The floor of a square room is tiled with square tiles. Along the two diagonals of room measuring 5x5 tiles there are nine tiles.
If there is 81 tiles along both diagonals, how many tiles are there on the floor?
The diagonal running from top left to bottom right will have one square in every column. So an n x n square will have n tiles in the diagonal.
The same will be true for the other diagonal, but if the dimensions of the square are odd, the diagonals will have a common tile at the centre.
n being even: Tiles in diagonal = 2n
n being odd: Tiles in diagonal = 2n - 1 (which itself is odd)
So if there are 81 tiles in the diagonal, n must be odd.
Solving 2n 1 = 81 2n = 82 n = 41.
Hence, there are 41 41 = 1681 tiles in the room.