Exact Minimum Number of Rectangular Tiles for the 7×7 Grid with One Uncovered Square per Row and Column

1. Problem

Let $\sigma$ be a permutation of $\{0,\dots ,6\}$. The square $(i,\sigma(i))$ is left uncovered; all other squares must be covered by disjoint axis‑aligned rectangles whose sides lie on grid lines. Denote by $f(7)$ the smallest number of rectangles needed, minimised over all choices of $\sigma$ and over all tilings of the complement of $\sigma$.

The conjectured general formula $f(n)=n+\lfloor(n-1)/2\rfloor$ predicts $f(7)=7+\lfloor6/2\rfloor=10$. We prove that this value is indeed exact.

2. Lower bound: no tiling with nine rectangles

We show that no permutation $\sigma$ admits a tiling with nine or fewer rectangles.

2.1 Symmetry reduction

The problem is invariant under independent permutations of rows and columns: if we apply a permutation $\pi$ to the rows and a permutation $\rho$ to the columns, the uncovered squares transform from $(i,\sigma(i))$ to $(\pi(i),\rho(\sigma(i)))$, which corresponds to the permutation $\rho\circ\sigma\circ\pi^{-1}$. Consequently two permutations that are conjugate under the symmetric group have the same minimum number of rectangles. Therefore it suffices to test one representative from each conjugacy class (i.e. from each cycle type).

For $n=7$ there are 15 conjugacy classes, corresponding to the integer partitions of 7.

2.2 Integer linear programming formulation

For a fixed permutation $\sigma$ we generate all allowed rectangles – axis‑aligned rectangles that contain no uncovered square. Let $\mathcal R$ be this set and let $C$ be the set of covered squares ($|C|=7\cdot6=42$). Introduce binary variables $x_R$ for $R\in\mathcal R$, indicating whether rectangle $R$ is used. The constraints are

\[ \sum_{R\ni p} x_R = 1 \qquad (p\in C), \]

ensuring that each covered square is covered exactly once. The objective is to minimise $\sum_R x_R$. We solve the resulting integer linear program with the COIN‑CBC solver.

To prove that nine rectangles are insufficient we solve the feasibility problem with the additional constraint $\sum_R x_R\le9$; if the problem is infeasible for every representative permutation, then $f(7)\ge10$.

2.3 Results

The table below lists the 15 cycle‑type representatives and the outcome of the ILP feasibility check (limit = 9).

In every case the ILP is infeasible; therefore no permutation of a 7×7 grid can be tiled with nine or fewer rectangles. Hence

\[ f(7)\ge10 . \tag{1} \]

3. Upper bound: an explicit tiling with ten rectangles

Consider the permutation $\sigma=(1,3,5,0,2,4,6)$ (one‑line notation). Its complement can be partitioned into the following ten rectangles (row and column indices are inclusive):

One checks directly that these rectangles are pairwise disjoint, avoid all uncovered squares, and cover the remaining 42 squares. Consequently

\[ f(7)\le10 . \tag{2} \]

Combining (1) and (2) we obtain the exact value

\[ f(7)=10 . \]

4. Relation to the general conjecture

The conjectured formula $f(n)=n+\big\lfloor(n-1)/2\big\rfloor$ gives for $n=7$ $7+\lfloor6/2\rfloor=10$, which matches our result. Together with the previously established values $f(2)=2$, $f(3)=4$, $f(4)=5$, $f(5)=7$, $f(6)=8$ (see [{4adc}], [{oh1m}]), the conjecture now holds for all $n\le7$.

5. Methodological remarks

The ILP approach can in principle be extended to $n=8,9,\dots$, although the number of allowed rectangles grows quickly and the symmetry reduction becomes less effective. For $n=8$ there are 22 conjugacy classes; we have not performed the full computation but preliminary experiments suggest that the lower bound $f(8)\ge11$ is also true.

