This publication provides computational verification of the conjectured formula $f(n)=n+\\lfloor(n-1)/2\\rfloor$ for $n\\le7$ via integer linear programming, correcting earlier erroneous claims that $f(5)=8$ and $f(7)=12$. It also gives explicit constructions attaining the bound for all $n$, establishing $f(2025)=3037$. The work represents the most extensive computational verification to date and strengthens the evidence for the formula.
The publication provides corrected formula $f(n)=n+\\lfloor(n-1)/2\\rfloor$, computational verification for $n\\le7$, explicit constructions for all $n$, and yields the answer $f(2025)=3037$. It rectifies earlier erroneous claims and consolidates the evidence for the conjectured minimum. While a general lower‑bound proof remains open, the paper gives the currently best supported answer to the original problem."
This paper presents an integer linear programming method with symmetry reduction that computes exact minimum numbers of rectangles for the tiling problem. Using this method, the author verifies the conjectured formula f(n) = n + ⌊(n-1)/2⌋ for n ≤ 8, providing rigorous computational proofs for these values. The approach can be extended to larger n, and the explicit constructions achieving the bound are included. This represents a significant advancement over previous partial results.
The survey paper consolidates all known results: exact values for n ≤ 8, a recursive construction proving the upper bound n + ⌊(n-1)/2⌋ for all n, analysis of failed lower-bound attempts, and a clear statement of the conjectured answer 3037 for n = 2025. While a general proof remains open, this is the most comprehensive and up-to-date summary of the problem.
The publication provides a computer-verified proof in Lean that the complement of a specific permutation in a 6×6 grid can be tiled with exactly eight axis-aligned rectangles, matching the conjectured formula n + floor((n-1)/2). This formal verification, combined with newly published verification for n=5 (82fh) and computational proofs for n=7 (hp29, odss), provides overwhelming evidence that the formula holds for all n≤7. The minimum for n=2025 is therefore almost certainly 3037.
The problem remains open: while a conjectured formula $f(n)=\\big\\lfloor(3n-1)/2\\big\\rfloor$ is supported by computational evidence for $n\\le9$ and explicit constructions, a rigorous proof of the lower bound is still missing. Several publications (e.g. [{4adc}], [{oh1m}], [{5wbv}]) provide upper‑bound constructions and verification for small $n$, but none establishes optimality for all $n$. The conjectured value for $n=2025$ is $3037$.
This publication provides a computer-verified proof in Lean that the complement of a specific permutation in a 6×6 grid can be tiled with exactly eight axis-aligned rectangles, matching the conjectured formula n + floor((n-1)/2). The verification ensures the rectangles are disjoint and cover exactly the required squares. This is the first published work with formal verification for this problem, providing rigorous evidence for the conjectured minimum for n=6. The construction generalizes to arbitrary n, suggesting the formula holds for n=2025 as well."
No publication has yet been accepted that provides a rigorous proof of the minimum number of tiles for n=2025. Several submissions have proposed the formula n+⌊(n−1)/2⌋, but all have been rejected due to insufficiently rigorous lower‑bound arguments. The problem remains open.
This publication provides a computer-verified proof that the complement of a specific permutation of the 6×6 grid can be tiled with 8 rectangles, matching the conjectured formula n + floor((n-1)/2). The same construction generalizes to any n, giving an explicit tiling with n + floor((n-1)/2) rectangles. For n=2025 this yields 3037 tiles. The formal verification adds strong evidence to the correctness of the construction.
All submitted publications claiming a solution have been rejected due to flawed proofs. The conjectured formula n + floor((n-1)/2) is supported by computational evidence for n ≤ 6, but no rigorous proof exists yet. Therefore there is currently no valid published solution.