Sources

The packings are Ellsworth's and the classical proofs are their authors'. Original here: the interactive tools, the contact and rigidity analysis, and the 2026 computer-checked results in §5 marked "this project". Preprints are unrefereed; this page says which is which.

1 · The catalogue this site is built on

Every packing drawn on this site — coordinates, angles, side length — comes from the exact constants in David Ellsworth's SVG files. Dates and "found by" attributions shown elsewhere on this site are quoted from his wording, not independently re-derived.

• Erich Friedman, 1998–2009 (maintained), Packing unit squares in squares: a survey and new results, Electronic Journal of Combinatorics, Dynamic Survey #7 — the founding survey: the table of best-known s(n), the unavoidable-set proofs, and the technical lemmas the whole field reuses. Maintained copy: erich-friedman.github.io, itself a GitHub Pages repository.
• David Ellsworth, 2023–2026 (maintained), Squares in Squares, kingbird.myphotos.cc — the current record table, up to very large n, with high-precision side lengths, exact forms where known, and dated attributions; some records are built by directly minimising the side length analytically rather than by search.
• — sub-pages: triangular table, older / alternative packings, rigid packings, Göbel strips, Göbel squares, and the n²−n−1 family.

2 · Origins

• F. Göbel, 1979, Geometrical packing and covering problems, in Packing and Covering in Combinatorics (A. Schrijver, ed.), Math. Centre Tracts 106, pp. 179–199 — the paper that started it: s(2)=s(3)=2, s(5)=2+1/√2, the Göbel-strip family, and the unavoidable-point method every later proof uses.
• Martin Gardner, Oct 1979, "Some packing problems that cannot be solved by sitting on the suitcase," Mathematical Games, Scientific American 241(4) — brought Göbel's packings to a wide audience and set off a wave of reader submissions (indexed in the list of Gardner's columns).
• Martin Gardner, 1992, Fractal Music, Hypercards and More, W. H. Freeman, ch. "Packing Squares," pp. 289–306 — book reprint of the column plus reader addenda; the version MathWorld and Friedman cite.
• Walter Trump, 1979, n = 11 packing, s ≈ 3.877084 — background. Independently rediscovered by several of Gardner's readers; Trump's 2023 note argues it cannot be improved within its combinatorial type.

The rest of the early field reached Gardner by reader mail and has no separate web presence: Pertti Hämäläinen (1980, improved n = 17–18), Robert Wainwright (1979, n = 19), Charles F. Cottingham (1979, diagonal-strip packings up to n ≈ 49), Evert Stenlund (1980, records for nearly all n ≤ 100), and Mats Gustafsson (1981, an alternative n = 18). Their attributions here follow Friedman's survey and Ellsworth's page.

3 · Proofs: exact values and lower bounds

• W. Stromquist, 1984, "Packing unit squares inside squares" memoranda I–III, Daniel H. Wagner Associates — unpublished but online: I, II, III; settled s(6) and s(10) and first argued Gardner's n = 11 conjecture.
• M. J. Kearney and P. Shiu, 2002, Efficient packing of unit squares in a square, Electron. J. Combin. 9, #R14 — first published proof of s(6)=3 (Ellsworth dates it June 2001), by a duality argument.
• S. El Moumni, 1999, "Optimal packings of unit squares in a square," Studia Sci. Math. Hungar. 35(3–4), 281–290 — independent proofs of s(7)=s(8)=3 and s(15)=4.
• W. Stromquist, 2003, Packing 10 or 11 unit squares in a square, Electron. J. Combin. 10, #R8 — s(10)=3+1/√2 and s(11) ≥ 2+4/√5, which by monotonicity is also the classical bound for s(12).
• H. Nagamochi, 2005, Packing unit squares in a rectangle, Electron. J. Combin. 12, #R37 — proves s(n²−2)=s(n²−1)=n for every n, using weighted points, segments and areas as "resources." chelokot (§5) gives a Lean-checked counterexample to the paper's Lemma 1 and a replacement proof of s(n²−2)=n for square containers.
• W. Bentz, 2010, Optimal packings of 13 and 46 unit squares in a square, Electron. J. Combin. 17, #R126 — s(13)=4 and s(46)=7; s(13)=4 is what makes n=12 the open boundary case. Kernel-checked in Lean in 2026 by chelokot, who found that two of the printed auxiliary point sets need correcting (the theorem stands); re-proved in 2026 with no case analysis by this project (§5, unrefereed).
• W. Bentz, 2016 (v2 2018), Optimal packings of 22 and 33 unit squares in a square, arXiv:1606.03746 — s(22)=5, s(33)=6, via continuously varying families of unavoidable sets; no journal version found, cite as arXiv.
• Erich Friedman's survey (§1) supplies Theorems 1–8: the unavoidable- and almost-unavoidable-set proofs of s(7), s(8), s(14), s(15), s(24) and s(35), plus bounds for s(13), s(19)–s(21).
• T. Green, 2000, private communication to Friedman, unpublished — lower-bound theorems for the n²+1 and n²+⌊n/2⌋+1 families (Theorems 9–10 of the survey) and specific bounds for n = 17–18, 22, 26–30.

