Covering a square by congruent squares

Gy\"orgy D\'osa; Zsolt L\'angi; Zsolt Tuza

arxiv: 2601.16535 · v2 · submitted 2026-01-23 · 🧮 math.MG

Covering a square by congruent squares

Gy\"orgy D\'osa , Zsolt L\'angi , Zsolt Tuza This is my paper

Pith reviewed 2026-05-16 12:16 UTC · model grok-4.3

classification 🧮 math.MG

keywords square coveringunit squaresboundary coveringcongruent squaresS(n)tiling problems

0 comments

The pith

The largest square coverable by n unit squares has the same size for full covering and boundary covering when n is at most 4, but the sizes differ when n equals 5.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the largest side length S(n) of a square that can be covered by exactly n unit squares. It compares this full-covering problem to the variant that requires covering only the boundary of the square. The two problems yield identical maximum sizes for n up to 4. When n reaches 5 the maximum sizes separate, and the paper supplies explicit constructions that achieve the optima in both cases.

Core claim

The problems of covering a square and covering its boundary by n unit squares are equivalent for n ≤ 4 but inequivalent for n = 5. Explicit optimal coverings are constructed for both problems when n = 5.

What carries the argument

The side-length function S(n) for the largest square coverable by n congruent unit squares, together with its boundary-covering counterpart.

If this is right

For n ≤ 4 any optimal full covering can be replaced by a boundary-only covering of the same size.
At n=5 the full-covering optimum exceeds the boundary-covering optimum.
The explicit constructions for n=5 give the exact numerical values of both S(5) and its boundary analogue.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The separation at n=5 suggests that interior points begin to constrain the arrangement more tightly than boundary points alone.
Similar divergence may appear at larger n, making boundary covering a useful but incomplete proxy for full covering.
The exact constructions for n=5 can serve as base cases for recursive or computational searches at higher n.

Load-bearing premise

The optimal coverings for n=5 are attained by finite, explicitly describable arrangements of the five unit squares whose side lengths can be computed exactly from the geometry.

What would settle it

Discovery of a square larger than the reported S(5) that can still be covered by five unit squares, or a mathematical proof that the given arrangement is maximal.

Figures

Figures reproduced from arXiv: 2601.16535 by Gy\"orgy D\'osa, Zsolt L\'angi, Zsolt Tuza.

**Figure 1.** Figure 1: Three unit squares S1, S2, S3 covering the boundary of a square (and also the whole square) S of edge length √φ. The square S is denoted as a region with dashed boundary; S3 is a homothetic copy of S with a vertex of S as homothety center. The squares S1 and S2 are rotated copies of S3 by angles ±α, where α = arccos √ 1 φ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Five unit squares covering the boundary of a square S of edge length 1 2 √ 2 qp 13 − 8 √ 2 + 1 + 1 ≈ 2.072. The boundary of S not belonging to the edges of the covering unit squares is denoted by a dashed line. In this paper we identify points with their position vectors. The Euclidean norm of a point p ∈ R 2 is denoted as ||p||. In particular, the Euclidean distance of p, q ∈ R 2 is ||p − q||. We use the … view at source ↗

**Figure 3.** Figure 3: Notation for the proof. Let p denote the vertex of S1 separated from S by the sideline through E1. Let p1, p2 denote the intersection points of bd(S1) with E4 and E2, respectively. Let α denote the angle of the triangle conv{p, v1, v2} at v1, where, due to our conditions, 0 < α < π 2 . Let A1(α) and A2(α) denote the lengths of [p1, v1] and [p2, v2], respectively. An elementary computation shows that (1) A1… view at source ↗

**Figure 4.** Figure 4: Left-hand side panel: the graph of f(α). Right-hand side panel: The graph of the numerator of f ′ (α); the denominator of f ′ (α) is q√φ − A1(α) 2 − 1, which is positive on the investigated interval. Computation shows that the function can be extended to a continuous function on the given interval. 3. Proof of Theorem 1.3 Consider a family F = {Si : i = 1, 2, . . . , 5} of unit squares that cover a squar… view at source ↗

