[ Tiling with dominoes and tiles ]  Problem 78476 Topics: [ Tiling with dominoes and tiles $]$

A sheet of grid paper of size $5 \times$ n is filled with cards of size $1 \times 2$ such that each card completely occupies two adjacent cells. Each card has the numbers 1 and -1 written on it. It is known that the products of the numbers in the rows and columns of the resulting table are positive. For which $n$ is this possible?

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Since the product of the numbers in each column is positive, minus ones occur an even number of times in each column, and thus in the entire table. Therefore, the total number of cards is even, and the total number of cells in the table (which is twice as many) is divisible by 4. Thus, $5 n$ is divisible by 4; hence, $n$ is divisible by 4. We will now show that for any $n=4 k$, the required placement of cards is possible. For this, it is sufficient to place the cards on a sheet of size $5 \times 4$ (see figure) and repeat this arrangement $k$ times.

![](https://cdn.mathpix.com/cropped/2024_05_06_9fc5d17caeb61bf6ddabg-30.jpg?height=571&width=688&top_left_y=1118&top_left_x=685)

## Answer

For $n$ divisible by 4.

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