Question

My square patio is tiled with square tiles, all the same size. All the tiles are gray, except the tiles along the two diagonals, which are all yellow. (The corners are yellow, the center is yellow, and all the tiles along the diagonal in between are yellow.) If there are $89$ yellow tiles, how many gray tiles are there

Answer 1

AOPS ANSWER:

Suppose that we have a square with dimensions
`$e * e$`. The diagonals each have
`$e$\\` tiles. If
`$e$\\` is even, the number of yellow tiles is
`$2e$`, because the
`2` diagonals don't intersect. If
`$e$` is odd, then the number of yellow tiles is
`$e+e-1=2e-1$`. (We have to subtract
`1` because the center tile is counted
`2` times). We know that there are
`89` yellow tiles, which is an odd number, so
`$89=2e-1$`. This implies that
`$e = 45$`. So the dimensions of the floor are
`$45 * 45$`, or
`2025\\` square units. Since all of the other tiles are gray, there are
`$2025-89=\boxed{1936}$` gray tiles.

Answer 2

1936

Explanation:

We can start by removing the center tile, because all four diagonals over lap on that tile. We'll add it back later. 88/4=22. which means there are 22 tiles per half, 44+1(the center tile) total tiles per side, which means there are a total of 1936 tiles.