Mathematical Assignment: Mathematical Exploration: Sudoku

Mathematical exploration: Sudoku

Sudoku has been a very widely used and popular game ever since 2005. In order to solve a sudoku puzzle, the player needs to use both logic, as well as trial-and-error. Whether we notice it or not, there is a lot of math involved in the puzzle: combinatorics which is used in counting the valid sudoku grids, group theory used to delineate the concept of when two grids are equal, and computational complexity with thoughts to solving sudoku puzzles.

Overview

The game as it is now was invented by an American named Howard Garns in 1979, and was published by Dell Magazines as “numbers in place”. In 1984, Maki Kaji of Japan published it in the magazine of his puzzle company Nikoli. He gave “numbers in place” the

The grid is divided into nine 3x3 blocks. Some of the 81 cells are filled in with numbers from the set {1,2,3,4,5,6,7,8,9} (the more numbers placed, the lower level of difficulty, the less numbers placed, the higher level of difficulty). These filled-in cells are called givens. The objective of the game is to fill in all of the cells using the nine numbers so that every row, column, and block consists of each number only once.This constraint on the rows, columns, and blocks is called the One Rule.

The puzzle described above is called a Sudoku of rank 3. A Sudoku of rank n is and n2×n2 square grid, separated into n2 blocks, each of size n×n. The numbers used to fill the grid in are 1, 2, 3, ..., n2, and the One Rule is still in effect.

The following is an example of a completed Sudoku

If a cell ends up having only one possible entry, it is a "forced" entry that you should fill in.

Another way to solve the puzzle is to pick a number and a row, column, or block. Note all the cells in the row, column, or block in which the number can be placed without violating the One Rule. If the number can only be placed in one cell in the area,that is the cell that the player should fill in. Once this is completed, the number inserted can be removed from being an option for any of the other cells in the area.

These two strategies are usually not enough to completely fill in a Sudoku grid. Usually, the player needs more complicated analysis methods to complete it, and sometimes they need to make a guess and continue, backtracking if the guess has more than one of the same number in the same area. If none of the entries are forced, it would be easier to pick one. Pattern recognition and logical deductions are also used in Sudoku.

Counting