My nephew

+Raymond Douglas (aged 14) got me to play a very cool game of his invention today. My opponent was my 7-year-old son. We had to sit back to back. Also, he spent a bit of time explaining to each of us what to do, and it was important that we should not hear what he said to the other person.

Here were my instructions. I had a list of letters associated with the numbers 1 to 9. I can't remember exactly what they were, but it doesn't matter, as it just had to be some kind of encoding. So I had some kind of chart such as 1W, 2T, 3N, 4J, 5P, 6H, 7L, 8S, 9K.

We then took turns calling out letters. My rules were that I was not allowed to choose a letter that my son had already called out, and my aim was to end up having chosen three letters (amongst the possibly more than three that I ended up choosing) such that the corresponding numbers added up to 15.

The question is, what was my son doing all the while? It's fun to think about this, so here's the customary spoiler alert plus some space.

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The answer is that my son was playing noughts and crosses, and for him the letters stood for squares in a 3-by-3 grid. In other words, the add-up-to-15 game and noughts and crosses are

*isomorphic*. What I find beautiful about this fact is that the add-up-to-15 game is very simple and yet the isomorphism is not entirely trivial. Here is how it works. You fill up a 3-by-3 grid with numbers as follows.

6 1 8

7 5 3

2 9 4

There are two properties that this grid of numbers must satisfy. First, it must be a Latin square -- that is, all rows, columns and diagonals must add up to 15. Secondly, there should be no triple adding up to 15 that is not a row, column, or diagonal. Actually, the second follows from the first, because it so happens that the number of triples of numbers between 1 and 9 that add up to 15 is precisely 8, the number of rows+columns+diagonals.

An interesting aspect of the experience for me was that I instinctively used a certain amount of standard noughts-and-crosses strategy. For instance, I thought it would be good to choose the number 5 if possible, because it would be in the most triples. Also, at one point I got into a winning position by noticing that I could choose a number that would give me two routes to a triple, with my son not able to stop both of them. He had played badly that round. However, while I know how not to lose at noughts and crosses, I didn't manage to stop my son winning the next round, so the isomorphism had been enough to blind me to quite a lot of the structure of the usual game.