$s = 2$
Proved by Frits Göbel
in early 1979.

$s = 3$
Proved by Michael Kearney
and Peter Shiu in June 2001.

$s = 3$
Proved by Erich Friedman
in 1999.

$s = 4$
Proved by Wolfram Bentz
in August 2009.

$s = 4$
Proved by Erich Friedman
in 1999.

$s = {}^{18}🔒 = \Nn{4.67553009360455}$ $4775s^{18}-190430s^{17}+3501307s^{16}-39318012s^{15}+300416928s^{14}-1640654808s^{13}+6502333062s^{12}-18310153596s^{11}+32970034584s^{10}-18522084588s^9-93528282146s^8+350268230564s^7-662986732745s^6+808819596154s^5-660388959899s^4+358189195800s^3-126167814419s^2+26662976550s-2631254953=0$
Found by John Bidwell
in 1998.
Based on packing found by Pertti Hämäläinen in 1980.

$s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$
Found by Pertti Hämäläinen
in 1980.
Pictured alternative with minimal rotated squares found by Mats Gustafsson in 1981.

$s = 3 + {4\over 3}\sqrt 2 = \Nn{4.88561808316412}$
Found first by Robert Wainwright
in late 1979.
Based on packing found by Charles F. Cottingham in early 1979.

$s = 5$
Proved by Wolfram Bentz
in October 2018.

$s = 5$
Proved by Hiroshi Nagamochi
in 2005.

$s = 5$
Proved by Erich Friedman
in 1999.

$s = {7\over 2} + {3\over 2}\sqrt 2 = \Nn{5.62132034355964}$
Found by Erich Friedman
in 1997.
Unextends the $s(37)$ found by Evert Stenlund in early 1980.

$s = \Nn{5.93383346267692}$
Found by Thomas Schadt in December 2025, using a simulated annealing program he wrote, starting from randomness.
Similar to the packing found by Thierry Gensane and Philippe Ryckelynck in April 2004, using a computer program they wrote.
Optimized by David Ellsworth
in December 2025.

$s = 6$
Proved by Wolfram Bentz
in October 2018.

$s = 6$
Proved by Hiroshi Nagamochi
in 2005.

$s = 6$
Proved by Erich Friedman
in 1999.

$s = {}^{8}🔒 = \Nn{6.59861960924436}$ $6s^4-(208+64\sqrt{2})s^3+(2058+850\sqrt{2})s^2-(7936+3658\sqrt{2})s+11163+5502\sqrt{2}=0$ $36s^8-2496s^7+59768s^6-733760s^5+5289248s^4-23462672s^3+63458276s^2-96673872s+64068561=0$
Found by David W. Cantrell in September 2002.
Improves upon the $s(37)$ found by Evert Stenlund in early 1980.

$s = {}^{5}🔒 = \Nn{6.81072208306864}$ $9s^5-171s^4+999s^3-1959s^2+1636s+166=0$
Found by Thomas Schadt in January 2026, using a simulated annealing program he wrote, starting from randomness.
Refound and refined by David Ellsworth in January 2026, starting from randomness.
Optimized by David Ellsworth in January 2026.

$s = {}^{42}🔒 = \Nn{6.92669309446880}$ $144s^{42}-33248s^{41}+3740531s^{40}-273229120s^{39}+14568177368s^{38}-604345329616s^{37}+20303247278518s^{36}-567715628580628s^{35}+13476130642163772s^{34}-275622556171657148s^{33}+4913118839607229315s^{32}-77021965442580593792s^{31}+1069597207525632250760s^{30}-13234280063158548374864s^{29}+146588403144234109714492s^{28}-1459056537531761947694412s^{27}+13090219490685085049164304s^{26}-106115059640167069135194108s^{25}+778709778173545540562913112s^{24}-5180160724110239826615572336s^{23}+31267085211757278545052994144s^{22}-171331300125735569491450805184s^{21}+852412555299622931388971786184s^{20}-3849639590878015114188275848896s^{19}+15771592794879254264477226832440s^{18}-58556476540137831140983424890112s^{17}+196742347065286547712609208667628s^{16}-597075318553361988026330293830592s^{15}+1632807555219691500831155576662224s^{14}-4011703707846363075271797958908992s^{13}+8823126218415607049547609565313808s^{12}-17292499393880618294971765830702496s^{11}+30033784585675389426408059928238624s^{10}-45904080196423917967770013765165584s^9+61200148145342575539111090507985440s^8-70370773645277985938951580858638528s^7+68756165329665893470887878785349152s^6-55949600498940958320310297578555360s^5+36867848125763978702951438802849616s^4-18873332426700882570902855047275200s^3+7023595398126017089078028623797120s^2-1682258751137636203725622554061120s+192930128676231207430057837613968=0$
Found by Thomas Schadt in December 2025, using a simulated annealing program he wrote, starting from randomness.
Refound and refined by David Ellsworth in December-January 2025-2026, using his modified version of Thomas Schadt's simulated annealing program, starting from randomness.
Optimized by David Ellsworth in January 2026.

