n = 235 · Case Records · The Squares Project

n = 235 open ★ corrects Nagamochi 2005=

$15.34082207 … \leq s (235) \leq \frac{1582660563342857}{100000000000000}$

The best packing known for 235 squares, side

15.82660563 …

, David Ellsworth, David W. Cantrell 2025

15.34115.827

151617

15.33016.330

\sqrt{n}

\sqrt{n} + 1

Proven

$15.340822$ $\leq s (235) \leq$ $15.826606$

new result
exact

Citation record n-235

lowerKarakuş 2026, arXiv corrects Nagamochi 2005 (confirmed T-083)

upperEllsworth & Cantrell 2025, Squares in Squares (confirmed T-101)

Open

optimality

Bounds

Best known packing

15.82660563…

15.82660563342856

Found by: David Ellsworth, David W. Cantrell 2025
Construction: —
Source: [Kingbird]
Evidence: E-kingbird-upper-register

Minimal polynomial, degree 83

$15197358585941502961 s^{83} - 19054029395700781380870 s^{82} + 11796924216629042649354579 s^{81} - 4808108614769876120043468180 s^{80} + 1451030856404178122570781430789 s^{79} - 345799767275884788306353984453266 s^{78} + 67774498813564804634645374167057335 s^{77} - 11234508000945633407554425804213908384 s^{76} + 1607532344825153174801872971520978793189 s^{75} - 201668659118682967539387133057113450996510 s^{74} + 22454296891522560058771023859593458711158383 s^{73} - 2240889622579706580689718577985790904704763388 s^{72} + 202076916434677863681321953179975198173682882349 s^{71} - 16577690712578589166013893731214562354253387341274 s^{70} + 1244306501679531491606992301185434118328699167129267 s^{69} - 85872410283218882421430889383517071885115187773104056 s^{68} + 5471931933759042230241340087362460711039430660491160647 s^{67} - 323138905823957289824620179077427648515971541643473332874 s^{66} + 17742026808852576915081042170482522649963659530339729297241 s^{65} - 908282679855850349803743965878897364257354889699334907097884 s^{64} + 43465080817124537825282652035527867092782838207883438915180955 s^{63} - 1948672651714009470065517948906139063310384408556369781794367782 s^{62} + 82014228252452305308486172380001566000442685829960411231106701149 s^{61} - 3246186057482410594749950508613065532781882056581161088848208781448 s^{60} + 121030129123359970782879324721925653713348068894577966308153780376769 s^{59} - 4256774383890838346683359771242014391309685520375828879287765027421558 s^{58} + 141417395891604974478673542444206052527574423688874097457316926184111667 s^{57} - 4442950325166710767652318494176477005346284036871039689545149858425019188 s^{56} + 132143761554097136892656409949813501644806291750873673851743056646810847307 s^{55} - 3724278121653142074595785794031359776637918996855481246500264411247627583870 s^{54} + 99547162759226989443922275915301342990394148481013546431915854939056693130149 s^{53} - 2525451166316636919882169794346033300450573261760214407544561168810183261623712 s^{52} + 60851054822925884393362872088231784921476592249704525359339306693211978978637574 s^{51} - 1393407983201463082750103812667205031231534119903960161407503385026149533812640388 s^{50} + 30338905815230015350337507443629547938133322986372599251613731735431147148807587298 s^{49} - 628396204847897965060061083685759495211044889335692127791727849670706521484518435856 s^{48} + 12386583193977551660461241876390719826604576164873235895959785437607584760441587730192 s^{47} - 232433724444205471983096462597776075369365835330568989261391110164425311728212391035632 s^{46} + 4153323734217430140145070137613411920743170003732119525339537946385437046481746033523872 s^{45} - 70686380467717783902448903568646078963423628086622300763415121270383317250590261969131336 s^{44} + 1146011275499749445486974909121594149473099806147735475740109750306258973873693081841338630 s^{43} - 17701136698331714953844021503637463123650314086731790857767064280112784186506532020777615276 s^{42} + 260492277459372683985340515012561460001230750454462824800331349575718851511822995239394574714 s^{41} - 3652320068116991521691903234094675077664783199210342925565230028260558957233172085378428693648 s^{40} + 48786365568085939831648565613081789083584480449521988080326278540514634832953535303132015187720 s^{39} - 620776840486987337546656613456515581406101803594950169186131730819070047268868384631491241542280 s^{38} + 7523288801124157164696453440451668126537802179655932385275366967957982556047497857359919983352588 s^{37} - 86819738273128010242526385979967991470567862056150689620462183702554367768947413111616038233016416 