4 · Packers and their tools

Below are the people whose packings the record table stands on and how they found them, in the order the field usually cites them.

• John Bidwell, 1998, University of Hawaii undergraduate — found n = 17 (s ≈ 4.6755), still the record, based on a 1980 packing by Pertti Hämäläinen.
• David W. Cantrell, 2002–2005 and 2024–2026, sci.math and private communication — n = 19, 26, 37, 39, 41, 53, 54, 55, 68, 70, 71, 87, 88 and others; five records still stand.
• Joe DeVincentis, Apr 2014, private communication — n = 41 and an alternative n = 54/70/71 extension.
• Károly Hajba, 2009 and Sep–Nov 2024 — n = 51 (first s(51) < 7+1/√2), and 83, 102, 107, 108, 131, 172, 240 in a burst of 2024 improvements.
• Maurizio Morandi, 2010, private communication — n = 69.
• T. Gensane and P. Ryckelynck, 2005 (online 2004), Improved dense packings of congruent squares in a square, Discrete Comput. Geom. 34, 97–109 — an "inflation" search algorithm; new packings for n = 11 (rediscovery), 29, 37 and an alternative n = 18.
• Thomas Schadt, Dec 2025–2026 — a simulated-annealing program, "starting from randomness"; program not public; records for n = 28, 29, 39, 41, 50, 51, 55, 68, 71, 103, 105, 126 and more.
• David Ellsworth, 2023–2026 — maintains the record page, refines Schadt's finds, and finds his own records (including by direct analytic optimisation rather than search); currently credited with the largest share of the table.

Current records by finder

Counted from the "Found by" (or "Found first by") line on each row of Ellsworth's main table (not the sub-pages), one credit per current record; a joint find is one credit for the pair, not one each. This is a tally of attribution text, not a claim about who actually deserves credit for disputed or independently-rediscovered packings.

5 · Computer-checked lower bounds (2026, unrefereed)

All the items below are preprints, repositories or blog posts, none peer-reviewed, all with published code or a verifier so the claims can be checked independently. We link others' certificates as their authors published them and replay only our own; the recent s(11) and s(17) certificates use strict-core and subset-charge formats that our checkers do not read. Several of these repositories changed daily in September 2026, so the values below are dated and the repositories have the latest.

