Introductory Combinatorics
Introductory Combinatorics
5th Edition
ISBN: 9780136020400
Author: Richard A. Brualdi
Publisher: Prentice Hall
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Chapter 1, Problem 1E
To determine

To show: An mandn chessboard has a perfect cover by dominoes if and only if at least one of m and n is even.

Expert Solution & Answer
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Explanation of Solution

To show this, we have to add 3 cases here.

Case 1: Assume that one of mandn is even and the second one is odd. Here, cover rows with even number of slabs, that is, cover the entire chessboard.

For an example, consider a 4-by-3 chessboard.

Case 2: Assume that both mandn are even. Now cover rows with even number of slabs, That is cover the entire chessboard.

For an example, consider a 4-by-4 chessboard.

Case 3: Suppose that both mandn are odd, and the chessboard has a perfect cover.

Here, the numbers of two color slabs are different, because one domino covers exactly one of the each.

So in this case, the chessboard can’t have a perfect cover. This is a contradicts.

Thus, our assumption is wrong. When both mandn are odd, then chess board does not have a perfect cover.

In other words, if the chessboard has a perfect cover, at least one of mandn is even.

For an example, consider a 4-by-4 chessboard.

Thus, by cases 1, 2, and 3, and mandn chessboard has a perfect cover by dominoes if and only if at least one of mandn is even.

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