The Mathematics of Sudoku
firstname.lastname@example.org http://www.geometer.org/mathcircles (Preliminary) November 15, 2010
Sudoku is a (sometimes addictive) puzzle presented on a square grid that is usually 9 × 9, but is sometimes 16 × 16 or other sizes. We will consider here only the 9 × 9 case, although most of what follows can be extended to larger puzzles. Sudoku puzzles can be found in many daily newspapers, and there are thousands of references to it on the internet. Most puzzles are ranked as to difﬁculty, but the rankings vary from puzzle designer to puzzle designer. Sudoku is an abbreviation for a Japanese phrase meaning “the digits must remain single,” and it was in Japan that the puzzle ﬁrst became popular. The puzzle is also known as “Number Place.” Sudoku (although it was not originally called that) was apparently invented by Howard Garns in 1979. It was ﬁrst published by Dell Magazines (which continues to do so) but now is available in hundreds of publications. At the time of publication of this article, Sudoku is very popular, but it is of course difﬁcult to predict whether it will remain so. It does have many features of puzzles that remain popular: puzzles are available of all degrees of difﬁculty, the rules are very simple, your ability to solve them improves with time, and it is the sort of puzzle where the person solving it makes continuous progress toward a solution, as is the case with crossword puzzles. The original grid has some of the squares ﬁlled with the digits from 1 to 9 and the goal is to complete the grid so that every row, column and outlined 3 × 3 sub-grid contains each of the digits exactly once. A valid puzzle admits exactly one solution. 1 a b c d e f g h i 2 3 4 5 6 7 8 9
4 8 9 4 6 7 5 6 1 4 2 1 6 5 5 8 7 9 4 1 7 8 6 9 3 4 5 9 6 3 7 2 4 1
Figure 1 is a relatively easy sudoku puzzle. If you have never tried to solve one, attempt this Figure 1: An easy sudoku puzzle one (using a pencil!) before you continue, and see what strategies you can ﬁnd. It will probably take more time than you think, but you will get much better with practice. The solution appears in section 20. Sudoku is mathematically interesting in a variety of ways. Both simple and intricate logic can be
applied to solve a puzzle, it can be viewed as a graph coloring problem and it certainly has some interesting combinatorial aspects. We will begin by examining some logical and mathematical approaches to solving sudoku puzzles beginning with the most obvious and continuing on to more and more sophisticated techniques (see, for example, multi-coloring, described in section 9.2). Later in this article we will look at a few of the more mathematical aspects of sudoku. A large literature on sudoku exists on the internet with a fairly standardized terminology, which we will use here: • A “square” refers to one of the 81 boxes in the sudoku grid, each of which is to be ﬁlled eventually with a digit from 1 to 9. • A “block” refers to a 3 × 3 sub-grid of the main puzzle in which all of the numbers must appear exactly once in a solution. We will refer to a block by its columns and rows. Thus block ghi456 includes the squares g4, g5, g6, h4, h5, h6, i4, i5 and i6. (See ﬁgure 1.) • A “candidate” is a number that could possibly go into a square in the grid. Each candidate that we can eliminate from a square brings us closer to a solution. • Many arguments apply equally well to a row, column or block, and to keep from having to write “row, column or block” over and over, we may refer to it as a “virtual line.” A typical use of “virtual line” might be this: “If you know the values of 8 of the 9 squares in a virtual line, you can always deduce the value of the missing one.” In the 9 × 9 sudoku puzzles there are 27 such virtual lines. • Sometimes you would like to talk about all of the squares that cannot contain the same number as a given square since they share a row or column or block. These are...
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