This is How I Roll: Non-transitive Pips

April 29th, 2021 by munizao Leave a reply »

A set of three non-transitive dice has the property that if you roll two of them and compare numbers, the first tends to beat the second, the second beats the third, and the third beats the first. I recommend this article by James Grime, which describes how these dice work and covers some more recent variations on the theme.

Red beats blue and blue beats green, both on 7/12 of rolls. Green beats red on 25/36 rolls.

I recently acquired some Sicherman dice with indented pips, which inspired me to use acrylic paints to color the faces so that matched colors on a die roll indicate doubles, in the manner described in this post. Since I had a method for altering dice, I wondered if there was anything interesting I could do with standard pipped dice.

Non-transitive dice inspired an idea: what if we could color the pips with three different colors so that, with a single roll of one die, the results are non-transitive over the pip counts of the three colors? Some experimentation showed that it is in fact possible:

Unlike the usual non-transitive dice, ties occur on these. For a roll of 1, a tie between the two missing colors is inevitable, and for a roll of 2, either one color is used, and the other two tie, or two colors are used, and those two tie. It’s possible to have those be the only ties, but I prefer the symmetry of having each pair of colors give the same results, so I want one more tie. Here, white beats red, red beats blue, and blue beats white, all with a 3-2 record, with one tie.

The “standard” non-transitive dice shown at the top have the interesting property that if you roll them twice, the results become non-transitive in the opposite direction. I found that there are no individual non-transitive pip dice that have this property, but there are pairs of dice that work. For example, you could add this die to the one above:

This die works the same way as the previous one as an individual die. (White beats red, red beats blue, and blue beats white.) But when you roll both dice together, white beats blue, blue beats red, and red beats white.

Problem #49: There are sets of 4 and more non-transitive dice, where each beats the next in a cycle. Can there be a non-transitive pip die with four colors, or even five? Since the total number of pips is 21, the number of pips of each color would have to be unbalanced (or the remainder pip could use a neutral color.)

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