Three Squares in a Rectangle

[2608.13595] Three Squares in a Rectangle Skip to main content Search Submit Donate Log in Mathematics > Metric Geometry arXiv:2608.13595 (math) [Submitted on 29 Jul 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Three Squares in a Rectangle Authors: Haobo Yang View PDF HTML (experimental) Abstract: For $x\ge1$, let $G_3(x)$ be the maximum sum of the side lengths of three pairwise interior-disjoint, arbitrarily rotated squares contained in a $1\times x$ rectangle. We determine this function exactly: $G_3(x)=x+\tfrac12$ for $1\le x\le\tfrac32$, $G_3(x)=2$ for $\tfrac32\le x\le2$, $G_3(x)=x$ for $2\le x\le3$, and $G_3(x)=3$ for $x\ge3$. This completes the $n=3$ case of the rectangular square-packing question posed by Richard Stanley in a 2021 MathOverflow comment. The values for $\tfrac32\le x\le3$ follow from a strip theorem stating that three squares in $[0,1]\times[0,H]$, $H\ge2$, have total side length at most $H$. For $x\ge3$, the formula is immediate because each square has side length at most $1$. The range $1\le x\le\tfrac32$ is handled by combining the two-square theorem with an additional semi-perimeter estimate for a triangle whose two nonhorizontal sides have opposite slopes. In particular, the special case $x=2$ answers the question asked in the MathOverflow post. In connection with Erdős problem #106, we also prove that every five-square packing in the unit square with an axis-parallel guillotine cut has total side length at most $2$. Comments: Revised version. Determines G_3(x) exactly for all x >= 1, completing the n=3 case of the rectangular square-packing question posed by Richard Stanley in a 2021 MathOverflow comment. Adds the range 1 <= x <= 3/2; title and abstract updated Subjects: Metric Geometry (math.MG) ; Combinatorics (math.CO) MSC classes: 52C15, 52C10 Cite as: arXiv:2608.13595 [math.MG] (or arXiv:2608.13595v2 [math.MG] for this version) https://doi.org/10.48550/arXiv.2608.13595 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Haobo Yang [ view email ] [v1] Wed, 29 Jul 2026 02:47:10 UTC (11 KB) [v2] Thu, 27 Aug 2026 06:22:15 UTC (12 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.MG < prev | next > new | recent | 2026-08 Change to browse by: math math.CO References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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