Tabelle intransitive Spiele




andere Würfel

Sequenz-Würfel
Die „Sequenz-Würfel 5/6“ bestehen aus den drei unterschiedlichen Würfeln: Würfel „36“ hat die Zahlen: 3, 3, 3, 3, 3, 6 Würfel „25“ hat die Zahlen: 2, 2, 2, 5, 5, 5 Würfel „14“ hat die Zahlen: 1, 4, 4, 4, 4, 4	Die „Sequenz-Würfel 8/9“ bestehen aus den drei unterschiedlichen Würfeln: Würfel „52“ hat die Zahlen: 5, 5, 2, 2, 2, 2 Würfel „41“ hat die Zahlen: 4, 4, 4, 4, 1, 1 Würfel „33“ hat die Zahlen: 3, 3, 3, 3, 3, 3
Bei einem Wurf mit allen drei Sequenz-Würfeln beträgt die Wahrscheinlichkeit, eine Sequenz zu würfeln, 5/6.	Bei einem Wurf mit allen drei Sequenz-Würfeln beträgt die Wahrscheinlichkeit, eine Sequenz zu würfeln, 8/9.
Es sind intransitive Würfel: „36“ gewinnt gegen „25“ mit 21:15 „25“ gewinnt gegen „14“ mit 21:15 „14“ gewinnt gegen „36“ mit 25:11	Es sind intransitive Würfel: "52" gewinnt gegen "41" mit 20:16 "41" gewinnt gegen "33" mit 2:1 "33" gewinnt gegen "52" mit 2:1.
Sequenzen: 123: 15 x 234: 75 x 345: 75 x 456: 15 x	ebenso: 6, 6, 3, 3, 3, 3 5, 5, 5, 5, 2, 2 4, 4, 4, 4, 4, 4
Keine Sequenzen: 126: 3 x 135: 15 x 156: 3 x 246: 15 x	und: 6, 6, 3, 2, 2, 2 5, 5, 5, 4, 1, 1 4, 4, 4, 3, 3, 3

Schrat-Würfel
Die Schrat-Würfel wurden am 27. 3. 2009 von mgf.winkelmann erfunden, sie bestehen aus sieben Würfeln, die folgende Eigenschaften haben: Auf jedem Würfel gibt es nur drei unterschiedliche Symbole, jedes 14 mal. In 5/7 aller Fälle liegen gleiche Symbole auf gegenüberliegenden Seiten wie ungleiche Symbole!

Three-dice set with minimal alterations to standard dice

The following intransitive dice have only a few differences compared to 1 through 6 standard dice:

as with standard dice, the total number of pips is always 21
as with standard dice, the sides only carry pip numbers between 1 and 6
faces with the same number of pips occur a maximum of twice per die
only two sides on each die have numbers different from standard dice:

A: 1, 1, 3, 5, 5, 6
B: 2, 3, 3, 4, 4, 5
C: 1, 2, 2, 4, 6, 6

Like Miwin’s set, the probability of A winning versus B (or B vs. C, C vs. A) is 17/36. The probability of a draw, however, is 4/36, so that only 15 out of 36 rolls lose. So the overall winning expectation is higher.

Grime Dice

Here is a set of five non-transitive dice:

These dice use values from 0 to 9, as follows;

Red: 4 4 4 4 4 9
Yellow: 3 3 3 3 8 8
Blue: 2 2 2 7 7 7
Magenta: 1 1 6 6 6 6
Olive: 0 5 5 5 5 5

This set of five dice is similar to other sets of dice we have seen, in that we have a chain where Blue>Magenta>Olive>Red>Yellow>Blue.

However, we also have a second chain, where Red>Blue>Olive>Yellow>Magenta>Red.
Notice the first chain is ordered alphabetically, while the second chain is ordered by word-length.
In general, the chain in alphabetical order is stronger than the chain in order of word-length. But if your friend discovers you using one chain, you can switch to the other.
Overall, the average winning probability for one die is 63%.
If our original set of three non-transitive dice was like a game of `Rock, Paper, Scissors', this diagram is closer to the related, but more extreme, non-transitive game `Rock, Paper, Scissors, Lizard, Spock'.
But not only that, because if you consider subsets of these five dice you will find other non-transitive chains! In particular, the first three dice in the chain ordered by word-length; that's Red, Blue and Olive; is a copy of the optimal set of three non-transitive dice that I describe above. So with this set of five dice you get the optimal set of three dice for free! In fact, any three consecutive dice in the chain ordered by word-length makes a set of three non-transitive dice.
And not only that, because any four consecutive dice in the alphabetical chain makes a set of four non-transitive dice. The first four in this chain; that's Blue, Magenta, Olive and Red; is not a copy of the optimal set of four dice, but they do have the same average winning probability as Efron dice.
With two dice the chain ordered by word-length now flips so Magenta>Yellow>Olive>Blue>Red>Magenta.
On the other hand, the alphabetical chain stays the same - with one exception, the Red-Olive arrow reverses. This is to be expected since Red, Blue and Olive form a copy of the optimal set of three.
With two dice, the chain ordered by word-length is stronger than the alphabetical chain. The average winning probability for two dice is 59%.
However, the probability that two Red dice beats two Olive dice is close to 50%. If we treat these dice as equal we can now play two games simultaneously.
Invite two opponents to pick a die each, but do not volunteer whether you are playing a one die or two dice version of the game.
If your opponents pick two dice that are next to each other alphabetically then play the one die version of the game. Use the diagram for the one die game to pick a die that can beat each opponent.
If your opponents pick two dice that are next to each other in the chain ordered by word-length, then play the two dice version of the game. Again, use the diagram for the two dice game (that treats Red and Olive as equal) to pick a die that can beat each opponent.