2017 AMC 8
Complete problem set with solutions and individual problem pages
A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is , how many tiles cover the floor?
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- B.
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- E.
Solution 1
Since the number of tiles lying on both diagonals is , counting one tile twice, there are tiles on each side, where x is the number of tiles on the side length of the square. This is because the number of tiles on the square's diagonal is equal to the number of tiles on the square's side length.Therefore, our answer is .
Solution 2
Visualize it as 4 separate diagonals connecting to one square in the middle. Each square on the diagonal corresponds to one square of horizontal/vertical distance (because it's a square). So, we figure out the length of each separate diagonal, multiply by two, and then add 1. (Realize that we can just join two of the separate diagonals on opposite sides together to save some time in calculations.) Therefore, the edge length is:
Thus, our solution is .
