6. A tiling of an $n \times n$ square grid is formed using $4 \times 1$ tiles.
What are the possible values of $n$ ?
[A tiling has no gaps or overlaps, and no tile goes outside the region being tiled.]
SOlUTION
Each tile is of area 4 , so the area of the grid is a multiple of 4 . Hence $n$ is even. We consider two cases, determined by whether $n$ is a multiple of 4 or not.
$n=4 k$, for some positive integer $k$
In this case the grid can be tiled using $4 k^{2}$ tiles, with each row containing $k$ tiles. Figure 1 shows the case $k=3$.
$n=4 k-2$, for some positive integer $k$
In this case the grid cannot be tiled. To see this, colour the grid in a $2 \times 2$ chessboard pattern, as shown in Figure 2, where $k=3$.
Notice that, in whatever way a $4 \times 1$ tile is placed on the coloured grid, it will cover two cells of each colour. So if it were possible to tile the grid using $4 \times 1$ tiles, then they would cover an equal number of $1 \times 1$ squares of each colour.
However, there is an odd number of $2 \times 2$ squares in the grid, so that the number of $1 \times 1$ squares of one colour in the coloured grid is always four more than the number of $1 \times 1$ squares of the other colour.
Hence such a tiling is impossible.

Therefore the values of $n$ for which it is possible to tile an $n \times n$ grid with $4 \times 1$ tiles are the multiples of 4 (only).