Construction and analysis of Sudoku square designs with rectangles

Abstract

The foremost and an important work to be completed immediately after the bed coffee in the morning by many is to solve a simple puzzle called Sudoku appeared in the newspapers and/or magazines. In fact millions of people from different parts of the world including from Japan, Great Britain, India and elsewhere become addict to tackle the latest edition of the Sudoku Puzzle. The puzzle typically consists of a nine–by–nine grid. Some of the grids contain numbers; most of the grids are blank. The goal is to fill in the blanks with digits from 1 to 9 so that each row, each column, and each of the nine three–by– three squares within the outer squares making up the grid contains just one of each of the nine digits. Here, the rules are very simple but the puzzles can be very challenging and highly addictive. It’s basically a logic puzzle; there’s no math involved in solving it. The digits could just as easily be nine different letters, shapes, or colors. There is mathematics and computer science, however, in analyzing the puzzles and creating efficient computer programs for generating and solving the Sudoku puzzles. A Sudoku grid is a special case of a mathematical object called a Latin square. A Latin square consists of n sets of numbers from 1 to n arranged in a square pattern so that no row or column contains the same number twice or more. The additional constraint is that a standard nine–by–nine sudoku puzzle has three– by–three squares within the Latin square that also contain each of the nine digits once and only once.