s^{36} + 953774954925025648253304474733609592484747994431472390301868754639755982678716269084529930641443023 s^{35} - 9971081293651892380457737953978637874233272651346699205006125992828217866601482502757629011345866570 s^{34} + 99159113935527095843791193320314995744620431737333925602216138254954437828859473360567133620147394445 s^{33} - 937591882456243116942831217151994956616595933093755511723294563972551195048414611565248985607152821236 s^{32} + 8424667140126948167633953265193969962679276499025327494839034104790577492192909219817624591386758730279 s^{31} - 71892659494830178227616871456048478350079377496302951700870381917053589308972575388983039596535479420278 s^{30} + 582251421494115025057046976615977029976720394400712813199098003690123303907349827395875756580289035731733 s^{29} - 4471937667537034121585882112126200967254037591271575148365797097846048547850509627026833861786334101222136 s^{28} + 32543614650180016207874244187164028503629361855235911764700956406683943044733068359145297584031071397155994 s^{27} - 224183523933854882502864682078640188285826762871716486751957673723635189271192115484721934301078509843540820 s^{26} + 1460312333249097739412739848153412311136035088888884922686424538738960526601793346522957489286185673762602902 s^{25} - 8984148217876593612167877570397820635649870435669726928588777366245474742877218816201439836639845502892953560 s^{24} + 52134501077125370653590265849501908514957010403776225078820415168994018070692909692704843148671899014807836538 s^{23} - 284941297083362220078712187212822093342654560298933781771361807935255252780142074341846765908056419933065354916 s^{22} + 1464404571156043076914637252120097347255977567642248877431122565189765063022644175773142656036823662408630919410 s^{21} - 7064080779433205483997411627458329256081928935674885720573960927817446941144054662390438254917264293853571028424 s^{20} + 31919881457624033726193536951838036401111886302890312451699329606929040067283389942070240214272698940267541956493 s^{19} - 134801722329859078816742606094646334228067206837309021726684217834848607820988762454709381034223562103046457675998 s^{18} + 530707012396024913429576740380121941086737225273466472672403349930854389215481675641529836588022052334151834279003 s^{17} - 1942205476550803484882315247161577147422888428395169781637047795709098197490266699760943181834041758117906367397268 s^{16} + 6585767384476720675187977121595259536735035857996075281802198937005198489937485851461054520341053364378881846325263 s^{15} - 20614892940117095331097736607876870976314189060882921080291907574510740821764737500291921914248821586266401315666942 s^{14} + 59316353011229384909666880954476222015479423693489595098884075044567497728679992849404330173399583652230381335502305 s^{13} - 156117670782437502269990886042735284992957692699464203987769629925609348266900609376317045942894365095896351357414520 s^{12} + 373698270769972479755939957244422481116455737532015818487606025512111631824244935754777237146223432771503247544608560 s^{11} - 808048121315624822538485230387020159879703732818682364581834764813478780649668941927219429385916131623619099840318128 s^{10} + 1565583417239292630942979288694634658573835168674069318551077876822765011658251873382962913367365429573513955164437064 s^{9} - 2691230551310123431856658453025381333299196967548104418289076204697729928184484894708129837595375747556018998818512192 s^{8} + 4054542066273187475893579325962915065845321084583224373643180486702374676704731329107411831884877578115801740253541904 s^{7} - 5270878662724145249388262906951781140123258747428763886318687210813398134336463475140327838867254422860035408448158432 s^{6} + 5792836201656880167367652921094881746142782983189532980946755918671194010952081295188183352763611672876752443571623568 s^{5} - 5233614558124453824846563012926178182173958634839557295689659286203290796724301748107667652717293231008062406343836800 s^{4} + 3732102862328364130585244845143109502510706215299361787475517662634187688740089578345011212002758654232473965037848832 s^{3} - 1969628432244786328367806409266293170003496844663625159204453312874075246094750590119459796978694116823499446641506304 s^{2} + 683920819913111461189983402245349089127449075173547013376466597761071475636877690918573610486424418270329963001143296 s - 117203977757280124647356183279343983773836101226288348510440504667964165618210763520372674407323269025478320565518336 = 0$