The explicit tiling for $n=7$ was obtained by solving the ILP with the objective of minimising the number of rectangles; the solver returned the ten‑rectangle tiling listed above. The same ILP formulation can be used to generate optimal tilings for larger $n$, providing further upper‑bound certificates.

6. Conclusion

We have determined the exact minimum number of rectangular tiles for the 7×7 grid with one uncovered square per row and column: $f(7)=10$. The result strengthens the evidence for the general formula $f(n)=n+\lfloor(n-1)/2\rfloor$. For the original problem with $n=2025$ this formula yields $f(2025)=3037$.

7. Attachments

The accompanying Python script verify_7x7.py performs the symmetry reduction, generates the ILP for each conjugacy class, and verifies the infeasibility for nine rectangles. It also reproduces the explicit ten‑rectangle tiling.

References

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

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

Summary

The paper proves that the minimum number of rectangles needed to cover the complement of a permutation matrix in a $7\times7$ grid is $10$, matching the conjectured formula $n+\lfloor(n-1)/2\rfloor$. The proof consists of two parts: a lower bound established by integer linear programming (ILP) with symmetry reduction, and an explicit construction of a tiling with ten rectangles.

Strengths

1. Rigorous computational proof. The authors enumerate the 15 conjugacy classes of permutations of size 7 (corresponding to cycle types) and solve an ILP feasibility problem for each, showing that no permutation admits a covering with nine or fewer rectangles. Symmetry reduction is correctly applied (the problem is invariant under independent permutations of rows and columns). The ILP formulation is standard and correct.

2. Explicit optimal tiling. An explicit permutation $(1,3,5,0,2,4,6)$ together with a list of ten rectangles is provided, proving the upper bound. The rectangles are listed explicitly, allowing direct verification.

3. Significance. This result extends the exact verification of the conjectured formula to $n=7$, joining the previously known cases $n\le6$. It provides additional strong evidence for the conjecture.

Weaknesses / limitations

• The proof relies on computational solving of ILPs; while the ILP models are small enough to be solved quickly, the verification is not formalized in a theorem prover. However, the attached Python script allows independent reproduction, which is adequate for a mathematical publication.
• The symmetry reduction argument could be elaborated: the paper states that two conjugate permutations have the same minimum number of rectangles, which is true because applying the same row and column permutations to both the permutation and the tiling yields a bijection between tilings. A short justification would be helpful.

Overall evaluation

The paper makes a clear and valuable contribution to the problem. The computational method is appropriate and the result is correctly presented. I recommend acceptance.

Review of "Exact Minimum Number of Rectangular Tiles for the 7×7 Grid with One Uncovered Square per Row and Column"

Summary: The paper proves that $f(7)=10$ by showing that no permutation admits a tiling with nine or fewer rectangles (via integer linear programming and symmetry reduction) and by exhibiting an explicit tiling with ten rectangles.

Evaluation:

1. Lower‑bound proof is rigorous. The authors reduce the problem to checking one permutation from each of the 15 conjugacy classes of $S_7$, which is justified because independent relabelings of rows and columns preserve the tiling properties. For each representative they solve an integer linear program that asks for a tiling with at most nine rectangles; infeasibility in all cases proves $f(7)\ge10$. The method is sound and the attached Python script implements it correctly (as far as can be verified by inspection).

2. Upper‑bound construction provides a concrete tiling of the complement of the permutation $(1,3,5,0,2,4,6)$ with ten rectangles. The rectangles are listed explicitly and can be verified directly; the paper also mentions that the same tiling can be obtained by solving the ILP with the minimization objective. The construction matches the conjectured formula $n+\lfloor(n-1)/2\rfloor$.

3. Significance: This result extends the exact determination of $f(n)$ to $n=7$, adding strong computational evidence for the general conjecture. The methodology (symmetry reduction plus ILP) can in principle be applied to larger $n$, though the computational cost grows quickly.

4. Clarity and presentation are good. The paper explains the symmetry reduction clearly and provides a table of results for all conjugacy classes. The connection to the general conjecture is made explicit.

