Vol. 3, No. 1, 2010

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Nontrivial solutions to a checkerboard problem

Meaghan Heires, Ryan Jones, Futaba Okamoto, Willem Renzema and Matthew Roberts

Vol. 3 (2010), No. 1, 109–127
Abstract

The squares of an $m×n$ checkerboard are alternately colored black and red. It has been shown that for every pair $m,n$ of positive integers, it is possible to place coins on some of the squares of the checkerboard (at most one coin per square) in such a way that for every two squares of the same color the numbers of coins on neighboring squares are of the same parity, while for every two squares of different colors the numbers of coins on neighboring squares are of opposite parity. All solutions to this problem have been what is referred to as trivial solutions, namely, for either black or red, no coins are placed on any square of that color. A nontrivial solution then requires at least one coin to be placed on a square of each color. For some pairs $m,n$ of positive integers, however, nontrivial solutions do not exist. All pairs $m,n$ of positive integers are determined for which there is a nontrivial solution.

Keywords
$m \times n$ checkerboard, coin placement, trivial solution, nontrivial solution
Primary: 05C15
Milestones
Received: 13 September 2009
Revised: 23 December 2009
Accepted: 29 December 2009
Published: 20 April 2010

Communicated by Ron Gould
Authors
 Meaghan Heires Department of Mathematics Western Michigan University Kalamazoo, MI 49008 United States Ryan Jones Department of Mathematics Western Michigan University Kalamazoo, MI 49008 United States Futaba Okamoto Mathematics Department University of Wisconsin La Crosse, WI 54601 United States Willem Renzema Department of Mathematics Western Michigan University Kalamazoo, MI 49008 United States Matthew Roberts Department of Mathematics Western Michigan University Kalamazoo, MI 49008 United States