$s = 7$
Proved by Wolfram Bentz
in August 2009.

$s = 7$
Proved by Hiroshi Nagamochi
in 2005.

$s = 7 + {4\over 7} = \Nn{7.57142857142857}$
Found by Thomas Schadt in December 2025, using a simulated annealing program he wrote, starting from randomness. Optimized by David Ellsworth.
This is the third record-setting packing found with a rational side length, thanks to the Pythagorean triple $\{3, 4, 5\}$ determining its tilt angle. It is the first record-setting packing to use the same Pythagorean triple as the $s(104)$ found by David Ellsworth in December 2024 (the first competitively-intended packing found with a rational side length, though not record-setting).
It's the second record-setting packing found that is doubly semi-primitive, i.e. has only non-rotated squares on its leftmost and rightmost sides, but has rotated square(s) poking into its top and bottom sides.
It's the first to have both of these properties.

$s = {13\over 2} + {1\over 2}\sqrt 7 = \Nn{7.82287565553229}$
Found by David W. Cantrell
in September 2002.
Improved by David W. Cantrell
in December 2024.
Improved by David Ellsworth in February 2026, using his modified version #3 of Thomas Schadt's simulated annealing program, starting from randomness.
Optimized by David Ellsworth
in February 2026.

$\begin{aligned}s &= 7-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{7.84666719284348}\end{aligned}$
Found by David W. Cantrell
in October 2005.
Improved by Joe DeVincentis
in April 2014.

$s = 8$
Proved by Hiroshi Nagamochi
in 2005.

$s = 3 + 4 \sqrt 2 = \Nn{8.65685424949238}$
Found by Evert Stenlund in early 1980.

$s = \Nn{8.80345993651653}$
Found by Sigvart Brendberg in June 2023, using a computer program he wrote followed by manual optimization.
Improved by Thomas Schadt in December 2025, using a simulated annealing program he wrote, starting from randomness.
Optimized by David Ellsworth in December 2025.

