The Copson lecture

Yesterday I was in St Andrews to deliver the biennial Copson Lecture.

Edward Copson was a significant figure in mathematics at St Andrews. I was told that he had built the Mathematics Institute building himself, almost literally (helping out with building work where necessary). Many people there (and in many other places) knew him, or were taught by him.

What better way to commemorate a mathematician than with a lecture series?

So I had a pleasant train trip north – Scotland very frosty but clear except for the eerie experience of crossing the Forth Bridge in thick fog – a wet day in St Andrews with some interesting mathematical discussions, a splendid dinner after the lecture, and then a pleasant trip back south.

I was told that the audience would range from schoolchildren to professors (and so indeed it was). Copson’s subject was analysis (indeed, I had his book as a textbook when I was an undergraduate), and I didn’t feel competent to deliver such a talk on analysis. So I went for my current favourite, “Sudoku and Mathematics”. I have given this talk several times before, but unlike many talks I give, it doesn’t seem to become stale; I thought that it went better than ever this time. It helped that a large part of the audience were undergraduate mathematics students, who had met some of the concepts in their lectures.

The talk was made up of three parts. First, a short rant based on the popular perception that Sudoku is not mathematics (I suppose because you do not actually have to do arithmetic with the numbers). Indeed, the Independent says, “You solve the puzzle with reasoning and logic”, and that is as good a short definition of mathematics as I can come up with!

Then I trace the history of Sudoku. The roots lie in magic squares, studied by the Chinese more than two thousand years ago. Euler’s problem of the 36 officers arose from his new construction of magic squares from Graeco-Latin squares (his invention). We now call these “orthogonal Latin squares”. Euler wrote two papers on this topic, and in the second he specifically studied Latin squares, which could be regarded as Sudoku without the 3×3 subsquares.

Then attention turned to the statisticians. In the 1950s Behrens invented gerechte designs. One of these is an n×n square with the cells divided into n “regions” of n cells each; it is required to place the digits 1,…,n in the cells such that each digit occurs once in each row, column, and region. Completed Sudoku puzzles are examples. In the 1970s, Nelder defined a critical set in a Latin square, a subset of the entries which can be completed in only one way, but such that if any entry is deleted there is more than one completion.

All the ingredients were now in place, but it was Howard Garns, neither a mathematician nor a statistician but a retired architect in New York, who actually devised the puzzle we have now. After moving to Japan and then New Zealand it spread worldwide, and now no newspaper is complete without a Sudoku puzzle.

The third and most technical part of the talk involves a variant devised by Robert Connelly which he called “symmetric Sudoku”. This has a beautiful mathematical theory, involving affine geometry and perfect error-correcting codes, at the end of which one shows that (up to symmetry) there are only two filled grids satisfying the rules of symmetric Sudoku. By contrast, Jarvis and Russell found that there are about six billion filled grids for regular Sudoku, up to symmetry; their proof involves extensive computation.

The affine geometry in the argument is illustrated by the card game SET. This involves 81 cards, each carrying a picture with four attributes (number, shape, colour, and shading) each of which can take three possible values. A winning combination in SET is a set of three cards such that, for each attribute, the pictures shown are either all the same or all different. The heart of the argument is that these are precisely the lines of the 4-dimensional affine geometry over the field of three elements. Indeed, from my limited experience of playing SET, I found that the best way to practice is to pick two cards at random and figure out as quickly as possible the unique third card making a winning combination with them.

A couple of years ago, Gary Gordon and Liz McMahon visited our department. Their daughter is a champion SET player, and they spread the craze among some of the mathematicians. Many results and problems in finite geometry can be illustrated (at least in interesting special cases) with a pack of SET cards!

Some of the mathematical discussions with Nik Ruskuc are reported in a comment on my earlier posting entitled “Seen this before?” The surprising result there has become even more remarkable!


About Peter Cameron

I count all the things that need to be counted.
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