1. For which natural numbers $n$ can a square grid $n \times n$ be cut into $1 \times 1$ and $2 \times 2$ squares such that there are an equal number of squares of each size?
Answer: all $n>5$, multiples of 5.

Solution. Let the board can be cut into $m$ squares $1 \times 1$ and $m$ squares $2 \times 2$. Then $n^{2}=m+4 m=5 m$, from which $n^{2} \vdots 5$ and $n \vdots 5$, and $m=\frac{n^{2}}{5}$. Therefore, $n$ that are not divisible by 5 do not fit.

Let $n \vdots 5$. Notice that for $n \geqslant 10$ the board $n \times n$ can be cut into squares of size $10 \times 10$ and $15 \times 15$. Therefore, it is sufficient to check that $n=10$ and $n=15$ satisfy the condition of the problem, while $n=5$ does not. For such $n$, the maximum number of non-overlapping squares of size $2 \times 2$ that can fit on the board $n \times n$ is 25, 49, and 4, while 20, 45, and 5 are required, respectively. Thus, the case $n=5$ does not fit. For $n=10$ and $n=15$, 20 and 45 squares of size $2 \times 2$ can be cut out, and the remaining part of the board can be cut into single cells. Therefore, $n=10$ and $n=15$ satisfy the condition.