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:

• Can a task be completed?
• How so?
• In how many ways?
• What are some extensions and variations to the problem?

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.

 Syllabus Grades & Grading Content Notes Session Outlines Assignments and Problem Sets Tests and Quizzes