Problem 11 · 2017 AMC 8
Easy
Geometry & Measurement
perfect-squarecareful-counting
A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?
Show answer
Answer: C — 361 tiles.
Show hints
Hint 1 of 2
37 is odd. Two diagonals of an n×n square would be 2n tiles — an even number — unless they cross and share one tile. The odd count is telling you they overlap at the center.
Still stuck? Show hint 2 →
Hint 2 of 2
So diagonal tiles = 2n − 1 (subtract the one shared center tile). Solve for n, then the floor is n².
Show solution
Approach: the odd count reveals a shared center tile
- Each diagonal of an n×n floor has n tiles. For two diagonals to share a single center tile, n must be odd — and then they cover 2n − 1 tiles, not 2n. The fact that 37 is odd confirms this overlap is happening.
- Set 2n − 1 = 37, so n = 19.
- Total tiles: 192 = 361.
- Sanity check: 361 is a perfect square (only choices 324, 361, 1296, 1369 are squares), and 19 is odd, matching the single-overlap picture. You'll see 2n−1 again for any pair of crossing diagonals / overlapping lines.
Mark:
· log in to save