For which $n \geqslant 1$ can a square grid of size $n \times n$ be paved with pieces in the shape of $\mathrm{T}$ (each of which covers 4 cells of the grid) without them overlapping or overhanging? Rotations are allowed.
Since each T covers 4 squares, the grid contains an even number of squares, so $n$ is even. If we color the grid like a chessboard, there are therefore as many white squares as black squares. There are two types of $\mathrm{T}$: those that cover three white squares and one black square, and those that cover three black squares and one white square. Since there are as many white squares as black squares, there must be as many $\mathrm{T}$ of type 1 as $\mathrm{T}$ of type 2. Therefore, there is an even number of T. We then obtain that $n \times n$ is divisible by 8, so $n$ is divisible by 4. If $n$ is divisible by 4, it is possible to cover the grid: we can tile a $4 \times 4$ square with $4 \mathrm{~T}$ and repeat this construction in both directions. The grid is therefore tileable if and only if $n$ is divisible by 4.