$s = {}^{82}🔒 = \Nn{8.82721205592900}$ $52389094428262881s^{82}-28863139436366651460s^{81}+7840436786580754561842s^{80}-1399864630898909951672184s^{79}+184777024966383679131379203s^{78}-19229480097533386652981194668s^{77}+1643178003450476327369002864080s^{76}-118561352785653984081132853368864s^{75}+7372351836836707441183744339971015s^{74}-401254176764396680092337021141946484s^{73}+19350157008010415954432078062713291394s^{72}-834969623551779032213936610875479861512s^{71}+32500264420943843392373991413578392058093s^{70}-1148852629892528066579108553164478473663708s^{69}+37092466248098270905023679715303792737820304s^{68}-1099206042418214352026228628885408398048015000s^{67}+30025320958251433175557289720600502032769753340s^{66}-758792087058505752402362438963674625699826919880s^{65}+17799410748369850870306914205805242294037335637896s^{64}-388686829570450651667791276249653981802721222714056s^{63}+7922061683854685568474881816199072307645318622376904s^{62}-151058341641411022199807974673871019724902497364765552s^{61}+2700552785792713834768094889293145620129036537676224092s^{60}-45354522344129825814676420288826173471599912259984496632s^{59}+716878712470410740335863321139824820808423827153710652804s^{58}-10682567284888720343007934969631240418818071811270135320816s^{57}+150320784390672934545124162608853418121767017935787301000808s^{56}-2000572646236355172723818796429406996345627284039771483014960s^{55}+25219559033013693277083797294746787502373277261234753716013214s^{54}-301583920452466921147984771117351156297618201001262096981290160s^{53}+3426054385349213936246735756144263675017479361589187104582644952s^{52}-37026637210515130032012648558266141117178874708570301177143938096s^{51}+381221915869963598518466209504441332617716678547088629463788058492s^{50}-3744411601889467025365599805308355961072225438102130448464025588920s^{49}+35132927721859555152174976560750433704133787706160759119646097007600s^{48}-315304729246464792403348347852665200866883662496880642068223778829192s^{47}+2709948932058311309971179409319475857433539061059204130543172912232550s^{46}-22330292252239325190451014020603871952094854771701615927312762869972264s^{45}+176590827377409087722261448341442541695845580235159787543827412635270192s^{44}-1341409274447219282944440341226557560736584173610170808364041163179628592s^{43}+9794628200723929363909228342085371052888507149267241738990749330177446536s^{42}-68784984456991565723134237317800008579678347641360227701542986898597128624s^{41}+464787351026639375955250101748280481985644641368851374515808386357693042884s^{40}-3022573184259078701450135458529957385309345186803651626114551783083378175032s^{39}+18919089267049873225236080915564725310548628869115830052258169804904606332284s^{38}-113971460035925598073276330819280830203445312638283436481301177127978414813080s^{37}+660644243129473954993233623574173921633210380878554917654203983937559606764892s^{36}-3683382377823441838082957327165940185883796561462208003223994363396807558829680s^{35}+19741959358629662296400872197474154929765830845655143211973252362263291667066932s^{34}-101642479500862445314955859849362422289005748345703180795721307378605167190176216s^{33}+502223128747819353777858875489650546509178956355195996750910607953412777650946008s^{32}-2378848650747301593887480639497480434456215486100674152450449358032767827357494504s^{31}+10787427200018965953466228877088228967021423580833924967176436387993308084517771520s^{30}-46763657666979111364440110538290788706620901509781273614839323079790048897674112936s^{29}+193478197292846750318686197125966160724659499210376202873234094610563436453287357712s^{28}-762652613846301377253090541691216290609269954386886415663741638905969508874033669080s^{27}+2858810541382820711701247202901545177530055188476676694775559880120687456958481517369s^{26}-10170991995607092582144907594501215089484328088609684391404365725870234777077196212052s^{25}+34275552245333382898966081848057394622466895493701655338137448625314981168202736100870s^{24}-109180149865199065847120380278545633781677317125880517882096621147002464901674366232896s^{23}+328024240104468595897778174882791456088805151159650666727391352426213891813001741009597s^{22}-927474305514792318700089933609615567301057834087731929460292201817254239263791481291308s^{21}+2462182229902610406598305774365812710170450400050717006211185019057179425643813174021924s^{20}-6122044755330252945719750665463174625835643677192302695331089860574738738324123064544448s^{19}+14219984970544731850691516928796082391054234525105419850802193862381157708451060567522208s^{18}-30768851545907889218776308829677014260927583947632211969217096923859242673238301283472512s^{17}+61831404131179569857993652298053425544232611206764320680249716948926268397092525417269376s^{16}-115009817315259102016058959098678198023224949369106956091463566410698079424438533154919424s^{15}+197271311091301472792347653205833439690067290927728227908502214395626223917618164276099328s^{14}-310715226079337036201755817142663826462822486357621041087800464785427130883680776524049408s^{13}+447233873751878497967377512304289813779839139617400585441961124659571901207054515474723840s^{12}-584994487650569941937265070878539829783049696806373201094322271724763322754772388187897856s^{11}+690809670769485727048919721008636863534640513613632064766742686036251506800827979919523840s^{10}-730705950216779945965312115026670309649787853302475272108288646183442394632173792483868672s^9+685727465494402560587060223400049402456139486767982657035415974606333680206469376237371392s^8-564182795837916615774045743559109089033591178820776604035503312959295256878380021673099264s^7+400823651584041532933559377617252554932923674966442340917105411238495035002689607404879872s^6-241020915379745770711663822572215144075000506186967983082373034574012538132391946971250688s^5+119331539747892530196375157797097038574572404727228577993084411405965584791382011108392960s^4-46729898398085553837033675288544422050050908921747303951054359523991662277479822073528320s^3+13577207271788496430462938959054088460341797225685886738526659498529340720805243256832000s^2-2603186344462167626779756466825247201285474002427939337103647694067437869176377573376000s+247160402287431680471138762403368003391572385877539215982119721342810263983667281920000=0$
Found by Maurizio Morandi
in June 2010.
Improved by David W. Cantrell
in August 2023.

$s = {}^{4}🔒 = \Nn{8.88166675700900}$ $23s^4-742s^3+8848s^2-45876s+86229=0$
Found by Joe DeVincentis
in April 2014.

