+0

# counting problem

-1
55
2
+255

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?

Feb 21, 2021
edited by wiseowl  Feb 21, 2021

#1
+31281
+2

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

Feb 21, 2021
#2
+118069
+1

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

Feb 21, 2021