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.
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molto · 582 weeks ago
Link: http://www.math.uci.edu/~mfinkels/dice9.pdf
@jamesgrime · 582 weeks ago
Your question about how many dice we could have is interesting. I think constructing a set of 6 dice would be quite easy - the special thing about the set of 5 I made is that they have two chains and one reverses, which makes a three person game possible.
In fact the original paper, from 1965, says that there exists n, m-sided, nontransitive dice for all n and m. There are some interesting results about the weakest probability, and how big it can be, but no general construction.
Here's a more formal paper http://www.jstor.org/discover/10.2307/2974903?uid...
Best wishes for the rest of the course!