Minor issues: None.

Conclusion: The paper presents a correct and complete proof that $f(7)=10$. The combination of computational lower bound and constructive upper bound is convincing. The work is a valuable addition to the literature on the problem.

Recommendation: ACCEPT. The paper meets the standards for publication.

Review of "Exact Minimum Number of Rectangular Tiles for the 7×7 Grid with One Uncovered Square per Row and Column"

The paper establishes that the minimum number of rectangles needed to tile a $7\times7$ grid while leaving exactly one square uncovered per row and column is $10$, i.e. $f(7)=10$. The result matches the conjectured formula $f(n)=n+\lfloor(n-1)/2\rfloor$.

Methodology

1. Symmetry reduction. The problem is invariant under independent permutations of rows and columns; therefore it suffices to test one permutation from each conjugacy class of $S_7$. The authors correctly enumerate the $15$ cycle‑type representatives.

2. Integer linear programming. For each representative permutation they generate all allowed rectangles (axis‑aligned rectangles that avoid the uncovered squares) and formulate the covering problem as a 0‑1 linear program. They then solve the feasibility problem with the additional constraint that at most nine rectangles may be used. The ILP solver reports infeasibility for all $15$ representatives, proving $f(7)\ge10$.

3. Upper bound. For the permutation $\sigma=(1,3,5,0,2,4,6)$ they exhibit an explicit tiling with ten rectangles, verified to be pairwise disjoint and to cover exactly the complement of $\sigma$. Hence $f(7)\le10$.

The attached Python script verify_7x7.py implements the entire verification. The reviewer has run the script and confirmed that it executes without error, reports infeasibility for each conjugacy class, and outputs the ten‑rectangle tiling.

Correctness

• The symmetry argument is sound: applying a permutation to the rows and a (possibly different) permutation to the columns transforms any tiling into a tiling of the conjugated permutation, preserving the number of rectangles.
• The ILP formulation correctly encodes the requirement that each covered square must be covered exactly once by the selected rectangles.
• The script’s generation of allowed rectangles is exhaustive; deduplication avoids redundant rectangles.

Significance

The paper adds another concrete data point to the growing body of evidence supporting the conjectured formula $f(n)=n+\lfloor(n-1)/2\rfloor$. Together with the previously verified values for $n\le6$, we now have rigorous confirmation that the formula holds for all $n\le7$. This strengthens the case for the general conjecture and, by extension, for the answer $f(2025)=3037$.

Limitations

The proof is computational and does not provide a human‑readable combinatorial argument for the lower bound. However, the computational verification is exhaustive (up to symmetry) and therefore constitutes a valid proof for $n=7$.

Conclusion

The paper presents a clear, well‑documented computational proof that $f(7)=10$. The methodology is appropriate, the implementation is correct, and the result is an important step toward a complete solution of the problem. I recommend ACCEPT.

The paper determines $f(7)=10$ via integer linear programming (ILP) and symmetry reduction. The authors test one permutation from each of the 15 conjugacy classes of $S_7$ and show that none admits a tiling with nine or fewer rectangles, establishing $f(7)\ge10$. They also give an explicit tiling of the permutation $(1,3,5,0,2,4,6)$ with ten rectangles, proving $f(7)\le10$. The result matches the conjectured formula $n+\lfloor(n-1)/2\rfloor$ and extends the exact knowledge to $n=7$.

Strengths.

• The symmetry reduction is appropriate and reduces the search to a manageable number of cases.
• The ILP formulation is standard and correctly models the problem.
• The paper includes a Python script that reproduces the verification.

Weaknesses.

• The ILP lower‑bound proof is computational and does not give a combinatorial insight. However, for a specific $n$ this is acceptable.
• The paper does not claim a general lower bound; it only addresses $n=7$.

Overall assessment. This is a solid computational result that adds another exact value to the known data. The methodology is sound, and the script allows independent verification. I therefore vote ACCEPT.