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 n 2 xn 2 grid of the numbers 1-n, with n 2 nxn grids.

Example

Solution

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 p.pdf p.pdf American Scientist “Unwed Numbers” Brian Hayes e.aspx?id=3475&y=0&no=&content=true&pag e=4&css=print e.aspx?id=3475&y=0&no=&content=true&pag e=4&css=print