Miwins Dice made of titanium 

Designer  Dr. Michael Winkelmann 

Publisher  Arquus Verlag Vienna 
Years active  1994 
Players  1–9 
Age range  6+, depending on game 
Playing time  5–60 minutes depending on game 
Website  http://www.miwin.com/miwinsche_Wuerfel_html/miwin_Wuerfel_1.html 
Miwin's Dice were invented in 1975 by the physicist Michael Winkelmann. They consist of three different dice with faces bearing numbers from 1 to 9, with opposite sides summing to 9, 10, or 11. The numbers on each die give the sum of 30 and have an arithmetic mean of 5.
Contents 
Miwins's dice have 6 sides like standard dice, and each side shows different numbers. The standard set is made of wood; special designs are made of titanium (see picture) or other materials (gold, silver). The numbers (dots) on each die are colored blue, red or black.
Each die is named for the sum of its 2 lowest numbers.
Die III  with blue dots  1  2  5  6  7  9  
Die IV  with red dots  1  3  4  5  8  9  
Die V  with black dots  2  3  4  6  7  8 
Numbers 1 and 9, 2 and 7, and 3 and 8 are on opposite sides. Additional numbers are 5 and 6 on die III, 4 and 5 on die IV, and 4 and 6 on die V. The dice are designed in such a way that for every die there exists one that will win against it. A given die will have a higher number with a probability of 17/36, or a lower number with a probability of 16/36. III wins against IV, IV against V, and V against III. Such dice are known as nontransitive.
Each die is named for the sum of its lowest and highest number.
Die IX  with yellow dots  1  3  5  6  7  8  
Die X  with white dots  1  2  4  6  8  9  
Die XI  with green dots  2  3  4  5  7  9 
Each of the Dice has similar attributes like having no double number, the sum of the numbers is 30, and having each number from 1 to 9 two times spread over the 3 dice. This attribute characterize the implementation of intransitive Dice enabling all the different game variants. All the games need only 3 dice in difference to other theoretical nontransitive dice designed in view of mathematics such as Efrons Dice.^{[1]}
Because of these special attributes Miwin's Dice used also in the area of education. Miwin's Dice help to develop the mathematical highlights and enhances the ability to calculate probability as happened in the summer semester 2007 during a seminar at the University Siegen.
Since the middle of the eighties the press wrote about the games.^{[2]} Winkelmann presents his games also himself, for example in Vienna at the "Österrechischen Spielefest, "Stiftung Spielen in Österreich", Leopoldsdorf, where "Miwin's dice" 1987 won the prize "novel independent dice game of the year".
In 1989 the games were reviewed by the periodical "Die Spielwiese".^{[3]} At that time 14 alternatives of gambling and strategic games existed for Miwin's dice. Also the periodical "Spielbox" had in the category "Unser Spiel im Heft" (now known as "Edition Spielbox") two variants of games for Miwin's dice to be taken out of the magazine. It was the solitaire game 5 to 4 and the strategic game Bitis for two persons.
In 1994 Vienna's Arquus publishing house published Winkelmann's book "Göttliche Spiele",^{[4]} which contained 92 games, a master copy for 4 game boards, documentation about the mathematical attributes of the dice and a set of Miwin's dice. Now one can find about 120 game variants for free.^{[5]}
With Miwin's dice strategic games gambles are possible. Variants with both elements also exist. The intrinsic attributes of the dice cause well defined probabilities and mathematical phenomena.
Solitaire games and games for up to nine people beginning with the age of 6 available. Some of the games need a game board. Playing time is from 5 minutes to 60 minutes.
The probability for a given number with all 3 dice is 11/36, for a given rolled double is 1/36, for any rolled double 1/4. The probability to obtain a rolled double is only 50% compared to normal dice.
Cumulative frequency type III and IV 
Cumulative frequency type III and V 
Cumulative frequency type IV and V 
Cumulative frequency type III and IV and V = "MiwinDistribution" 
Removing the common dots of Miwin's Dice reverses intransitivity.
Miwin's dice allow to create several equal distributions. Adding a constant changes the range.
1 – 9 (rolling dice one time) P(19) = 1/9
take one of Miwins dice by random
0 – 80 (roll the dice 2 times) P(080) = 1/9² =
1/81
1st throw * 9  2nd throw
1st throw  2nd throw  Equation  Result 