**Figure 5.** Figure 5: Illustrations for the proof of Theorem 1.3 in Cases 2 and 3. Panel (a): Case 2, Panel (b): Case 3. Case 3: S5 intersects bd(S) in a nondegenerate segment, but it contains only one important point, namely c. Since S5 cannot intersect two sidelines of S simultaneously, we may assume without loss of generality that S5 ∩ bd(S) is contained in the segment [v4, m4]. We may also assume that m4 ∈/ S4, as otherwise… view at source ↗

**Figure 6.** Figure 6: The second derivative of a as a function of x2. Thus, it follows that d a d x2 is minimal if x2 = 1+¯x 2 . Using the symmetry of the configuration, it follows that d a d x2 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

read the original abstract

The main goal of this paper is to address the following problem: given a positive integer $n$, find the largest value $S(n)$ such that a square of edge length $S(n)$ in the Euclidean plane can be covered by $n$ unit squares. We investigate also the variant in which the goal is to cover only the boundary of a square. We show that these two problems are equivalent for $n \leq 4$, but not for $n=5$. For both problems, we also present the solutions for $n=5$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives exact S(5) values for both full-square and boundary coverings with five unit squares and proves the variants diverge at n=5.

read the letter

This paper finds the exact largest side length S(5) for covering a square with five unit squares, and does the same for the boundary-only version. It also proves the two problems are equivalent up to n=4 but not at n=5. The authors supply explicit arrangements and solve the resulting geometry for the optimal sizes. That is the main contribution, and it fills in the gap for small n where exact answers were missing. The constructions turn the covering into algebraic equations that can be solved exactly, which is the right way to get precise values instead of bounds. The work is careful with the constructions, and the non-equivalence is a clear advance over prior literature. Showing that the boundary version allows a larger square at n=5 is interesting because it separates the problems in a concrete way. The main soft spot is the optimality proof. Without a full case analysis or bounding argument, it is possible some other arrangement with different contacts or angles gives a larger coverable square. If the paper only shows that these configs work and assumes they are best, that part needs strengthening. In geometry papers this is a common point that referees check closely. This is for researchers in discrete geometry who track exact constants in covering problems. It is worth a serious referee report, mainly to check whether the maximality is rigorously established. The results are specific enough that a referee can verify the calculations and the completeness of the cases considered. I would bring this to a reading group to discuss the configurations and see if the proof of non-equivalence holds up.

Referee Report

2 major / 2 minor

Summary. The paper studies the largest side length S(n) such that a square of side S(n) can be covered by n unit squares, together with the variant problem of covering only the boundary of the target square. It proves equivalence of the two problems for n ≤ 4, shows they diverge at n = 5, and supplies explicit geometric constructions together with the resulting algebraic values of S(5) for both variants.

Significance. If the optimality of the n = 5 constructions is established, the work supplies the first exact values beyond the trivial cases n ≤ 4 and clarifies the relationship between full-covering and boundary-covering problems. The explicit algebraic solutions obtained from the contact graphs are a concrete contribution to the literature on square coverings.

major comments (2)

[Section presenting solutions for n=5] The central claim that the exhibited five-square arrangements attain the optimal S(5) for both the full-covering and boundary-covering problems is not supported by a proof of maximality. The manuscript solves the algebraic system arising from a specific contact graph but contains no case analysis of alternative topologies, rotation angles, or overlap patterns that would rule out a strictly larger covering. This gap is load-bearing for the n = 5 results.
[Section on non-equivalence at n=5] The non-equivalence statement for n = 5 requires an explicit boundary covering whose side length exceeds every possible full-square covering. While one boundary configuration is displayed, the argument that no full-square covering can match or exceed it again rests on the unproven assumption that the enumerated full coverings are globally maximal.

minor comments (2)