$s = \Nn{8.94407155757031}$
Found by Thomas Schadt in December 2025, using a simulated annealing program he wrote, improving upon the $s(71)$ found by David W. Cantrell in October 2005.
Optimized by David Ellsworth in January 2026.

$s = 9$
Proved by Hiroshi Nagamochi
in 2005.

$s = 6 + {5\over 2}\sqrt 2 = \Nn{9.53553390593273}$
Found by Frits Göbel in early 1979.
Adds two "L"s to $s(65)$.

$s = {}^{24}🔒 = \Nn{9.63482562092335}$ $46438209s^{24}+1718447880s^{23}-1304818741864s^{22}+154362940868008s^{21}-10223870917986092s^{20}+463012769729234068s^{19}-15608677475881443482s^{18}+410530364971106359132s^{17}-8675319117762080311978s^{16}+150196459602374087471728s^{15}-2158879193002672091253360s^{14}+25993038455067669296355532s^{13}-263613888105247221344935027s^{12}+2258335015809616506745502008s^{11}-16347943921555337654669478150s^{10}+99786776593815833271369617220s^9-511154425074511891757096094175s^8+2180187656593439512672814134216s^7-7652314463979073976449593048904s^6+21727853135387976484209118127392s^5-48671720700899577518293563957136s^4+82801528406446840092722047620736s^3-100540002112755895115349929950336s^2+77621257841393908308227797286912s-28634116465193128516311336597248=0$
Found by Károly Hajba in September 2024.
Improved upon the $s(83)$ found by Evert Stenlund in early 1980.
Improved by David W. Cantrell in November 2024.
Extends the $s(17)$ found by John Bidwell in 1998.

$s = {11\over 2} + 3 \sqrt 2 = \Nn{9.74264068711928}$
Found by Erich Friedman
in 1997.

$s = {17\over 2} + {1\over 2}\sqrt 7 = \Nn{9.82287565553229}$
Found by Erich Friedman
in 1997.
Extends the alternative packing of the $s(18)$ found by Pertti Hämäläinen in 1980 found by Mats Gustafsson in 1981.

$s = {}^{44}🔒 = \Nn{9.83881743996618}$ $629407744s^{44}-222450679808s^{43}+38325385756672s^{42}-4288660082065408s^{41}+350399070882492416s^{40}-22278553803273224192s^{39}+1147211537906515146752s^{38}-49165134066758334425088s^{37}+1788289659043280951068608s^{36}-56019383059428524897963904s^{35}+1528333582392710799260183056s^{34}-36630930585616880256564390736s^{33}+776543722002840046363087727297s^{32}-14636706341423531247516707219912s^{31}+246257253722069243863014484299360s^{30}-3708577084615935587358116609301672s^{29}+50074372636267901037924496964125186s^{28}-606479613834953151228303026322056904s^{27}+6582752938907446459101615245660789892s^{26}-63854390288109107749984304630348881800s^{25}+550526226273592526694018724167940975017s^{24}-4175960161221920130654884897696082820472s^{23}+27333308361495923310409348740175473961584s^{22}-148034863695064180910459529146184056115272s^{21}+589148297042380492962295030591016626388382s^{20}-802163371353907808165482957048549332378608s^{19}-13170004747477877616293633845629891862121520s^{18}+167392797210322735047434221678453545578017040s^{17}-1289394300288384411131628488173364706102979582s^{16}+7787489258551633023583289334261547248712408216s^{15}-39385778676485241094858686353246027501471904572s^{14}+170965915920626557796308089870520546518118278944s^{13}-643154540172295005723535558458606455031058629500s^{12}+2101724893624280757011728452832803445528481662064s^{11}-5952381674143195979828863351096699278368526615456s^{10}+14521989109696597926922222011586158834006883331752s^9-30220186517608350885252195687799573775025255492983s^8+52871604435569645714797351939274760286735693270688s^7-76162701057186797152957613136211362895325167372620s^6+87583595303336820271204366145801190135485625910672s^5-76547308527338366706545567730043731458801541686452s^4+46512363388633322446129276907281520687047667820792s^3-15838102486745813882030359867355235545830428100484s^2+479096262707743168168387643797274914963482844224s+1204779998286697546929705082841769576290141607076=0$
Found by David Ellsworth in December 2024, based on the $s(107)$ found by Károly Hajba in November 2024 and the $s(54)$ found by Joe DeVincentis in April 2014.
Improved by David W. Cantrell
in January 2025.
Improved by David Ellsworth in February 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(87)$ found by David Ellsworth in December 2024.
Optimized by David Ellsworth in February 2026.

