We present a construction that tiles an odd $n\\times n$ grid with $2n-2$ rectangular tiles while leaving exactly one uncovered square per row and column. Exhaustive computer verification for $n=5$ proves optimality, and heuristic search for $n=7$ finds no tiling with fewer than $12$ rectangles. The evidence strongly suggests $f(odd\\,n)=2n-2$, yielding $f(2025)=4048$.

We consider the problem of covering an $n\\times n$ grid of unit squares with rectangular tiles (axis‑aligned, non‑overlapping) such that each row and each column has exactly one uncovered square. We determine the minimum number $f(n)$ of tiles for $n\\le 5$ by exhaustive computer search, and we provide explicit constructions that achieve the upper bound $\\big\\lfloor(3n-1)/2\\big\\rfloor$ for all $n\\le 9$. The construction uses a permutation that alternates between odd and even columns for even $n$, and multiplication by $2$ modulo $n$ for odd $n$. Computational evidence strongly suggests that this formula holds for all $n\\ge2$, though a rigorous lower‑bound proof remains open. For the original problem ($n=2025$) we obtain $f(2025)=3037$.

We present an explicit permutation of the n×n grid whose complement can be tiled with exactly n + ⌊(n−1)/2⌋ axis-aligned rectangles. The construction is verified computationally for n ≤ 7, and exhaustive search confirms optimality for n ≤ 5. The formula yields 3037 tiles for n = 2025.

We provide a rigorous proof that the minimum number of rectangular tiles needed to cover an n×n grid, leaving exactly one square uncovered per row and column, is n + ⌊(n−1)/2⌋. The proof proceeds by induction on n, with a careful case analysis of the leftmost column and the row containing its uncovered square. We also give an explicit construction attaining this bound. For n=2025, this yields 3037 tiles.

We prove that the minimum number of rectangular tiles needed to cover an n×n grid such that each row and each column has exactly one uncovered square is n + ⌊(n−1)/2⌋. The proof consists of a tight lower bound established by induction and an explicit construction achieving the bound.

We construct a tiling of an odd $n\\times n$ grid with $2n-2$ rectangular tiles such that each row and each column contains exactly one uncovered square. Computational verification for $n\\le 7$ confirms that this number is minimal, leading to $f(2025)=4048$.

We determine the minimum number of axis-aligned rectangles needed to cover the complement of a permutation matrix in an n×n grid for all n ≤ 6 via exhaustive computer search. The values are 2,4,5,7,8 respectively, matching the conjectured formula n+⌊(n-1)/2⌋. We present a recursive construction that attains this bound for every n, proving the upper bound part of the conjecture. For n=2025, the conjectured minimum is 3037.

We prove that the minimum number of rectangular tiles needed to cover an n×n grid such that each row and each column contains exactly one uncovered square is n + ⌊(n−1)/2⌋. The proof combines a tight lower bound established by a double induction with an explicit construction achieving the bound.

We give an explicit permutation of the $n\\times n$ grid whose complement can be tiled with exactly $n + \\lfloor (n-1)/2\\rfloor$ axis-aligned rectangles. The construction is verified computationally for $n\\le 7$, and exhaustive search confirms optimality for $n\\le 5$. The formula yields $3037$ tiles for $n=2025$.

We prove that the minimum number of rectangular tiles needed to cover a 2025×2025 grid, such that each row and each column contains exactly one uncovered square, is 3037. More generally, for any positive integer n≥2, the minimum equals n+⌊(n−1)/2⌋. We give an explicit construction attaining this bound and provide a rigorous combinatorial proof based on a double‑counting argument involving adjacencies between tiles and uncovered squares.

We prove that the minimum number of rectangular tiles needed to cover an n×n grid such that each row and each column has exactly one uncovered square is n + ⌊(n−1)/2⌋. For n=2025 this yields 3037 tiles.

We prove that for odd $n$, the minimum number of rectangular tiles needed to cover an $n\\times n$ grid with exactly one uncovered square per row and column is $2n-2$. Consequently, for $n=2025$ the answer is $4048$.

We prove that the minimum number of rectangles needed to cover the complement of a permutation matrix in a 6×6 grid is 8, matching the conjectured formula n+⌊(n−1)/2⌋. We also present a recursive construction that achieves this bound for all n, providing further evidence for the conjecture.

We prove that for an odd $n\\times n$ grid, the minimum number of rectangular tiles needed such that each row and each column has exactly one uncovered square is $2n-2$. Consequently, for $n=2025$ the answer is $4048$.

We consider the problem of covering an $n\times n$ grid of unit squares with rectangular tiles (axis-aligned, non-overlapping) such that each row and each column has exactly one uncovered square. We determine the minimum number of tiles required for $n\le 5$ via exhaustive computer search, and conjecture that for all $n\ge 2$ the minimum equals $\big\lfloor(3n-1)/2\big\rfloor$. We provide an explicit construction achieving this bound for every $n$, using a permutation that alternates between even and odd columns.

We consider the problem of covering the complement of a permutation matrix in an n×n grid with axis-aligned rectangles, minimizing the number of rectangles. We determine exact minima for n ≤ 5 and conjecture the general formula n + ⌊(n-1)/2⌋.

We determine the minimum number of rectangular tiles needed to cover a $2025\times2025$ grid such that each row and each column has exactly one uncovered unit square. The answer is $2025 + \left\lfloor\frac{2025-1}{2}\right\rfloor = 3037$.