Peter Norvig presents a nice and simple way to solve every Sudoku puzzle in http://norvig.com/sudoku.html, and mention some ports of his effort. He picked python to implement his solver, and the Ruby ports are dead links.
Let's begin by copying the introduction as-is:
First we have to agree on some notation. A Sudoku puzzle is a grid of 81 squares; the majority of enthusiasts label the columns 1-9, the rows A-I, and call a collection of nine squares (column, row, or box) a unit and the squares that share a unit the peers. A puzzle leaves some squares blank and fills others with digits, and the whole idea is:
A puzzle is solved if the squares in each unit are filled with a permutation of the digits 1 to 9.
That is, no digit can appear twice in a unit, and every digit must appear once. This implies that each square must have a different value from any of its peers.
A1 A2 A3| A4 A5 A6| A7 A8 A9 4 . . |. . . |8 . 5 4 1 7 |3 6 9 |8 2 5
B1 B2 B3| B4 B5 B6| B7 B8 B9 . 3 . |. . . |. . . 6 3 2 |1 5 8 |9 4 7
C1 C2 C3| C4 C5 C6| C7 C8 C9 . . . |7 . . |. . . 9 5 8 |7 2 4 |3 1 6
---------+---------+--------- ------+------+------ ------+------+------
D1 D2 D3| D4 D5 D6| D7 D8 D9 . 2 . |. . . |. 6 . 8 2 5 |4 3 7 |1 6 9
E1 E2 E3| E4 E5 E6| E7 E8 E9 . . . |. 8 . |4 . . 7 9 1 |5 8 6 |4 3 2
F1 F2 F3| F4 F5 F6| F7 F8 F9 . . . |. 1 . |. . . 3 4 6 |9 1 2 |7 5 8
---------+---------+--------- ------+------+------ ------+------+------
G1 G2 G3| G4 G5 G6| G7 G8 G9 . . . |6 . 3 |. 7 . 2 8 9 |6 4 3 |5 7 1
H1 H2 H3| H4 H5 H6| H7 H8 H9 5 . . |2 . . |. . . 5 7 3 |2 9 1 |6 8 4
I1 I2 I3| I4 I5 I6| I7 I8 I9 1 . 4 |. . . |. . . 1 6 4 |8 7 5 |2 9 3
Every square has exactly 3 units and 20 peers. For example, here are the units and peers for the square C2:
A2 | | | | A1 A2 A3| |
B2 | | | | B1 B2 B3| |
C2 | | C1 C2 C3| C4 C5 C6| C7 C8 C9 C1 C2 C3| |
---------+---------+--------- ---------+---------+--------- ---------+---------+---------
D2 | | | | | |
E2 | | | | | |
F2 | | | | | |
---------+---------+--------- ---------+---------+--------- ---------+---------+---------
G2 | | | | | |
H2 | | | | | |
I2 | | | | | |
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