Sudoku

Introduction

• In this presentation I will cover the Sudoku puzzle, some basics of its complexity as well as specifically discussing the complexity of order 2 and order 3 Sudoku puzzles. I will also show and discuss the beginnings of NDFSMs for order 2 Sudoku puzzles and order 3 Sudoku puzzles to determine if a solution is correct.

Rules

• Most commonly, a sudoku puzzle is a 9x9 grid of the numbers 1-9 where in each row, column, and 3x3 grid each number is only used once.

• This is an “order 3” sudoku – an order n sudoku would be an n2xn2 grid of the numbers 1-n, with n2 nxn grids.

Example7 3 6 2 4

3 5 8

6 8 4 7 5

5 6 2 9

7 8

5 6 2 9

7 3 2 1 9

4 1 7

6 8 1 5 4

Solution1 5 7 3 8 9 6 2 4

3 2 4 5 6 7 9 8 1

9 6 8 1 4 2 3 7 5

8 7 3 4 5 6 2 1 9

4 9 2 7 1 8 5 6 3

5 1 6 2 9 3 7 4 8

7 3 5 8 2 4 1 9 6

2 4 9 6 3 1 8 5 7

6 8 1 9 7 5 4 3 2

How complex is it?

• For an order 3 sudoku you just have to be able to count to 9, so how hard are they really?

• How many different answers can there be?

Order 2 sudoku

• For order 2 sudoku puzzles there are 288 possible answers

• When symmetries are considered there are actually only 2 distinct puzzles with the remainder being some variation

Order 3 sudoku

• For order 3 sudoku puzzles there are 6,670,903,752,021,072,936,960 possible combinations

• Symmetrical operations only reduce this to 3,546,146,300,288

Beginnings of an order 2 DFSM

Basics of an order 3 DFSM

More complex data structure

• 2 dimensional array for checking– Number the columns, rows, and interior grids– Boolean

• 2 dimensional array for solving– Number the columns, rows, and interior grids– Each cell has a linked list of possible values– Some sort of relationship among the rows,

columns, and grids to identify what cells are affected by a change in each

Conclusion

• If you can solve sudoku puzzles you’re a genius!

• Both a human or computer would take a different approach to solve or verify a solution, as FSMs are probably not the best way to approach the problem

References

• “A Pencil-and-Paper Algorithm for Solving Sudoku Puzzles” J.F. Crook http://www.ams.org/notices/200904/tx090400460p.pdf

• American Scientist “Unwed Numbers” Brian Hayes http://www.americanscientist.org/issues/issue.aspx?id=3475&y=0&no=&content=true&page=4&css=print