The combinatorics of chessboards

The classic combinatorial problem know as The n-Queens Problem is to find the number of arrangements of n queens on an n x n chessboard such that no queen attacks another. In addition to numerous papers on the topic, the problem has many extensions. Examples include The Toroidal n-Queens Problem: To find the number of arrangements of n queens on a toroidal n x n chessboard such that no queen attacks another, The Cylinder n-Queens Problem: To find a similar solution for a cylindrical n x n chessboard; The Minimum Queens Problem: To place fewer than n queens on an n x n chessboard so that none attacks another, but so that they also together attack every unoccupied cell; The Reflecting Queens Problem; The Queens on an Infinite Chessboard Problem; and many others. The classic problem and each of its variations contains unsolved problems. In this paper, I present a new method for generating solutions to the classic problem using quasi-groups and I offer yet another extension to the problem, The Queens Problem on a Partial Chessboard, which is to arrange more than n queens on an n x n chessboard with m cells blocked such that no queen attacks another. Under what conditions do such arrangements exist? How many blocked cells are needed to yield solutions? I also present a computer simulation of the n-Queens Problem and of the Queens Problem on a Partial Chessboard that is a useful tool for mathematicians who study these problems.