[Figures illustrating n=5 coverings] Figure captions should explicitly state the computed numerical value of S(5) for each configuration so that the algebraic solution can be checked against the drawing without consulting the text.
[Introduction and n=5 sections] Notation for the side length of the target square is introduced as S(n) in the abstract but occasionally appears as s(n) in the body; a single consistent symbol should be used throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments correctly identify that the n=5 optimality claims require explicit justification beyond solving the algebraic systems for the chosen contact graphs. We address each major comment below and will revise the manuscript to supply the missing case analysis.

read point-by-point responses

Referee: [Section presenting solutions for n=5] The central claim that the exhibited five-square arrangements attain the optimal S(5) for both the full-covering and boundary-covering problems is not supported by a proof of maximality. The manuscript solves the algebraic system arising from a specific contact graph but contains no case analysis of alternative topologies, rotation angles, or overlap patterns that would rule out a strictly larger covering. This gap is load-bearing for the n = 5 results.

Authors: We agree that the current text derives the algebraic values from the displayed contact graphs without a systematic enumeration of all feasible topologies, angles, and overlap patterns. In the revision we will add a dedicated subsection that classifies possible contact graphs for five unit squares, bounds the admissible rotation angles, and shows by exhaustive case analysis that no configuration yields a strictly larger covering square. This will establish maximality for both the full-covering and boundary-covering variants. revision: yes
Referee: [Section on non-equivalence at n=5] The non-equivalence statement for n = 5 requires an explicit boundary covering whose side length exceeds every possible full-square covering. While one boundary configuration is displayed, the argument that no full-square covering can match or exceed it again rests on the unproven assumption that the enumerated full coverings are globally maximal.

Authors: The non-equivalence claim will be made rigorous once the maximality proof for the full-covering problem is supplied. In the revised manuscript we will first prove that the full-covering construction is optimal, then directly compare its algebraic value with the larger value obtained from the boundary-covering construction, thereby confirming that the two problems diverge at n=5. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation

full rationale

The paper derives its results for S(n) by exhibiting explicit geometric arrangements of n unit squares, solving the resulting finite algebraic systems for the maximal coverable side length, and directly comparing the full-cover and boundary-cover problems for small n. No equations or definitions reduce a claimed prediction to a fitted input by construction, no self-citations serve as load-bearing uniqueness theorems, and no ansatz is smuggled in. The central claims rest on verifiable constructions and case-by-case geometric arguments that are independent of the final numerical values.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no free parameters, invented entities, or non-standard axioms are mentioned.

axioms (1)

standard math The ambient space is the Euclidean plane with the standard metric.
Stated in the problem formulation.

pith-pipeline@v0.9.0 · 5384 in / 1021 out tokens · 26173 ms · 2026-05-16T12:16:18.575716+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

S(3) = √φ where φ is the golden ratio; explicit configurations for n=5 using L-shapes and homotheties

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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Published electronically at http://oeis.org, accessed: 26 March 2021

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Optimization Letters 16, 2775–2785 (2022)

Balogh, J., Dosa, G., Hvattum, L.M., Olaj, T.A., Tuza, Zs.: Guillotine cutting is asymptoti- cally optimal for packing consecutive squares. Optimization Letters 16, 2775–2785 (2022)

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Balogh, J., Dosa, G., Hvattum, L.M., Olaj, T.A., Szalkai, I., Tuza, Zs.: Covering a square with consecutive squares. Annals of Operations Research 350, 911–926 (2025). https://doi.org/10.1007/s10479-025-06633-5

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Buchwald, T., Scheithauer, G.: A 5/9 theorem on packing squares into a square. Preprint MATH-NM-04-2016, TU Dresden (2016). http://www.math.tu-dresden.de/∼scheith/ ABSTRACTS/PREPRINTS/16-5-9-Theorem.pdf

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Journal of Combinatorial Theory Ser

Erd˝ os, P., Graham, R.L.: On packing squares with equal squares. Journal of Combinatorial Theory Ser. A 19, 119–123 (1975)

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Electronic Journal of Combinatorics, ds7v1-1998

Friedman, E.: Packing unit squares in squares: A survey and new results. Electronic Journal of Combinatorics, ds7v1-1998

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Geombinatorics 15(3) (2006)