9  9  9 times 9  9  72 
9  1  9 times 9  1  80 
1  9  9 times 1  9  0 
2  9  9 times 2  9  9 
2  8  9 times 2  8  10 
8  4  9 times 8  4  68 
1  3  9 mal 1  3  6 
This variant provides numbers from 0  80 with a probability of (1/9)², 81 = 9²
1st throw = 9 gives 10 * 2nd throw  10 all others 10 * 1st throw + 2nd throw  10
1st throw  2nd throw  Equation  Result 

9  9  10 times 9  10  80 
9  1  10 times 1 10  0 
8  4  10 times 8 + 4  10  74 
1  3  10 times 1 + 3  10  3 
This variant provides numbers from 0  80 with a probability of (1/9)², 81 = 9²
Both throws with 9 gives 0 1st throw = 9 and 2nd throw not 9 gives 10 * 2nd throw 1st throw = 8 gives 2nd throw all other give 10 * 1st throw  2nd throw
1st throw  2nd throw  Equation  Result 

9  9    0 
9  3  10 times 3  30 
8  4  1 times 4  4 
5  9  5 times 10 + 9  59 
0 – 90 (throw 3 times) P(090) = 8/9³ = 8/729
To obtain an equal distribution with numbers from 0  90 throw 3 times.
1st throw = 9, 3rd throw is not 9 gives 10 * 2nd throw (10, 20, 30, 40, 50, 60, 70, 80, 90) 1st throw is not 9 gives 10 times 1st throw plus 2nd throw 1st throw is equal to the 3rd throw gives 2nd throw (1, 2, 3, 4, 5, 6, 7, 8, 9) All dice equal gives 0 All dice 9 repeat the procedure
1st throw  2nd throw  3rd throw  Equation  Result 

9  9  not 9  10 times 9  90 
9  1  not 9  10 times 1  10 
8  4  not 8  10 times 8 + 4  84 
1  3  not 1  10 times 1 + 3  13 
7  8  7  78 gives 8  8 
4  4  4  three equals  0 
9  9  9  repeate   
This gives 91 numbers from 0  90 with the probability of 8 / 9³, 8 * 91 = 728 = 9³  1
0 – 103 (throw 3 times) P(0103) = 7/9³ = 7/729
This gives 104 numbers from 0  103 with the probability of 7 / 9³,
7 * 104 = 728 = 9³  1
0 – 728 (throw 3 times) P(0728) = 1/9³ = 1/729
This gives 729 numbers from 0  728 with the probability of 1 / 9³
Creating a number system with base 9:
(1st throw  1) * 81 + (2nd throw  1) * 9 + (3rd throw  1) * 1 gives a maximum from: 8 * 9² + 8 * 9 + 8 * 9° = 648 + 72 + 8 = 728 (throw  1) because we have only 9 digits ( 0,1,2,3,4,5,6,7,8 )
1st throw  2nd throw  3rd throw  Equation  Result 

9  9  9  8 * 9² + 8 * 9 + 8  728 
4  7  2  3 * 9² + 6 * 9 + 1  298 
2  4  1  1 * 9² + 4 * 9 + 0  117 
1  3  4  0 * 9² + 3 * 9 + 3  30 
7  7  7  6 * 9² + 6 * 9 + 6  546 
1  1  1  0 * 9² + 0 * 9 + 0  0 
4  2  6  3 * 9² + 1 * 9 + 5  257 
This gives 729 numbers (0  728), each with a probability of 1 / 9³ = 1 / 729 728 = 9³  1
Variant  Equation  number of variants 

one throw with 3 dice, types don't mind    135 
one throw with 3 dice, type is an additional attribute  (135 – 6 * 9) * 2 + 54  216 
1 throw with each type, type is not used as attribute  6 * 6 * 6  216 
1 throw with each type, type is used as attribute  6 * 6 * 6 * 6  1296 
3 throws, random selection of one of the dice for each throw, type is not used as attribute  9 * 9 * 9  729 
3 throws, random selection of one of the dice for each throw, type is used as attribute:
Variant  Equation  number of alternatives 

III, III, III / IV, IV, IV / V, V, V  3 * 6 * 6 * 6  648 
III, III, IV / III, III, V / III, IV, IV / III, V, V / IV, IV, V / IV, V, V  6 * 3 * 216  + 3888 
III, IV, V / III, V, IV / IV, III, V / IV, V, III / V, III, IV / V, IV, III  6 * 216  + 1296 
= 5832 
5832 = 2 x 2 x 2 x 9 x 9 x 9 = 18³ numbers are possible.
