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

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Answer 1

Question

Chapter 1, Problem 1E

To determine

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

Expert Solution & Answer

Explanation of Solution

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

Case 1: Assume that one of m and n 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 m and n 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 m and n 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 m and n 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 m and n is even.

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

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

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Answer 2

Question

Chapter 1, Problem 1E

To determine

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

Expert Solution & Answer

Explanation of Solution

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

Case 1: Assume that one of m and n 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 m and n 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 m and n 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 m and n 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 m and n is even.

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

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

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Answer 3

Question

Chapter 1, Problem 1E

To determine

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

Expert Solution & Answer

Explanation of Solution

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

Case 1: Assume that one of m and n 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 m and n 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 m and n 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 m and n 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 m and n is even.

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

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

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Question

Chapter 1, Problem 1E

To determine

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

Expert Solution & Answer

Explanation of Solution

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

Case 1: Assume that one of m and n 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 m and n 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 m and n 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 m and n 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 m and n is even.

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

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

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Chapter 1, Problem 1E

To determine

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

Expert Solution & Answer

Explanation of Solution

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

Case 1: Assume that one of m and n 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 m and n 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 m and n 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 m and n 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 m and n is even.

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

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

Answer 6

Chapter 1, Problem 1E

To determine

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

Expert Solution & Answer

Explanation of Solution

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

Case 1: Assume that one of m and n 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 m and n 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 m and n 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 m and n 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 m and n is even.

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

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

Answer 7

Chapter 1, Problem 1E

To determine

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

Answer 8

Expert Solution & Answer

Explanation of Solution

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

Case 1: Assume that one of m and n 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 m and n 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 m and n 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 m and n 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 m and n is even.

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

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

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Ch. 1 - Prob. 1E Ch. 1 - Prob. 2E Ch. 1 - Imagine a prison consisting of 64 cells arranged...Ch. 1 - Verify that there is no magic square of order 2. Ch. 1 - Use de la Loubère’s method to construct a magic...Ch. 1 - 12. Use de la Loubère’s method to construct a...Ch. 1 - Construct a magic square of order 6. Ch. 1 - Show that the result of replacing every integer a...Ch. 1 - Let n be a positive integer divisible by 4, say n...Ch. 1 - Show that there is no magic cube of order 2.

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

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

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