Illinois State University Mathematics Department
MAT 305: Combinatorics Topics for K8 Teachers 
Perfect Covers of Chess Boards 



We will explore the problem of finding the perfect covers of an nbym rectangular chess board with dominoes measuring 2by1 units. A perfect cover means that there are no gaps or overlaps when we cover the board entirely with the dominoes.
We first consider two questions:
We will work to determine the restrictions on m and n and encourage multiple ways to justify our results. We will then consider the second question, again exploring the strategies used.
To extend the problem, we can look at a pruned 8by8 chess board. Does a perfect cover exist for such a board?
Another extension is to consider perfect covers of a jbykbyl rectangular prism, using a 1by1by2 domino.
We'll use the perfect cover problem and use our exploration to identify fundamental components of a problemsolving approach to combinatorics problems:
In using these key questions to exemplify the type of investigations that will underscore course activities, we will emphasize the need to justify our efforts as we progress in solving a problem. We will talk about the need to consider or search for elegant and creative ways to approach problems.