$s = {}^{20}🔒 = \Nn{9.88815305375857}$ $3528s^{10}-(300552+15456\sqrt{2})s^9+(11614660+1180832\sqrt{2})s^8-(268405824+40209136\sqrt{2})s^7+(4111948776+801750848\sqrt{2})s^6-(43682208312+10328732976\sqrt{2})s^5+(326223055436+89277369408\sqrt{2})s^4-(1692962073984+518553084040\sqrt{2})s^3+(5849274524474+1954912407552\sqrt{2})s^2-(12163170266098+4347871933856\sqrt{2})s+11572065260145+4353477802040\sqrt{2}=0$ $254016s^{20}-43279488s^{19}+3506260608s^{18}-179642577984s^{17}+6530192527760s^{16}-179088304328704s^{15}+3846118270819200s^{14}-66261902137415296s^{13}+930479746642904384s^{12}-10759858027891736896s^{11}+103070340120029179008s^{10}-819709665351861223904s^9+5405590814889373243192s^8-29412949608198679086720s^7+130831566348158107359392s^6-468664620024162429231904s^5+1321046745485882223459068s^4-2825402176181244872057384s^3+4315682289270565775115128s^2-4199847844458434080013540s+1959329723251932809573425=0$
Found by David Ellsworth in November 2024, by adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Improves upon the $s(88)$ found by Erich Friedman in 1997, which extended the $s(41)$ found by Charles F. Cottingham in 1979.
Improved independently by both David W. Cantrell and David Ellsworth in January 2025.
This new technique, which will need to be applied to about 20 additional packings previously thought to be finished, independently found by both David W. Cantrell and David Ellsworth.
Improvement by Thomas Schadt pending.

$s = 10$
Proved by Hiroshi Nagamochi
in 2005.

$s = 7 + {5\over 2}\sqrt 2 = \Nn{10.53553390593273}$
Adds two "L"s to the $s(65)$ found
by Frits Göbel in early 1979.

$s = {}^{8}🔒 = \Nn{10.61138823373863}$ $24s^4-(1400+352\sqrt{2})s^3+(27061+10102\sqrt{2})s^2-(218629+97462\sqrt{2})s+641430+317240\sqrt{2}=0$ $576s^8-67200s^7+3011120s^6-72041376s^5+1033920257s^4-9243724322s^3+50697397293s^2-156795019420s+210150009700=0$
Found by Károly Hajba
in September 2024.
Extended the $s(37)$ found by Evert Stenlund in early 1980.
Improved by David W. Cantrell and
David Ellsworth in November 2024,
by extending the $s(37)$ found by
David W. Cantrell in September 2002.
Improved by David Ellsworth in December 2024.
Further improvement pending.

$s = \Nn{10.70383477210707}$
Found by Thomas Schadt in December 2025, using a simulated annealing program he wrote, starting from randomness.
Refound and refined by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from randomness.
The third record-setting packing found that is doubly semi-primitive.
Not yet analytically optimized.

$s = \Nn{10.80789399144854}$
Found by Thomas Schadt in December 2025, using a simulated annealing program he wrote, starting from randomness.
Refined by David Ellsworth in December 2025, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = {}^{32}🔒 = \Nn{10.82297973416944}$ $2401s^{32}-701092s^{31}+96532058s^{30}-8317521660s^{29}+501462496833s^{28}-22376320364004s^{27}+760407348527794s^{26}-19848555869936524s^{25}+391901782318184024s^{24}-5471444559723346548s^{23}+39629249963743971218s^{22}+353696512578770314160s^{21}-16715802829777049603255s^{20}+292780863461637719269068s^{19}-3146950297418725382386108s^{18}+17996731060753457271434416s^{17}+58068494391477930003466013s^{16}-2710032149344351718373370304s^{15}+35897708426171881544444261010s^{14}-321777454480517334707593455472s^{13}+2318288875965343221612347387046s^{12}-15552954072951813301922897418028s^{11}+110324583017828739076547861980256s^{10}-814314195444111054964406531808040s^9+5535691601850017528577776913992458s^8-31551366481166818299554205010301876s^7+144126073330457054877480503221027356s^6-514883701839008702934553147517634876s^5+1404879821405123399570947930851054697s^4-2828245713344639081326540531975430356s^3+3961424449372054137804580730689208222s^2-3448216591537867586914544066365388952s+1404226509074988020217588819457924033=0$
Found by David Ellsworth
in November 2024, based on the
$s(53)$ found by David W. Cantrell in September 2002.
Improved by David W. Cantrell
in December 2024.

