The Structure of Reduced Sudoku Grids and the Sudoku Symmetry Group

Siín K Jones ,Paul A Roach,Stephanie Perkins

doi:10.1155/2012/760310

Abstract

A Sudoku grid is a constrained Latin square. In this paper a reduced Sudoku grid is described, the properties of which differ, through necessity, from that of a reduced Latin square. The Sudoku symmetry group is presented and applied to determine a mathematical relationship between the number of reduced Sudoku grids and the total number of Sudoku grids for any size. This relationship simplifies the enumeration of Sudoku grids and an example of the use of this method is given.

Highlights

A Sudoku grid, Sx,y, is a n × n array subdivided into n minigrids of size x × y where n xy
A similar relationship is developed here between the total number of Sudoku grids and the number of reduced Sudoku grids. Such relationships have previously been given for “NRCSudoku” 7 and “2-Quasi-Magic Sudoku” 8 where the focus is the symmetry groups for these structures; symmetry groups have been defined for S3,2 9, S3,3 10, and the symmetry group of Sn,n is given in this paper
The Sudoku symmetry group, S, containing symmetry operations applicable to Sx,y consists of all homomorphisms of the structure of a Sudoku grid

Summary

Introduction

A Sudoku grid, Sx,y , is a n × n array subdivided into n minigrids of size x × y where n xy. A Latin square, or rectangle, is reduced if the values in the first row and column are in the natural order 1. Stones 6 surveys some well-known and some more recent formulae for Latin rectangles, their usefulness, and means to obtain approximate numbers. A similar relationship is developed here between the total number of Sudoku grids and the number of reduced Sudoku grids. Such relationships have previously been given for “NRCSudoku” 7 and “2-Quasi-Magic Sudoku” 8 where the focus is the symmetry groups for these structures; symmetry groups have been defined for S3,2 9, S3,3 10, and the symmetry group of Sn,n is given in this paper

The Sudoku Symmetry Group

Reductions

Example of the Enumeration Technique

Comparison of the Number of Known and Reduced Sudoku Grids