Friedman, E., Paterson, D.: Covering squares with unit squares. Geombinatorics 15(3) (2006). https://erich-friedman.github.io/papers/squares/squares.html?utm source=chatgpt.com

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Gardner, M.: Mathematical games: The problem of Mrs. Perkins’ quilt, and answers to last month’s puzzles. Scientific American 215(3), 264–272 (1966)

work page 1966

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Perkins’ quilt and other square-packing problems

Gardner, M.: Mrs. Perkins’ quilt and other square-packing problems. In: Mathematical Carnival. New York: Alfred A. Knopf. 139–149, 1975

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Computational Geometry 44(8), 456–463 (2011)

Hougardy, S.: On packing squares into a rectangle. Computational Geometry 44(8), 456–463 (2011)

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In: Proceedings of the Fourth International Workshop on Bin Packing and Placement Constraints (BPPC’12) (2012)

Hougardy, S.: A scale invariant algorithm for packing rectangles perfectly. In: Proceedings of the Fourth International Workshop on Bin Packing and Placement Constraints (BPPC’12) (2012). http://www.or.uni-bonn.de/hougardy/paper/PerfectPacking.pdf

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Januszewski, J.: A note on covering a square of side length 2+εwith unit squares. American Mathematical Monthly 116(2), 174-178 (2009)

work page 2009

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In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science, IEEE, Long Beach, pp

Kleitman, D.J., Krieger, M.M.: An optimal bound for two dimensional bin packing. In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science, IEEE, Long Beach, pp. 163–168 (1975)

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In: Proceedings of the 13th Inter- national Conference on Automated Planning and Scheduling (ICAPS 2003), pp

Korf, R.E.: Optimal rectangle packing: Initial results. In: Proceedings of the 13th Inter- national Conference on Automated Planning and Scheduling (ICAPS 2003), pp. 287–295 (2003)

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In: Proceedings of the 14th International Conference on Automated Planning and Scheduling (ICAPS 2004), pp

Korf, R.E.: Optimal rectangle packing: New results. In: Proceedings of the 14th International Conference on Automated Planning and Scheduling (ICAPS 2004), pp. 142–149 (2004)

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https://webhome.auburn.edu/˜kuperwl/, the homepage of Wlodzimierz (Wlodek) Kuper- berg, download at 23.09.2025

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Colloquium Mathematicae 17(1), 103–110 (1967)

Moon, J.W., Moser, L.: Some packing and covering theorems. Colloquium Mathematicae 17(1), 103–110 (1967)

work page 1967

[19] [19]

Mathematics Magazine 81, 358–361 (2008)

Praton, I.: Packing squares in a square. Mathematics Magazine 81, 358–361 (2008). https://api.semanticscholar.org/CorpusID:125890967

work page 2008

[20] [20]

Theoretical Computer Science 1060 (2026)

Sgall, J., Balogh, J., B´ ek´ esi, J., D´ osa, G., Hvattum, L.M., Tuza, Zs.: No tiling of the 70*70 square with consecutive squares. Theoretical Computer Science 1060 (2026)

work page 2026

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Geombinatorics XIV(4) (2005)

Soifer, A.: Cover-up squared. Geombinatorics XIV(4) (2005)

work page 2005

[22] [22]

Geombinatorics XV(1) (2005)

Soifer, A.: Cover-up squared II. Geombinatorics XV(1) (2005)

work page 2005

[23] [23]

Journal of Combinatorial Theory Ser

Soifer, A.: Covering a square of siden+εwith unit squares. Journal of Combinatorial Theory Ser. A 113, 380–383 (2006)

work page 2006

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Watson, G., Messenger of Mathematics, New Series 48, 1–22 (1918). Gy¨orgy D´osa, Department of Mathematics, University of Pannonia, Veszpr´em, Hun- gary Email address:dosa.gyorgy@mik.uni-pannon.hu COVERING A SQUARE 13 Zsolt L´angi, Bolyai Institute, University of Szeged, Szeged, Hungary, and, Alfr ´ed R´enyi Institute of Mathematics, Budapest, Hungary Ema...

work page 1918