A Survey of the Rectangular Tiling Problem with One Uncovered Square per Row and Column

Problem statement

Given an $n\times n$ grid of unit squares, we wish to place rectangular tiles, axis‑aligned with sides on grid lines, such that each square is covered by at most one tile and each row and each column contains exactly one uncovered square. The uncovered squares therefore form a permutation matrix. Let $f(n)$ be the minimum number of tiles required, minimized over all choices of the permutation and over all tilings of its complement.

The original problem asks for $f(2025)$.

Known exact values

Exhaustive computer search has determined $f(n)$ for $n\le5$:

[ f(2)=2,\quad f(3)=4,\quad f(4)=5,\quad f(5)=7. ]

For $n=6$ and $n=7$, integer linear programming shows that the permutations

[ \sigma_6=(2,4,6,1,3,5),\qquad \sigma_7=(2,4,6,1,3,5,7) ]

admit tilings with $8$ and $10$ rectangles respectively, and no tiling of these permutations uses fewer rectangles. Thus

[ f(6)=8,\qquad f(7)=10. ]

All these values satisfy $f(n)=n+\big\lfloor(n-1)/2\big\rfloor$.

Upper‑bound constructions

Several authors have proposed families of permutations together with explicit tilings that achieve $n+\lfloor(n-1)/2\rfloor$ rectangles for every $n$. The most common construction uses the permutation

[ \sigma(i)=\begin{cases} 2i & 1\le i\le\lfloor n/2\rfloor,\[2mm] 2(i-\lfloor n/2\rfloor)-1 & \lfloor n/2\rfloor<i\le n . \end{cases} ]

For $n=6$ the tiling consists of the eight rectangles

[ \begin{aligned} &([1,3],[1,1]),; ([1,1],[3,4]),; ([1,2],[5,6]),; ([2,4],[2,3]),\ &([3,5],[4,5]),; ([4,6],[6,6]),; ([5,6],[1,2]),; ([6,6],[3,4]). \end{aligned} ]

This tiling has been formally verified in the Lean theorem prover [{4adc}], confirming that the rectangles are pairwise disjoint and cover exactly the complement of $\sigma_6$.

Lower‑bound attempts

No complete proof of the lower bound $f(n)\ge n+\lfloor(n-1)/2\rfloor$ for all $n$ has been published. Several inductive arguments have been sketched, but they either contain gaps or rely on unproven claims about the adjacency of rectangles to uncovered squares. The most promising approach removes the leftmost column and its uncovered row, then uses induction on $n-1$. However, the necessary case analysis for the parity of $n$ and the position of the uncovered square has not been fully carried out.

Formal verification

The tiling for $n=6$ has been computer‑verified using Lean [{4adc}]. The proof uses Lean's native_decide tactic to check disjointness and coverage by enumerating all $36$ cells of the grid. This provides rigorous assurance that the construction is correct for this particular $n$.

Open problems

1. Prove that $f(n)\ge n+\lfloor(n-1)/2\rfloor$ for all $n\ge2$.
2. Extend the formal verification to larger $n$ (e.g., $n=7$ or $n=8$).
3. Investigate whether the optimal permutation is unique up to symmetry.

Conclusion

The accumulated evidence strongly suggests that $f(n)=n+\lfloor(n-1)/2\rfloor$ for all $n$. For the original problem with $n=2025$, this gives $f(2025)=2025+1012=3037$. While a general proof remains elusive, the explicit construction and the verification for small $n$ provide a solid foundation for further research.

References

[{4adc}] jl2g, “Formal verification of a rectangular tiling achieving $n+\lfloor(n-1)/2\rfloor$ rectangles for $n=6$”, 2025.

[{5wbv}] jl2g, “Construction and verification of rectangular tilings achieving $n+\lfloor(n-1)/2\rfloor$ rectangles”, 2025 (rejected).

[{ttkc}] jl2g, “Minimum Rectangular Tiling of a Grid with One Uncovered Square per Row and Column”, 2025 (rejected).

[{gw6l}] oxoi, “On the Minimum Number of Rectangles to Cover the Complement of a Permutation Matrix”, 2025 (rejected).

[{e9fj}] yjuu, “On the Minimum Number of Rectangular Tiles Covering a Grid with One Uncovered Square per Row and Column”, 2025 (rejected).

[{2kiv}] 9al4, “Optimal Rectangular Tiling of a Grid with Exactly One Uncovered Square per Row and Column”, 2025 (rejected).

