In a comment to the Lecture 9's blog someone has alerted me to the fact that a set of 3 non-transitive dice can be bought from Maths Gear. They have sides like this:

Die A: 333336

Following some links on that page I have found this interesting article by James Grime (of the Millennium Mathematics Project) in which he says lots more about non-transitive dice. He has devised a set of 5 non-transitive dice. In another interesting article there is a video in which James discusses non-transitive dice with David Spiegelhalter.

Reading this made me wonder more about optimal designs. If dice were 2-sided (i.e. coins with a number on each side) then non-transitivity clearly is impossible. But what about 3-sided dice? Is a set of three 3-sided non-transitive dice possible? I quickly found one: A: 333, B: 225, C:441. Is there a set of 4 non-transitive 3-sided dice? What is the greatest number of 6-sided non-transitive dice possible? Is it 5, as found by James Grime, or could we have 6 non-transitive dice?

James Grime has now added a comment below.

Die A: 333336

Die B: 222555

Die C: 144444 $P(A>B)=P(B>C)=21/36$ and $P(C>A)=25/36$.

These are optimal, in that they maximize the smallest winning probability. However, there are other dice for which the sum of the 3 winning probabilities is greater.

Die A: 114444

Die B: 333333

Die C: 222255 $P(A>B)=P(B>C)=24/36$ and $P(C>A)=20/36$.

Following some links on that page I have found this interesting article by James Grime (of the Millennium Mathematics Project) in which he says lots more about non-transitive dice. He has devised a set of 5 non-transitive dice. In another interesting article there is a video in which James discusses non-transitive dice with David Spiegelhalter.

Reading this made me wonder more about optimal designs. If dice were 2-sided (i.e. coins with a number on each side) then non-transitivity clearly is impossible. But what about 3-sided dice? Is a set of three 3-sided non-transitive dice possible? I quickly found one: A: 333, B: 225, C:441. Is there a set of 4 non-transitive 3-sided dice? What is the greatest number of 6-sided non-transitive dice possible? Is it 5, as found by James Grime, or could we have 6 non-transitive dice?

James Grime has now added a comment below.