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# A Pencil And Paper Algorithm For Solving Soduku

4765 words - 20 pages

A Pencil-and-Paper
Algorithm for Solving
Sudoku Puzzles
J. F. Crook

T

he puzzle Sudoku has become the passion of many people the world over in
the past few years. The interesting fact
about Sudoku is that it is a trivial puzzle
to solve. The reason it is trivial to solve
is that an algorithm exists for Sudoku solutions.
The algorithm is a tree-based search algorithm
based on backtracking in a tree until a solution is
found.
If all a person needs to do is sit down at their
personal computer, punch in the numbers given in
the puzzle, and then watch a computer program
compute the solution, we can reasonably ask why
a person would bother to struggle to solve Sudoku
puzzles. ...view middle of the document...

For example,
Sheldon (2006, p. xiv) names them partnerships,
whereas Mepham (2005, p. 9) names the concept number sharing. Here, we will use the name
preemptive sets, which is more precise from a
mathematical point of view. The theory developed
here applies to Sudoku boards of all sizes.1
1

Sudoku boards can be classiﬁed into regular and nonregular boards. The formula for regular Sudoku boards
is: Let m denote the width and height of a Sudoku subboard, where m ≥ 2. Then the width and height of a
regular Sudoku board is m2 . The sizes of regular subboards and boards are given for a few values of m in the
following table:

Deﬁnition of the Sudoku Board

Subboard Width
and Height
m
2
3
4
5
6

Sudoku is played on a 9 × 9 board. There are
eighty-one cells on the board, which is broken
J. F. Crook is professor emeritus of computer science,
Winthrop University, Rock Hill, SC. His email address is
crookj@winthrop.edu.
I thank John Hohn, Kathryn Cooper, David Gill, and
Libby Neely for reading my paper. I also thank Loretta
Nethercot for giving me my ﬁrst two Sudoku books—on
Christmas Day, 2005, and on my seventieth birthday in
2007.

460

Board Width
and Height
m2
4
9
16
25
36

Number of Cells
on the Board
m2 × m2
16
81
256
625
1296

The most common nonregular Sudoku board is the
6 × 6, which consists of six nonoverlapping 2 × 3 subboards. The newspaper USA Today publishes 6 × 6
puzzles regularly.

Notices of the AMS

Volume 56, Number 4

Figure 1. The Sudoku board.

Figure 2. Example of a Sudoku puzzle.

Preemptive Sets and the Occupancy
Theorem
The single most important tool for solving Sudoku
puzzles is based on the deﬁnition of the solution
of a Sudoku puzzle.
Deﬁnition 1 (Sudoku Solution). The solution of
a Sudoku puzzle requires that every row, column, and box contain all the numbers in the set
[1, 2, . . . , 9] and that every cell be occupied by one
and only one number.
This deﬁnition implies that no row, column, or
box will have any number repeated. An example
of a Sudoku puzzle is shown in Figure 2. The more
diﬃcult puzzles can only be solved eﬃciently by
writing down in each empty cell the possible numbers that can occupy the cell. This list of possible
numbers for each cell is called the markup of the
cell. The markup of the example puzzle in Figure
2 is shown in Figure 3.
We now deﬁne preemptive sets, which is the
primary tool for solving Sudoku puzzles up to
the point where either (1) a solution is found or
(2) continuation requires randomly choosing one
of two or more numbers from the markup of an
empty cell.
Deﬁnition 2 (Preemptive Sets). A preemptive set
is composed of numbers from the set [1, 2, . . . , 9]
and is a set of size m, 2 ≤ m ≤ 9, whose numbers are potential occupants of m cells exclusively,
where exclusively means that no other numbers in
the set [1, 2, . . . , 9] other than the members of the
preemptive set are potential occupants...

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