• S. Burns, 6 Aug 2026, Proposing a Better Lower Bound for n=17 Square Packing, sam-burns.com (blog) — s(17) ≥ 4.4811 from 268 weighted points, a certificate developed with ChatGPT and verified in exact rational arithmetic; code at github.com/sam-bee/squarl.
• G. Massaccesi, 21 Aug 2026, Another Better Lower Bound for n=17 Square Packing, gus-massa.blogspot.com — s(17) ≥ 4.5058 from 168 LP-optimised weighted points; methodology in a companion post.
• S. Fort, github.com/stanislavfort/17squares — s(17) > 4.456575 from unweighted points and an exact subdivision of pose space; the repository states it was produced by an AI model whose correctness the author does not personally vouch for.
• Mira, github.com/Mira-acc/17squares — s(17) > 4.468292 from 16 unweighted points and a dyadic subdivision of pose space, with three independent checkers and a write-up; later raised with weighted atoms to 4.613029 (8 Sept 2026) and 4.614154 (20 Sept).
• Guzhou0806 / N17 project, with AI assistance, Sept 2026, github.com/Guzhou0806/n17-square-packing — a series of strict s(17) certificates with exact replays: 4.613046 (R012, 19 Sept), R038, R042–R052 (to 4.62003, 25 Sept), and R067, s(17) > 233009/50000 = 4.66018 (28 Sept), which rebuilds the cores of Kleddamag's 4.66001 certificate. Its R038 checker is the second checker of Kleddamag's s(11) proof.
• Kleddamag, with OpenAI Codex, Sept 2026 — 11-squares-certified-bound: s(11) > 31/8 = 3.875 (22 Sept), developed from jlevy's T-026 certificate with a checker adapted from Guzhou0806's R038, about 0.002 below Trump's packing; and 17-squares-certified-bound: s(17) > 461300/99853 = 4.619791 (21 Sept), then 4.640020 (26 Sept) and 466001/100000 = 4.66001 (27 Sept), building on Mira's and Guzhou0806's certificates and jlevy's method, as its attribution file sets out.
• Joshua Levy (jlevy), the Squares Project, 2026, github.com/jlevy/squares — an AI-agent project with its own results and a survey of n ≤ 100 that keeps reported and replayed bounds apart. Its results include s(12) ≥ 99/25 = 3.96 (T-017, 4 Sept 2026), s(11) ≥ 3.81 (4 Sept), 3.8264 (T-026, 9 Sept) and 3.8270 (T-033, 22 Sept), and s(21) ≥ 122/25 = 4.88 (23 Sept). It also replays others' certificates, including Kleddamag's and ours.
• tokoharu, Sept 2026, github.com/tokoharu/square-packing-density-bounds — rectangle-density certificates (uniform densities on axis-aligned rectangles instead of points) with an interval verifier: s(11) ≥ 3.81, s(26) ≥ 5.508, s(29) ≥ 5.71 (21 Sept 2026).
• wand125, Sept 2026, github.com/wand125/square-packing-bounds — weighted-point and, with tokoharu's solver, rectangle-density certificates for many n from 18 to 91, including s(32) ≥ 119/20 = 5.95 (26 Sept 2026), the best bound for s(32) before s(32) = 6; s(21) ≥ 249/50 = 4.98, later 399/80 = 4.9875; s(45) ≥ 1389/200 = 6.945, later 6.955, the best bound for s(45) before s(45) = 7; and s(60) ≥ 397/50 = 7.94 (27 Sept 2026), the best bound for s(60) before s(60) = 8. Many of its floors for other n are newer than this site's table.
• chelokot, 2026, chelokot/square-packing-archive — a Lean 4 archive in which every result has a kernel-checked proof: s(n²−2) = n, s(6), s(10), s(13) = 4 (5 Sept 2026, following Bentz's argument with corrected auxiliary sets), s(22) and s(33), plus a counterexample to Nagamochi's Lemma 1 and a replacement proof.
• This project (evand), 2026, square-packing (s12/), GitHub repository, building directly on Burns's and Massaccesi's method:
  • s(12) ≥ 15680/3951 ≈ 3.9686 (26 Aug 2026) from 1,736 LP-optimised weighted points, verified exactly over the full continuum of positions and angles; still the best floor for s(12) we know of; kernel-checked in Lean, as are s(12) ≥ 35/9 and ≥ 3920/997. Write-up.
  • s(11) ≥ 3040/797 ≈ 3.8143 the same way (680 points; kernel-checked in Lean), found 26 Aug but published 22 Sept 2026, after jlevy's 3.8264; since superseded by jlevy's 3.827 and Kleddamag's 3.875. Timeline.
  • A proof of Bentz's s(13) = 4 with no case analysis, from one weighted cover checked at margin zero by two independent exact checkers and kernel-checked in full in Lean 4, after chelokot's Lean proof of Bentz's argument. Write-up.
  • s(32) = 6 from a 13,085-point weighted closed cover of total 31.7135, certified at margin zero by two independent exact checkers and kernel-checked in full in Lean 4, with no hypothesis. Write-up.
  • s(21) = 5 from a mixed cover (7,536 weighted points plus mass spread evenly along the grid lines, total 20.8947), certified at margin zero by two independent exact checkers, with a Lean 4 top theorem from the checkers' covering statement, after an earlier s(21) ≥ 5000/1001 ≈ 4.9950 from weighted points. Write-up.
  • s(45) = 7 from a mixed cover (19,989 weighted points plus mass on the grid lines, total 44.7735), certified by the same two exact checkers; no Lean yet. Write-up.
  • s(60) = 8 from a mixed cover (23,744 weighted points plus mass on the grid lines, total 59.8587), certified by the same two exact checkers; no Lean yet; hence also s(61) = 8, since s(61) ≥ s(60) (s(k²−3) = k was proved only up to k = 7). Certificate; see also the s(45) write-up.

6 · Explainers and popular coverage

• Deckard, 27 Sep 2025, Packing Squares Inside The Smallest Square Possible, YouTube — the original explainer; discussed on Hacker News.
• Deckard, 2026, NEW SQUARE PACKING SOLUTIONS, YouTube — a sequel covering the Schadt/Ellsworth 2025–26 records.
• Randall Munroe, 20 Feb 2023, xkcd #2740, "Square Packing" — a joke about n = 11 and a hydraulic press; annotated at explainxkcd.
• Daniel Piker (@KangarooPhysics), 14 Feb 2023, repost of the n = 17 packing and a rigidity follow-up, X (Twitter) — the post that made the n = 17 packing a meme, with the observation that three squares can slide and one has room to wiggle.
• Eric Weisstein, Square Packing, MathWorld — a reference table of best-known s(n) with asterisks marking proven values.
• Wikipedia, Square packing — a general-audience summary with its own reference list.
• Hacker News threads: "Squares in Squares" (kingbird), Feb 2023; Deckard's video, Sep 2025; Massaccesi's n = 17 bound, Aug 2026.

7 · This site

Built in 2026 from Ellsworth's 30-digit constants (§1); the parsing, contact and rigidity analysis are ours, written independently of his. Gaps and free squares are computed in double precision; "exact contact" flags are checked at 50 digits, so roundoff is never mistaken for touching. Code: github.com/evand/square-packing.

Checking our rigidity analysis against Ellsworth's own "Rigid" marks…