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Chapter 1: Problem 46

Prove or disprove that you can use dominoes to tile a \(5 \times 5\) checkerboard with three corners removed.

Short Answer

Expert verified
You cannot tile the 5x5 checkerboard missing three corners because there is an imbalance in the remaining squares鈥� color.

Step by step solution

01

Understand the Domino and Checkerboard

A domino covers exactly two squares on a checkerboard. A standard checkerboard has alternating colored squares.
02

Calculate the Total Number of Squares

A standard 5x5 checkerboard has 25 squares. Removing three corners means 25 - 3 = 22 squares remain.
03

Determine If Domino Tiling is Possible

Since each domino covers 2 squares, an even number of squares are needed to use all dominoes. As 22 is even, it might be possible.
04

Analyze the Difference in Color

In a 5x5 checkerboard, there are 13 of one color and 12 of another color. Removing three corners, which are two of one color and one of the other, results in an imbalance: - If whites removed: 11 whites, 12 blacks.- If blacks removed: 12 whites, 11 blacks.Therefore, an imbalance in color always exists.
05

Conclusion

Since each domino must cover one white and one black square, the imbalance in color (11 vs. 12) makes it impossible to tile the modified checkerboard with dominoes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

checkerboard tiling
In a typical checkerboard tiling problem, the board consists of alternating colored squares, usually black and white, in a grid pattern. When working with dominoes, each domino covers two adjacent squares on the board.

The underlying concept here involves systematically arranging the dominoes so that they fully cover all the squares without any overlaps or gaps. In our specific example with a 5x5 checkerboard, there are 25 squares, but with three corners removed, we are left with only 22 squares.

Since a single domino covers two squares, the problem boils down to whether 22 squares can be perfectly covered using dominoes. This leads us to our next concept: the even and odd properties of numbers.
even and odd properties
The properties of even and odd numbers are fundamental in determining the possibility of tiling a checkerboard with dominoes. In this problem, we saw that after removing three corners, we are left with 22 squares.

Since each domino covers two squares, to completely tile the board, the total number of remaining squares must be even. Here, 22 is an even number, suggesting it might be possible to tile the board.

However, merely having an even number of squares isn't sufficient for successful tiling, especially in modified boards like ours. This is because the color arrangement of the remaining squares affects the tiling pattern, which brings us to our next key concept: color imbalance.
color imbalance
Checkerboards typically have an equal distribution of two colors. On a 5x5 checkerboard, there are 13 squares of one color and 12 of another. However, when we remove three corners, we disturb this balance.

To understand the importance of color balance, remember that each domino, placed on an unaltered checkerboard, will cover one square of each color. This pattern is essential for tiling; otherwise, you'll end up with mismatched pairs.

Removing three corners not only reduces the number of squares but also introduces a color imbalance. Depending on which corners are removed, we are left with either 11 whites and 12 blacks or vice versa. This imbalance implies that no matter how you try to arrange the dominoes, one color will always be left unpaired. Consequently, it becomes impossible to completely tile the 5x5 checkerboard with three corners removed using dominoes.

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