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Unwed Numbers

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

Brian Hayes

Counting Solutions

In the search for general principles, a first step is to generalize the puzzle itself. The standard 81-cell Sudoku grid is not the only possibility. For any positive integer n, we can draw an order-n Sudoku grid with n 2 rows, n 2 columns and n 2 blocks; the grid has a total of n 4 cells, which are to be filled with numbers in the range from 1 to n 2. The standard grid with 81 cells is of order 3. Some publishers produce puzzles of order 4 (256 cells) and order 5 (625 cells). On the smaller side, there's not much to say about the order-1 puzzle. The order-2 Sudoku (with 4 rows, columns and blocks, and 16 cells in all) is no challenge as a puzzle, but it does serve as a useful test case for studying concepts and algorithms.

Order-2 Sudoku...Click to Enlarge Image

How many Sudoku solutions exist for each n? To put the question another way: Starting from a blank grid—with no givens at all—how many ways can the pattern be completed while obeying the Sudoku constraints? As a first approximation, we can simplify the problem by ignoring the blocks in the Sudoku grid, allowing any solution in which each column and each row has exactly one instance of each number. A pattern of this kind is known as a Latin square, and it was already familiar to Leonhard Euler more than 200 years ago.

Consider the 4 x 4 Latin square (which corresponds to the order-2 Sudoku). Euler counted them: There are exactly 576 ways of arranging the numbers 1, 2, 3 and 4 in a square array with no duplications in any row or column. It follows that 576 is an upper limit on the number of order-2 Sudoku. (Every Sudoku solution is necessarily a Latin square, but not every Latin square is a valid Sudoku.) In a series of postings on the Sudoku Programmers Forum, Frazer Jarvis of the University of Sheffield showed that exactly half the 4 x 4 Latin squares are Sudoku solutions; that is, there are 288 valid arrangements. (The method of counting is summarized in the illustration on the next page.)

Moving to higher-order Sudoku and larger Latin squares, the counting gets harder in a hurry. Euler got only as far as the 5 x 5 case, and the 9 x 9 Latin squares were not enumerated until 1975; the tally is 5,524,751,496,156,892,842,531,225,600, or about 6 x 1027. The order-3 Sudoku must be a subset of these squares. They were counted in June 2005 by Bertram Felgenhauer of the Technical University of Dresden in collaboration with Jarvis. The total they computed is 6,670,903,752,021,072,936,960, or 7 x 1021. Thus, among all the 9 x 9 Latin squares, a little more than one in a million are also Sudoku grids.

It's a matter of definition, however, whether all those patterns are really different. The Sudoku grid has many symmetries. If you take any solution and rotate it by a multiple of 90 degrees, you get another valid grid; in the tabulations above, these variants are counted as separate entries. Beyond the obvious rotations and reflections, you can permute the rows within a horizontal band of blocks or the columns within a vertical stack of blocks, and you can also freely shuffle the bands and stacks themselves. Furthermore, the numerals in the cells are arbitrary markers, which can also be permuted; for example, if you switch all the 5s and 6s in a puzzle, you get another valid puzzle.

When all these symmetries are taken into account, the number of essentially different Sudoku patterns is reduced substantially. In the case of the order-2 Sudoku, it turns out there are actually only two distinct grids! All the rest of the 288 patterns can all be generated from these two by applying various symmetry operations. In the order-3 case, the reduction is also dramatic, although it still leaves an impressive number of genuinely different solutions: 3,546,146,300,288, or 4 x 1012.

Does the large number of order-3 Sudoku grids tell us anything about the difficulty of solving the puzzle? Maybe. If we set out to solve it by some kind of search algorithm, then the number of patterns to be considered is a relevant factor. But any strategy that involves generating all 6,670,903,752,021,072,936,960 grids is probably not the best way to go about solving the puzzle.

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