



6,3,3,3,3,3 Nonuniform P(6)=1/6, P(3)=5/6 Nontransitive die Grand Illusions 
5,5,5,2,2,2 3x(5,2) K=3, a=3, b=1 Nontransitive die Grand Illusions 
5,5,5,1,1,1 3x(5,1) K=3, a=4, b=3 Nontransitive die Grand Illusions 


4,4,4,4,4,1 Nonuniform P(4)=5/6, P(1)=1/6 Nontransitive die Grand Illusions 
4,4,4,4,0,0 Nonuniform P(4)=2/3, P(0)=1/3 Nontransitive 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 nonuniform distributions.


2,2,1,1,1,1,0,0 Nonuniform 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 Nonuniform 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.




Elongated triangular prism HABA 
Elongated triangular prism Abraham Neddermann 
Roundedoff triangular prism GameScience 
Roundedoff triangular prism Crystal Caste 



Hexahedron 2 x 13 K=2 
Hexahedron 2 x 13 K=2 
Deltoidal Icositetrahedron D24, 8 x 13 1..3 pips K=8 DTotal by A. Simkin / GameScience 
Not yet found: D12, 4x(13)
This is an unfair die:

3,3,2,2,2,1 Nonuniform, Dragonland Game? 
Dice with three numbers other than 1..3 are used as cheaters or in special games.


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 


6,6,5,5,1,1 2x(6,5,1) K=2, nonlinear 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, nonlinear 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, nonlinear HABA Schleckermaul 



5,5,4,4,1,1 2x(5,4,1) K=2, nonlinear 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 


4,4,4,3,3,2 Nonuniform, Formula Dé Jeux Descartes Also used in Dragonland game? 
4,3,1 K=1, nonlinear 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 nonuniform distributions.



3,3,2,1,1,0 Nonuniform P(3)=1/3, P(2)=1/6 P(1)=1/3, P(0)=1/6

3,2,2,1,1,0 Nonuniform P(3)=1/6, P(2)=1/3 P(1)=1/3, P(0)=1/6 
3,2,1,1,1,0 Nonuniform P(3)=1/6, P(2)=1/6 P(1)=1/2, P(0)=1/6 Babylon 5 Defense Grid Die 

3,2,2,2,1,1,1,0 Nonuniform 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.




Regular Tetrahedron 
Isosceles Tetrahedron 
Scalene Tetrahedron friz / Shapeways 
Modified square prism Crystal Caste (left) Bear Cub Machine (right) 



Octahedron D8, 2 x 14 K=2 
Dodecahedron D12, 3 x 14 K=3 Koplow 
Deltoidal Icositetrahedron D24, 6 x 14 K=6 Numbers in triangle DTotal by A. Simkin / GameScience 


Spherical D4 This die is not numbered, but it could be used to generate numbers 1..4 
Spinner D4 
Not yet found: D20, 5x(14)
Unfair dice generating 1..4 (Sicherman):


4,3,3,2,2,1 Nonuniform 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 Nonuniform 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.



500,200,200,100,50,50 Nonuniform, P(100)=1/6, P(50)=1/3 Monopoly Dice Game 
400,300,250,200,200,200 Nonuniform, P(250)=1/6, P(200)=1/2 Monopoly Dice Game 
9,9,8,7,6,6 Nonuniform P(7)=1/6, P(6)=1/3




8,7,7,6,6,5 Nonuniform P(8)=1/6, P(7)=1/3 P(6)=1/3, P(5)=1/6 
6,5,4,4,3,3 Nonuniform 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 


5,5,4,4,3,2 Nonuniform P(5)=1/3, P(4)=1/3 P(3)=1/6, P(2)=1/6 
5,4,4,3,3,2 Nonuniform Averaging die This die has the same average as a “regular” D6, namely 3.5 

5,4,3,3,2,2 Nonuniform 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.

Pentagonal Trapezohedron D10, 2 x 04 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.


Elongated pentagonal prism Abraham Neddermann 
Pentagonal Trapezohedron D10, 2 x 15 K=2 a=1, b=0 
Not yet found: D20, 4x15
This die is not fair:

Triangular Prism. Not fair. GameScience 
There are some D10 with all numbers printed twice. These dice are fair.


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:

8,8,7,7,6,6,5,4 Nonuniform 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.

40,32,24,16,8,8 Nonuniform P(16)=1/6, P(8)=1/3 


20,15,10,5,5,0 Nonuniform P(5)=1/3, P(0)=1/6 
12,11,10,10,9,2 Nonuniform P(9)=1/6, P(0)=1/6 


6,6,5,4,3,2 Nonuniform P(3)=1/6, P(2)=1/6 Cheater, 1 replaced by 6 
6,5,4,2,0,0 (blank,blank) Nonuniform P(6)=1/3, P(5)=1/6, P(4)=1/6, P(3)=1/6, P(2)=1/6 


5,4,3,2,1,1 Nonuniform 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 Nonuniform P(5)=1/6, P(3)=1/6, P(2)=1/6, P(1)=1/3, P(0)=1/6 Babylon 5 
The most common shape of this family is the cube (hexahedron).


Hexahedron numbered 
Hexahedron pipped 
Not yet found: D12, 2 x 05
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, 6sided antiprism, 6sided prism, dodecahedron, deltoidal icositetrahedron




Hexahedron pipped 
Hexahedron numbered 
Modified 6sided Antiprism Crystal Caste (left) Hasbro / Monopoly (right) 
Football 6sided prism Hasbro (Monopoly) 



Dodecahedron D12, 2x16 K=2 Hasbro (Monopoly) 
Deltoidal Icositetrahedron D24, 4 x 16 K=4 Numbers in square DTotal by A. Simkin / GameScience 
Spinner D6 
Other shapes:





Sphere 
Flattened Sphere 
Concave

Crooked Not fair 
Jumping Not fair 




Elliptical Not fair 
Tactile 
Crystal pips* 
Holes 
There is a large number of D6 with numbers other than 1..6. These include trick dice, backgammon dice, educational dice, mathematical dice (nontransitive and Sicherman) and special dice designed for games. Most of these dice are hexahedra, and there are also a few dodecahedra.



971,872,773,377,278,179 Trick die 
960,762,663,564,366,168 Trick die 
960,663,564,366,267,168 Trick die 



954,855,756,657,558,459 Trick die 
913,814,616,517,418,319 Trick die 
902,803,704,605,506,209 Trick die 



840,741,642,543,345,147 Trick die 
780,681,483,384,285,186 Trick die 
690,591,492,393,294,195 Trick die 



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 



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 



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 



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? 



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? 



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 



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? 



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 



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 



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 

12,11,10,9,8,7 K=1, a=1, b=6 



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 

12,9,8,7,3,1 



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 



11,8,7,5,3,0 
10,9,8,7,6,5 K=1, a=1, b=4 
10,9,8,7,6,0 



10,9,7,6,5,3 
10,8,8,8,7,7,6,6,5,5,5,3 D12, nonuniform 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 



9,8,8,8,7,7,6,6,5,5,5,4 D12, nonuniform 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 



9,8,5,4,3,2 
9,8,5,4,3,1 Nontransitive die Miwin 
9,7,6,5,2,1 Nontransitive die Miwin 



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, nonuniform 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 



8,7,7,6,6,6,5,5,4,3 D10, nonuniform 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 Nontransitive 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. 


8,7,4,3,2,1 8UR4L1L7LR2R3R 
8,7,4,2,1,0 8UL7R2U4LL1R0R 


8,6,5,4,3,1 Sicherman die Grand Illusions 
8,6,5,4,3,1 Sicherman die GameStation 



8,6,6,6,5,5,4,4,3,3,3,1 Same numbers as Sicherman die D12, nonuniform 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, nonuniform 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 



7,5,4,3,2,1 
6,5,4,3,2,0 (blank) 
5,5,4,4,3,2,1,0 D8, nonuniform P(5)=1/4, P(4)=1/4, P(3)=1/8, P(2)=1/8, P(1)=1/8, P(0)=1/8 



5,4,3,2,1,0 K=1, a=1, b=1 
+5,+4,+3,+21,2 
6,4,2,1,3,5 Odd Negative 



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) 
There are only very few dice with 7 different numbers.
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.

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 


Elongated heptagonal prism Abraham Neddermann 
Pentagonal Prism. Not fair 
Not yet found: D14, 2x(17)

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 
There are many D8 on the market, mostly octahedral numbered 1..8. Besides that there are very few other dice.



Octahedron 
Modified 8sided Antiprism Crystal Caste 
Deltoidal Icositetrahedron D24, 3 x 18 K=3 Numbers in diamond DTotal by A. Simkin / GameScience 
Wanted: D8 pipped
Not yet found: D16, 2x(18)



128,64,32,16,8,4,2,1 D8, nonuniform 2^(X1) 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, nonuniform 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 
There are only a few dice with nine different numbers in my collection. There is also a heptagonal prism (missing in my collection).


45,44,35,34,30, 29,25,25,21,20 D10, nonuniform 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, nonuniform 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 


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, Nonuniform P(9)=P(8)=…=P(2)=1/10 P(0)=1/5 Cheater, 1 replaced by 0 
Missing in my collection 
Heptagonal prism Not fair 
In addition to the “classical” pentagonal trapezohedron, there are also a decagonal
edged cigar, a 10sided antiprism, an icosahedron and a rhombic triacontahedron with numbers 0..9. There are also two types of flattened octahedra, which are not fair.





