Illinois State University Mathematics Department
MAT 409: Topics in Algebra and Combinatorics for K8 Teachers 




Combinatorics in
the Real World



Perfect
Covers of Chess Boards


The Pigeonhole
Principle


Summation
and Product Notation 

Basic Counting Strategies


Course Information







Assignment #1
Read
Early SendIns


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 in small groups 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 among the groups.
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 dominolike prism.
We'll carry out an activity designed to remix the student groups and to encourage you to get to know your colleagues. We will next return to 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.