Sudoku and Candidate
The Mathematics of Sudoku
Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles (Preliminary) November 15, 2010
1 Introduction
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
Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles (Preliminary) November 15, 2010
1 Introduction
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