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Unwed Numbers

The mathematics of Sudoku, a puzzle that boasts "No math required!"

Brian Hayes

A few years ago, if you had noticed someone filling in a crossword puzzle with numbers instead of letters, you might well have looked askance. Today you would know that the puzzle is not a crossword but a Sudoku. The craze has circled the globe. It's in the newspaper, the bookstore, the supermarket checkout line; Web sites offer puzzles on demand; you can even play it on your cell phone.

Sudoku puzzles...Click to Enlarge Image

Just in case this column might fall into the hands of the last person in North America who hasn't seen a Sudoku, an example is given on the opposite page. The standard puzzle grid has 81 cells, organized into nine rows and nine columns and also marked off into nine three-by-three blocks. Some of the cells are already filled in with numbers called givens. The aim is to complete the grid in such a way that every row, every column and every block has exactly one instance of each number from 1 to 9. A well-formed puzzle has one and only one solution.

The instructions that accompany Sudoku often reassure the number-shy solver that "No mathematics is required." What this really means is that no arithmetic is required. You don't have to add up columns of figures; you don't even have to count. As a matter of fact, the symbols in the grid need not be numbers at all; letters or colors or fruits would do as well. In this sense it's true that solving the puzzle is not a test of skill in arithmetic. On the other hand, if we look into Sudoku a little more deeply, we may well find some mathematical ideas lurking in the background.

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