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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 21 yellow tiles, how many gray tiles are there?

 Jun 23, 2023
 #1
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Let n be the number of rows of tiles in the patio. Then the number of yellow tiles is n+(n−1)=2n−1. Since there are 21 yellow tiles, we have 2n−1=21, so n=11.

The number of gray tiles is the total number of tiles minus the number of yellow tiles, which is n^2−(2n−1) = n^2−2n+1 = 112−2∗11+1 = 122​.

 Jun 23, 2023
 #6
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Thanks, that's right!

Guest Jun 25, 2023
 #2
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21-1 for the center, 

20/4 for the diagonals from the center to each corner. 5+5+1 for side length, 11^2 for total number of tiles, 121-21 yellow tiles for 100 gray tiles

 Jun 24, 2023
 #5
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That's right :)

Guest Jun 24, 2023
 #7
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That's the way I was going to do it, when I saw you had already posted this. 

 

1 in the middle leaves 20.  Run them out to each corner and it's 5 to each corner.   

 

That makes the square 11 tiles wide, thus the square comprises 121 tiles total.  

 

121 total, minus the 21 yellow tiles, leaves 100 gray tiles.   

.

Bosco  Jun 30, 2023
 #3
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Let n be the number of rows of tiles in the patio. Then the number of yellow tiles is n+(n−1)=2n−1. Since we know that this number is equal to 21, we have 2n−1=21, so n=11.

The number of gray tiles is the total number of tiles minus the number of yellow tiles, so it is n2=112=121​.

 Jun 24, 2023
 #4
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Let n be the number of tiles per side of the patio. Then the number of yellow tiles is n+(n−1)=2n−1. Since this is equal to 21, we have 2n−1=21, so n=11.

The number of gray tiles is then (n−2)^2 = (11−2)^2 = 9^2 = 81​.

 Jun 24, 2023

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