## Sudoku and Kakuro

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In recent years many people have played Sudoku, a number game in which a nine-by-nine grid is filled with the digits 1-9 so that each row, each column, and each group of nine squares contains exactly one of each digit.

(Puzzle drawn by Lawrence Leonard Gilbert, and released to the public domain.)

The game appears to have been invented by Howard Garnes, a contributor for Games Magazine: the puzzles began appearing there in 1979 under the title Number Place. They became popular in Japan and then, only a few years ago, in the United States. The puzzles lend themselves not just to recreation, but to mathematical thought: How many different grids are possible? What is the fewest number of squares that can be filled in initially but that still give rise to a unique solution? What is the greatest number of squares that can be filled in that don’t give rise to a unique solution?*

You can find more about the history of Sudoku in The New York Times article here; more mathematical information on Wolfram MathWorld here (including the little tidbit that Sudoku puzzles appears in an episode of Num3rs) or in the MAA Focus article “The Sudoku Epidemic” here. By looking around the web, too, you can find all sorts of variations on Sudoku: letters instead of numbers, smaller (4 by 4) or larger (16 by 16) grids, rectangular grids, overlapping grids, cubes….

Fans of Sudoku may also enjoy Kakuro, a number puzzle with a different twist. In Kakuro, the digits 1-9 are also used, but each row or column has to add up to a certain sum written above or to the left of the column. Within each sum a digit can be used only once, although not every digit is used in every sum. In the sample puzzle below, looking at the upper left-hand portion, the top row would have two distinct digits that add to 16, while the left-most column would have three distinct digits that add to 23.

In solving Kakuro it is helpful to keep track of unique combinations (for example, the only three distinct digits to add to 7 are 1, 2, and 4 in some order). Some people (like TwoPi) find that the extra challenge of Kakuro makes it more interesting than Sudoku; others (like Ξ) have yet to solve a puzzle without help and are primarily interested in finding the easiest versions. (What is the smallest possible puzzle with a unique solution?**)

*Answers: 6,670,903,752,021,072,936,960 different grids; 17 squares, many of which can be seen in G. Royle’s collection here; 77 squares

** Mark Huckvale hints here that it is 5 by 5, at least to be interesting.

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### 3 Responses to “Sudoku and Kakuro”

1. GaussGirlRibby06 Says: