Illinois State University Mathematics Department


MAT 305: Combinatorics Topics for K-8 Teachers



Perfect Covers of Chess Boards

Description and Exploration
An Exemplary Combinatorics Problem Situation

We will explore the problem of finding the perfect covers of an n-by-m rectangular chess board with dominoes measuring 2-by-1 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:

  1. What restrictions are there are n and m?
  2. How many ways are there to cover an n-by-m board?

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 8-by-8 chess board. Does a perfect cover exist for such a board?

Another extension is to consider perfect covers of a j-by-k-by-l rectangular prism, using a 1-by-1-by-2 domino.

We'll use the perfect cover problem and use our exploration to identify fundamental components of a problem-solving 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.


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