+1678 My 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 81 yellow tiles, how many gray tiles are there?
+37146 Take out center tile then 80 tiles form four diagonal halves
so diagonal is 41 tiles
and each side is 41 tiles
total tiles = 41 x 41 = 1681 tiles all together subtract the 81 yellow tiles leaves 1600 grey tiles
+129852 The number of grey tiles can be found thusly :
Let n be the number of rows of tiles
The number of yellow tiles = n + (n -1) = 81
So
2n - 1 = 81
2n = 82
n = 41 rows
Number of grey tiles = ( n - 2) ( n - 1) + (n - 1) = ( n - 1) ( n - 2 + 1) = ( n - 1) ( n -1) =
(n -1)^2 = (41 -1) ^2 = 40^2 = 1600
