n- Queen Problem

Abstract

The n-queen problem is one of the problems that can be solved using mathematical concepts, (where  is a positive integer), where there are many methods to solve this problem as backtracking search technique[2, 9] and using global parallel genetic algorithm[3] and dynamic programming[10] and using DFS and BFS algorithms[12]. In this paper, we have studied some computation related to the chessboard, and we computed the number of squares associated and disassociated with square  on a chessboard of size  according to the queen's movement throught the theorms (1),(2) and (3) and apply it by the implementation on example for a chessboard of size , to represent this problem algebraically we used some concepts of computation in commutative algebra to solve the n-queen problem which are the concept of graded and the standard graded of rings, the hilbert series of graded rings, the dimension and multiplicity of the ring, additionally to the concept of the combinatorial dimension of an affine algebras. we connected between queen's movement on the chessboard and the graph by the definition of the queen's graph and coloring it in special case. using the computer algebra system SINGULAR[15], combinatorial dimension and we found the maximum number of queens can be placed on a chessboard so that no two of them attack one another in general case.