How to tile an infinite sheet of graph paper with dominoes $2 \times 1$ so that each grid line cuts only a finite number of dominoes?

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The dividing lines of areas divided into different types of dominoes are diagonal lines.

## Solution

Draw two diagonal lines $y=x$ and $y=-x$, which divide the infinite sheet into four parts. Divide the upper and lower parts into vertical dominoes, and the left and right parts (including the remaining parts of the lower and upper sections) into horizontal dominoes. Then, a vertical grid line can only cut through horizontal dominoes, and there are only a finite number of them along each vertical line. The same applies to horizontal grid lines and vertical dominoes.

Consider coverings of a chessboard with dominoes, each containing two adjacent cells.

Which coverings are more numerous - those that contain the domino a1-a2, or those that contain the domino b2-b3?

## Hint

Correspond each covering that contains the domino b2-b3 to some covering that contains the domino a1-a2.

## Solution

Exclude from consideration all coverings that contain both the domino a1-a2 and the domino b2-b3. Consider the remaining coverings. Call a covering of type A if it contains the domino a1-a2, and a covering of type B if it contains the domino b2-b3. We will show that there are more coverings of type A.

Consider some covering of type B. Since it does not contain the domino a1-a2, the cell a1 is covered by the domino a1-b1. Therefore, the cell a2 can only be covered by the domino a2-a3. Thus, in a covering of type B, the left bottom $2 \times 3$ corner is covered by the dominoes b2-b3, a1-b1, a2-a3. Correspond to this covering of type B a covering of type A, in which the left bottom $2 \times 3$ corner is covered by the dominoes a1-a2, b1-b2, a3-b3, and the rest of the board is covered in the same way as in the original covering of type B. Thus, each covering of type B is associated with a covering of type A, and different coverings of type B correspond to different coverings of type A. However, a covering of type A consisting only of vertical dominoes does not correspond to any covering of type B. Therefore, there are more coverings of type A.

## Answer

There are more coverings that contain the domino a1-a2.