Pentagonal Trapezohedron numbered 
Pentagonal Trapezohedron pipped 
Decagonal Edged Cigar Phi Sports 
Modified 10sided Antiprism Crystal Caste 
Icosahedron D20, 2 x 09 K=2 


Rhombic Triacontahedron D30, 3 x 09 +9..+0,9..0,9..0 K=3 
Truncated Octahedron Not fair. 
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).



Pentagonal Trapezohedron 
Modified 10sided Prism Bear Cub Machine 
10 sided spinner 
Not yet found: D20, 2x(110)
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.



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 



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 



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 

19,18,18,17,17,16,16,15,15,15,14,14,14,13,13,13,12,12,11,10 D20, nonuniform 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 

15,14,14,13,13,12,12,11,11,11,10,10,10,9,9,9,8,8,7,6 D20, nonuniform 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 
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).

25,24,23,22,21,21,21,20,20,20,20,19,19,19,19,18,18,17,16,15 D20, nonuniform 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 

11,10,9,8,7,6,5,4,3,2,1 K=1, a=1, b=0 Elongated hendecagonal prism Abraham Neddermann 
There are many different shapes for dice numbered 1..12: pentagonal dodecahedron, rhombic dodecahedron, 12sided antiprism, and deltoidal icositetrahedron (with each number printed twice).




Pentagonal Dodecahedron 
Rhombic Dodecahedron AskAstro 
Modified 12sided Antiprism Crystal Caste 
Deltoidal Icositetrahedron D24, 2 x 112 K=2 Numbers in pentagon DTotal by A. Simkin / GameScience 
This die can be used to roll minutes from 0 to 55 in steps of 5:

55,50,…5,0 K=1, a=5, b=5 
Dice with 13 different numbers are very rare. This one is numbered 1..13.

Elongated tridecagonal prism Abraham Neddermann 
This heptagonal trapezohedron is numbered 1..14.

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)
Dice with 15 different numbers are very rare. This one is numbered 1..15.

Elongated pentadecagonal prism Abraham Neddermann 
The octagonal bipyramid with 16 faces exists both with “regular” and with hexadecimal numbers (0..F).


Octagonal Bipyramid 
Octagonal Bipyramid Hexi Die 
There are two icosahedral cheater dice:


20,20,19,18,…3,2 Nonuniform 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 Nonuniform P(1)=1/10, P(19)=…=P(2)=1/20 average 201/20=10.05 Cheater, 1 replaced by 20 Truant 
The most popular die numbered 1..20 is the icosahedron, but there is also a 20sided antiprism.


Icosahedron 
Modified 20sided Antiprism Crystal Caste 
There is only one die in my collection with 20 different numbers other than 1..20:

57,51,50,46,45,43,38,35,32,29,24,20,19,16,13,11,07,04,02,00 

49,47,46,44,39,37,36,32,28,27,24,19,18,16,15,14,13,9,5,1 Pic6 Lotto Dice 

49,46,43,40,37,34,31,28,25,22,21,19,16,13,10,7,4,3,2,1 Pic6 Lotto Dice 

48,47,46,43,42,41,39,38,35,34,33,31,30,29,26,20,17,8,4,2 Pic6 Lotto Dice 

48,44,42,38,37,36,34,33,32,31,29,28,27,26,24,23,22,12,6,3 Pic6 Lotto Dice 

47,45,44,41,38,35,32,29,26,23,22,21,20,17,14,11,8,7,6,5 Pic6 Lotto Dice 

45,43,42,41,40,39,36,33,30,27,25,24,23,21,18,15,12,11,10,9 Pic6 Lotto Dice 
There are two shapes of dice with 24 different numbers, tetrakishexahedron and deltoidal icositetrahedron.



Tetrakishexahedron 
Deltoidal Icositetrahedron Franck Dutrain 
Deltoidal Icositetrahedron Numbers in center of face DTotal by A. Simkin / GameScience 
Rhombic triacontahedron, numbered 1..30:

Rhombic Triacontahedron 
Dekaeptagonal Trapezohedron, numbered 1..34:

Dekaeptagonal Chessex 
Two different shapes for dice numbered 0..49, the eikosipendegonal
trapezohedron and a flattened sphere:


Eikosipendegonal Trapezohedron GameScience Big 50 Topper 
Flattened Sphere Not fair Alan Davies

The Zocchihedron, a spherical die numbered 1..100:

Zocchihedron Not fair GameScience 
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:



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 



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 Nonuniform 



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 Nonuniform 


1/3,1/6,1/6,1/12,1/12,1/12 Nonuniform 
1/6,1/6,1/8,1/8,1/9,1/9 



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 



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 


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 