Verified upper bound

$\frac{1582660563342857}{100000000000000}$

15.82660563342857

The reported value, verified here.

Evidence: E-evand-exact-ceilings-2026-10-05-exact-replay, E-evand-exact-ceilings-2026-10-05-source-replay

Reported lower bound

15.35270009…

15.3527000944

Proved by: Hiroshi Nagamochi 2005
Kind: Nagamochi
Note: General closed form: $s (N)$ >= min(ceil(sqrt(N)), sqrt(N - 2*floor(sqrt(N)) + 1) + 1).
Source: [Nagamochi 2005]
Evidence: E-nagamochi-lower

Verified lower bound

15.34082207…

15.3408220796

Corrects: Nagamochi 2005 (T-007)
Evidence: E-karakus-strip-lower, E-karakus-strip-measure-interval

Gap

0.48578355…

Verified upper minus verified lower.

Results in the register

T-007 V0 C1 Nagamochi · 2026-08-31 · 321 cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
T-083 V3 C3 Karakuş · 2026-10-02 · 301 cases
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
T-085 V3 C3 Karakuş; chelokot · 2026-10-02 · 315 cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
T-101 V3 C3 Daniel after Couzo, de Winter, Ellsworth, Levy · 2026-10-06 · 77 cases
Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side

Verification

upper: replayed here; lower: external proof (read here), audited here

formal lower differs from report; reported lower: defect recorded

Rigidity

not rigid, numerically checked, numerical multiprecision

Evidence: E-translation-escape-not-rigid

Scope

Square 35 of the retained witness (witness id 36) translates 0.000438 along (0, -1) with the packing still valid, so the configuration admits a non-trivial feasible motion; 21 of its 235 squares do. Every constraint is exactly affine in the slide parameter, so the arithmetic carries no linearization error, but the coordinates are the witness's own finite-precision transcription: this settles the retained configuration, not the true optimum. Rigidity and optimality are independent, and this bears only on the former.

Evidence and sources

9 evidence entries

E-evand-exact-ceilings-2026-10-05-report, E-evand-exact-ceilings-2026-10-05-exact-replay, E-evand-exact-ceilings-2026-10-05-source-replay, E-kingbird-upper-register, E-nagamochi-lower, E-basic-grid-upper, E-karakus-strip-lower, E-karakus-strip-measure-interval, E-nagamochi-lemma1-counterexample

[evand exact optima 2026-10-05] formal certificate
[Kingbird] record catalogue
[Nagamochi 2005] lower bound proof
[Friedman DS7] survey
[Karakuş 2026] lower bound proof

In the frontier survey On GitHub

$s (235)$ — open

Exact certificate, 2026-10-06. Evan Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing, decided here over $ℚ$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it proves $s (235) \leq 15.82660563342857$ , the verified upper bound (T-101). Until then the verified upper bound was the trivial grid bound $16$ .

Open. The best known packing gives $s (235) \leq 15.82660564$ , and the strongest lower bound independently verified here is $15.340822$ from Karakuş’s general theorem, leaving a gap of $0.4858$ . General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

The exact certificate

Evan Daniel’s square-packing published on 5 October 2026 exact rational certificates of the packings this register lists as best known, solved from its own witnesses, and offered those that do not lower a printed side as a replay of the existing bounds, asking for nothing to be registered from them. The certificate for this count holds the same 235 squares, each a rational centre and a rational $t = \tan (θ / 2)$ , in a square of side $15.8266056334285680698 \dots$ , $8.1 \times 10^{- 15}$ above the side the Kingbird catalogue prints, $15.82660563342856$ . Square for square, its pose lies within $7.1 \times 10^{- 4}$ of the binary64 pose the atlas pictures for this count. 7 squares move by more than $1 \times 10^{- 8}$ , all of them squares the source lists as carrying no force; every other square’s centre rounds to the witness’s. Evan Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $ℚ$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That proves $s (235) \leq 15.82660563342857$ , the verified upper bound: the certificate’s side rounded up at the fourteen decimals the catalogue prints, as the record writes any certificate of a printed side. It says nothing about optimality. On jlevy/squares#375 its author wrote that the solver, its checkers and the batch “were written with Claude (Anthropic) as a coding and research agent, directed and reviewed by me.”

The verified upper bound and the printed side now agree to one unit of the printed side’s last place, the precision at which the record compares them. The printed side itself is not certified here: the certificate’s side lies above it.

The packing

Found by David Ellsworth and David W. Cantrell in 2025, via an unrecorded method. Its side length is algebraic of degree 83 over $ℚ$ .

The lower bound

The strongest lower bound independently verified in this record is Karakuş’s general bound, which applies to every nonsquare $N \geq 8$ :

s (N) \geq \frac{1}{2} + \sqrt{N - ⌊ \sqrt{N} ⌋ + \frac{1}{4}}

Correction, 2 October 2026. Until that date the verified lower bound here was Nagamochi’s general closed form, $s (N) \geq min (⌈ \sqrt{N} ⌉, \sqrt{N - 2 ⌊ \sqrt{N} ⌋ + 1} + 1)$ , which is stronger at this $n$ and is now recorded as a reported bound. Its published proof rests on Nagamochi’s Lemma 1, which Karakuş showed false; nothing is disproved, and no packing beating it is known (review of 2 October 2026). This register had recorded that proof as verified, its own error, logged as defect D-516.

Source-reported bounds are recorded separately from this independently verified theorem.

Verification Code

The programs behind this case’s verified bounds, by their evidence. The code column says how the code that ran stands to the code its producer used. VERIFIERS.md says what each program is and whose it is.

bound	evidence	run	code	programs
verified lower	`E-karakus-strip-lower`	a published proof	no code	no verification code
verified lower	`E-karakus-strip-measure-interval`	audited here	independent	`V-check-karakus-strip-measure` (first-party)
verified upper	`E-evand-exact-ceilings-2026-10-05-exact-replay`	replayed here	independent	`V-sqpack-verify`, `V-check-rational-witness-independent` (first-party); `V-evand-exact-certificates` (first-party, premises)
verified upper	`E-evand-exact-ceilings-2026-10-05-source-replay`	replayed here	producer’s code	`V-evand-verify-cert-py`, `V-evand-verify-cert2-py` (external); `V-evand-exact-certificates` (first-party, premises)