Consider a grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile. Determine the minimum number of tiles Matilda needs to place so that each row and each column of the grid has exactly one unit square that is not covered by any tile.
tiles (or the specific minimum from the official solution).
tiles (or the specific minimum from the official solution).
tiles (or the specific minimum from the official solution).
tiles (or the specific minimum from the official solution).
Hint 1: The uncovered cells form a permutation matrix (one per row, one per column). So 2025 specific cells are left uncovered.
Hint 2: Each rectangular tile is axis-aligned and covers a contiguous block. The uncovered cells 'interrupt' potential large tiles.
Hint 3: Count the minimum tiles by analyzing how the permutation pattern creates mandatory tile boundaries.
Step 1: The uncovered cells form a permutation matrix: exactly one uncovered cell per row and per column. So there are exactly 2025 uncovered cells, forming a permutation of (where is uncovered in row ).
Step 2: The remaining cells must be covered by non-overlapping rectangles aligned to the grid.
Step 3 (Lower bound): Each tile covers a contiguous rectangle. The uncovered cells 'break' rows and columns, creating natural segments. In each row , the uncovered cell at column splits the row into at most 2 segments, requiring at least 1 tile if or 2 tiles otherwise. But tiles can span multiple rows.
Step 4: A careful counting argument (or a dual argument using the number of 'corners' created by the permutation) gives the minimum number of tiles.
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