Yesterday Tim Gowers gave a really nice talk at Harvard about intransitive dice. The talk was both interesting to math faculty and also accessible to undergraduates (and also to math enthusiasts like me).

The subject of the talk – properties of intransitive dice – was based on a problem discussed on Gowers’s blog earlier in 2017. Here’s one of the blog posts:

One of Tim Gowers’s blog posts on intransitive dice

Part of the reason that I wanted to attend the talk is that we have played with non-transitive dice previously and the kids seemed to have a lot of fun:

Here’s the punch line from that project:

The starting example in Gowers’s talk was due to Bradley Efron. Consider the four 6-sided dice with numbers:

A: 0,0,4,4,4,4

B: 3,3,3,3,3,3

C: 2,2,2,2,6,6

D: 1,1,1,5,5,5

If you have little dice rolling competition in which the winner of each turn is the die with the highest number, you’ll run across the following somewhat surprising expected outcome:

A beats B, B beats C, C beats D, and D beats A.

The question that interested Gowers was essentially this -> Is the situation above unusual, or is it reasonably easy to create intransitive dice?

Although the answering this question probably doesn’t create any groundbreaking math, it does involve some fairly heavy lifting, and I think the details in the talk are not accessible (or interesting) to kids. Still, though, the general topic I think does have questions that could be both fun and interesting for kids to explore.

In discussing a few of the ideas that I think might be interesting to kids, I’ll use a constraints that Gowers imposed on the dice he was studying. Those are:

(i) The numbers on each side of an n-sided can be any integer from 1 to n

(ii) The sum of the numbers must be (n)(n+1)/2

I’ll focus just on 6-sided dice for now. One question that kids might find interesting is simply how many different 6-sided dice are there that meet these two criteria above? Assuming I’ve done my own math right, the answer is that there are 32 of them:

Next, it might be interesting for kids to play around with these dice and see which ones have lots of wins or lots of losses or lots of draws against the other 31 dice. Here’s the win / draw / loss totals (in the same order as the dice are listed above):

So, for clarity, the 4th die on the list – the one with numbers 1,1,3,5,5,6 – wins against 16 other dice, draws with 7 (including itself), and loses to 9. Not bad!

The die three down from that – the one with numbers 1, 2, 2, 4, 6, 6 – has the opposite results. Such poor form 😦

It certainly wasn’t obvious to me prior to running the competition that one of these two dies would be so much better than the other one. Perhaps it would be interesting for kids to try to guess ahead of time which dice will be great performers and which will perform poorly.

Also, what about that one that draws against all the others – I bet kids would enjoy figuring out what’s going on there.

Once I had the list, it wasn’t too hard for me to find a set of three intransitive dice. Choosing

A -> 2, 2, 3, 3, 5, 6

B -> 1, 1, 3, 5, 5, 6, and

C -> 1, 2, 4, 4, 4, 6

You’ll see that A beats B on average, B beats C, and C beats A.

It is always fun to find problems that are interesting to professional mathematicians and that are also accessible to kids. A few ideas I’ve found from other mathematicians can be found in these blog posts:

Amazing math from mathematicans to share with kids

10 more math ideas from mathematicians to share with kids

I think exploring intransitive dice will allow kids to play with several fun and fascinating mathematical ideas. I’m going to try a project (a computer assisted project, to be clear) with the kids this weekend to see how it goes.