# Miwins dice: Wikis

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# Encyclopedia

Designer Miwins Dice made of titanium Dr. Michael Winkelmann Arquus Verlag Vienna 1994 1–9 6+, depending on game 5–60 minutes depending on game 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.

## Description

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.

Miwins dice IX, X, XI

### Second set of Miwin's dice: IX, X, XI

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

## Mathematical attributes

design of numbers of Miwins dice III, IV, V
design of numbers of Miwins dice IX, X, XI

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.

## Games

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.

## Features

• 1/3 of the sum of dots of all dice can be divided by 3 without carry over.
• 1/3 of the sum of dots of all dice can be divided by 3 having a carry over of 1.
• 1/3 of the sum of dots of all dice can be divided by 3 having a carry over of 2.

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.

## Reversed intransitivity

Removing the common dots of Miwin's Dice reverses intransitivity.

## Equal distribution of random numbers

Miwin's dice allow to create several equal distributions. Adding a constant changes the range.

1 – 9 (rolling dice one time) P(1-9) = 1/9

take one of Miwins dice by random

0 – 80 (roll the dice 2 times) P(0-80) = 1/9² = 1/81

## Variants 0 - 80

### 1st Variant

1. ) Take one dice by random, roll it and lay it back: 1st throw!
2. ) Take one dice by random, roll it and lay it back: 2nd throw!

1st throw * 9 - 2nd throw

Examples
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

### 2nd Variant

1. ) Take one dice by random, roll it and lay it back: 1st throw!
2. ) Take one dice by random, roll it and lay it back: 2nd throw!

1st throw = 9 gives 10 * 2nd throw - 10 all others 10 * 1st throw + 2nd throw - 10

Examples
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

### 3rd Variant

1. ) Take one dice by random, roll it and lay it back: 1st throw!
2. ) Take one dice by random, roll it and lay it back: 2nd throw!

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

Examples
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

## Other distributions

0 – 90 (throw 3 times) P(0-90) = 8/9³ = 8/729

To obtain an equal distribution with numbers from 0 - 90 throw 3 times.

1. ) Take one dice by random, roll it and lay it back: 1st throw!
2. ) Take one dice by random, roll it and lay it back: 2nd throw!
3. ) Take one dice by random, roll it and lay it back: 3rd throw!

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

Examples
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(0-103) = 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(0-728) = 1/9³ = 1/729

This gives 729 numbers from 0 - 728 with the probability of 1 / 9³

1. ) Take one dice by random, roll it and lay it back: 1st throw!
2. ) Take one dice by random, roll it and lay it back: 2nd throw!
3. ) Take one dice by random, roll it and lay it back: 3rd throw!

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 )

Examples
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

## Combinations of numbers with Miwin's dice type III IV and V

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.

## Notes

1. ^ http://www.miwin.com/ click "Miwin'sche Würfel 2", then check attributes
2. ^ Austrian paper "Das Weihnachtsorakel, Spieltip "Ein Buch mit zwei Seiten", the Standard 18.Dez..1994, page 6, Pöppel-Revue 1/1990 page 6 and Spielwiese 11/1990 page 13, 29/1994 page 7
3. ^ 29/1989 page 6
4. ^ The book on the German version of Amazon
5. ^ Winkelmann's homepage