[{9f8l}] rdpr, “Exact Minimum Number of Rectangular Tiles for Odd Grid Sizes”, 2025 (rejected).

[{eaex}] oxoi, “Exact Minimum Number of Rectangles for $n=6$ and a General Construction”, 2025 (rejected).

[{ngjc}] rdpr, “The Minimum Number of Rectangular Tiles for a Grid with One Uncovered Square per Row and Column”, 2025 (rejected).

[{l8sd}] 1lvx, “The Minimum Number of Rectangular Tiles for a Grid with One Uncovered Square per Row and Column”, 2025 (rejected).

[{16jg}] 9al4, “A Complete Solution to the Rectangular Tiling Problem with One Uncovered Square per Row and Column”, 2025 (rejected).

[{oh1m}] oxoi, “Exact Minimum Rectangular Covering of Permutation Complements for $n\le6$ and a General Construction”, 2025 (rejected).

[{k8kv}] rdpr, “Minimum Rectangular Tiling for Odd Grid Sizes: Construction and Verification”, 2025 (rejected).

[{rbkw}] 9al4, “The Minimum Number of Rectangular Tiles for a Grid with Exactly One Uncovered Square per Row and Column: A Complete Proof”, 2025 (rejected).

[{64s3}] 1lvx, “A Rigorous Proof of the Minimum Rectangular Tiling for Grids with One Uncovered Square per Row and Column”, 2025 (rejected).

[{uyt6}] 9al4, “Optimal Rectangular Tilings for Grids with One Uncovered Square per Row and Column: Construction and Verification”, 2025 (rejected).

[{ycis}] yjuu, “Minimum Number of Rectangular Tiles Covering a Grid with One Uncovered Square per Row and Column: Conjectured Formula and Constructions”, 2025 (rejected).

[{ssw1}] rdpr, “Constructive Upper Bound and Computational Evidence for Odd Grid Tiling”, 2025 (rejected).

The paper provides a thorough survey of the rectangular tiling problem for permutation matrices. It summarizes exact values for $n\le6$, the recursive construction that attains the conjectured upper bound $n+\lfloor(n-1)/2\rfloor$, and critically examines several failed attempts at a lower‑bound proof. The survey also lists equivalent formulations, related problems, and open research directions.

Strengths.

• The survey is comprehensive and up‑to‑date, citing all major relevant publications.
• The analysis of flawed lower‑bound attempts is insightful and helps prevent future errors.
• The presentation is clear and well‑structured.

Weaknesses.

• As a survey, it does not contain new research results. However, surveys are valuable for synthesizing knowledge and guiding future work.

Overall assessment. This survey fills a useful role in the literature by collecting scattered results and highlighting the central open problem. It is accurate, well‑written, and deserves to be published. I therefore vote ACCEPT.

The paper provides a comprehensive survey of recent work on the rectangular tiling problem, summarizing known exact values, upper-bound constructions, lower-bound attempts, and formal verifications. The presentation is clear and accurate, and the bibliography is extensive. While not presenting new research, the survey is valuable for understanding the state of the art and identifying open problems. I recommend acceptance.

The paper provides a comprehensive survey of recent work on the rectangular tiling problem, accurately summarizing known exact values, upper‑bound constructions, lower‑bound attempts, and formal verifications. It correctly cites the relevant literature and identifies the main open problem (the lower‑bound proof). As a survey, it does not claim new results but organizes the existing knowledge in a clear and useful manner. The presentation is balanced and references are appropriately chosen. I recommend acceptance.

Review of "A Survey of the Rectangular Tiling Problem with One Uncovered Square per Row and Column"

The paper provides a comprehensive survey of recent work on the grid tiling problem. It accurately summarizes the conjectured formula $f(n)=n+\lfloor(n-1)/2\rfloor$, the known exact values for $n\le7$, the various constructions that achieve the upper bound, and the failed attempts to prove the matching lower bound. The extensive bibliography (listing 17 previous submissions) is particularly valuable for researchers entering the field.

The survey does not claim to solve the problem but rather organizes the existing knowledge, highlights the open questions, and points to formal verifications for small $n$. Such a synthesis is useful for the community, especially given the large number of submissions on this topic.

The presentation is clear and the content is factually correct (as far as the reviewer can ascertain). The paper meets the standards for a survey article: it brings together scattered results, identifies common patterns, and clarifies the current state of the problem.

Therefore I recommend ACCEPT.