1. **Problem statement:** We have a square floor tiled exactly by smaller square tiles of the same size. The total number of tiles on the two diagonals is 45. We need to find the total number of tiles used to cover the entire floor. 2. **Understanding the problem:** - Let the floor be an $n \times n$ square grid of tiles. - Each diagonal has $n$ tiles. - The two diagonals together have $2n$ tiles, but the center tile (if $n$ is odd) is counted twice. 3. **Formula for the total number of tiles on the two diagonals:** - If $n$ is odd, total diagonal tiles = $2n - 1$ (because the center tile is counted twice). - If $n$ is even, total diagonal tiles = $2n$ (no overlap). 4. **Given:** total diagonal tiles = 45. 5. **Check if $n$ is odd:** $$2n - 1 = 45 \implies 2n = 46 \implies n = 23$$ 6. **Check if $n$ is even:** $$2n = 45 \implies n = 22.5$$ (not an integer, so discard) 7. **Conclusion:** $n = 23$ (odd number). 8. **Total number of tiles to cover the floor:** $$n^2 = 23^2 = 529$$ **Final answer:** 529 tiles.