## Symmetric Sudoku

Here is a beautiful thing I wanted to share.

It is a solution to “Symmetric Sudoku”, a puzzle invented by Robert Connelly, with the digits replaced by colours. Each colour occurs, not only once in each row, once in each column, and once in each 3×3 subsquare (as in ordinary Sudoku), but once in each “broken row” (this is a set or 3 minirows at the same positions in each of the three 3×3 squares in a fat column), once in each “broken column” (similarly defined), and once in each “location” (a location being the nine small squares which lie in the same relative positions in the nine 3×3 squares).

Robert proved that there are just two essentially different solutions to symmetric Sudoku. The proof is beautiful, using ideas from finite geometry and coding theory. You can read about it in the American Mathematical Monthly 115 (2008), 383-404.

The picture was produced by David Spiegelhalter, whom you will recognise from the Horizon programme on Infinity earlier this year. He is probably going to make a stained glass piece to this design. If you want a real treat, take a look at some of his stained glass.

And thanks to him for sending a copy of the picture!

## About Peter Cameron

I count all the things that need to be counted.
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### 2 Responses to Symmetric Sudoku

1. Andy says:

That’s beautiful – I wonder if anything is known about general nxn symmetric sudoku?

• Not much. They do exist for prime power orders (see the Monthly article). I don’t imagine that the 16×16 will be determined any time soon. As you will see from the article, there is something special and wonderful about 9×9 here.

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