Prove that the $1*1$ tile should be in the center We have a $5*5$ room and we want to fill it with $8$,$3*1$ tiles and $1$,$1*1$ tile.Prove that the $1*1$ tile should be in the center.
My attempt:I colored it with three colors every $1*3$ tile should fill one of each color because the tiles with the color $B$ are one more than the others the $1*1$ tile should be in one of the tiles with the color $B$.The center is also colored with $B$ but there are also other tiles colored with $B$.I don't know how to do to show that the $1*1$ tile should be in the center that is colored with $B$.Any hints?

 A: Hint: You're on the right track. Note that your particular coloring isn't the only way to color like this -- you can, for example, start with $ABCAB$ from right to left in the first row instead. Can you narrow down the possibility of the position of the $1\times 1$ tile using different colorings?
A: Obviously such a coloring cannot directly tell you that the $1\times 1$ piece has to be in the centre, since the centre has the same color of many other squares. But such a coloring tells you for sure that the $1\times 1$ piece has to be over a square marked with $B$. Now take a similar coloring, but with the diagonals being colored by $B'$ being parallel to the main diagonal of the square. For a similar argument, the $1\times 1$ piece has to fall over a square marked by $B'$, but the only square having color $B$ in the original coloring and color $B'$ in the secondary coloring is the centre of the square.
At last, you just have to prove that by placing the $1\times 1$ piece at the centre of the square, the remaining part can be tiled by $3\times 1$ pieces:
 