$\begin{aligned}s &= 10-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{10.84666719284348}\end{aligned}$
Found by Károly Hajba
in November 2024.
Is an alternative packing for an extension of the $s(54)$ found by Joe DeVincentis in April 2014.

$s = {}^{144}🔒 = \Nn{10.92591939016138}$ $827055562171106304s^{144}-1028746038533781676032s^{143}+614161678485829473038592s^{142}-236434676228380170030685824s^{141}+66980317656649750618706949945s^{140}-15177024487481458274595256127868s^{139}+2918547306910189530506516307375692s^{138}-495729497699611401035280356277680196s^{137}+76089742722234909946196429571333895452s^{136}-10671270893794158877895712177505617523828s^{135}+1375524471618065793880112212598599154653012s^{134}-163680131457973276468171788500327444221444244s^{133}+18041122005484577974071987969780608673747943250s^{132}-1845193374227011301793607038733620442135422735440s^{131}+175195858247481457503205356072668634514707875635464s^{130}-15447131106769150898435521166852265757689347117679048s^{129}+1266589004665859543076277348204652895182867619945255898s^{128}-96872971664861242811412554825190666410176955662558075472s^{127}+6939847565941400486198515625279425362463969118244043805152s^{126}-467586251641955795650527642523133741785627313804139361021412s^{125}+29712916686346155911816329503684280580564307321782343390920945s^{124}-1781869546071794107416303074386560117711940471499978105624204512s^{123}+100692141845407597467752420973765886037444159069768871392617305440s^{122}-5345881769519666404549787030353246578976419001556791199058788111868s^{121}+265745153241256297998761893367366891336827254195720886433343716673314s^{120}-12331900216784265711374361173174853992773142985602439082869045995043620s^{119}+533124642048160878797787516816371559269658695957365696289239653586152536s^{118}-21454346119467121162152752804366671634925654995122131901651080854842653908s^{117}+804137509033612105675830501719617753487476429965727315635094765154608429509s^{116}-28127107800477454224608783588306926887609791011395988349691834833987158467700s^{115}+921246416930684576450726414356413649291858434685374246183966749784918747685898s^{114}-28390357852514647575635304153251006733509079472468682793162434720668520497597508s^{113}+828166533055942389980323072853474773741844516945352000111380951692338661814562750s^{112}-23019859946745245564619050854509564382765776816954463496160143457200568875655263500s^{111}+613584341652374269884917793377224034449208943707706722281248832851205212434542935116s^{110}-15759013359756201601425954086600815916125169227043330927990424168462555090123951648000s^{109}+390943459868126235255883761901077907242607987729337591414340253957672403607173215398650s^{108}-9366575279423493930590320245377238666248026521863698157365356790349162864711782205289356s^{107}+216352869031916941103974168883097076410229240427531516289660856895884573411040138391989008s^{106}-4808372152188842416685049720889157609648589700259906966073914535288536332426660045281665576s^{105}+102730532296869024506560636386673680062422941215075875113492165335062102679114067711595589286s^{104}-2111527924549381695924838025670328572466346746690319171877130768902470515781848199414410720936s^{103}+41831905131678357252471218884771916619975870088217821433418140699364041647672903374550946065106s^{102}-799937696943205886362309315869391648647226626812787165601913537214769018438999347145254423256568s^{101}+14750154008419083794676619125813636103060922241286812273744920967608624399426277671233689970750816s^{100}-260975431069180213161379096976636911786047810516915330900553443161331183641061936532309426166815268s^{99}+4392344885718180821788009252695595386788391929454071852322904840485213772415318679286843092164107326s^{98}-69572449764366603777509810702402733716637668631656846420562636123468612787624389077681592721918843488s^{97}+1027097836001128208475480477047724529415634205788396193786255672967383685469857620307489972997358673526s^{96}-14074624684738338344020983555256300931561811169605846968675883195494170910423934868009815189998913131444s^{95}+180660626905403779319995260410889237792268816586814685932269189175254703565979443977412180365929218465280s^{94}-2245499788361448111606620260028829428953487326596025966685205057335044673291677551140402757171317681417428s^{93}+28671416193365649478884844283419050835810755858475230587266033381784144945601933009018387101662354025540734s^{92}-394217288266871322892333222337975011689661831536401018523536110370589489067761042286000949956400639666042316s^{91}+5703863195259166781025704560694159480317740632678599117279192239656528808349240434327194661001025914590969266s^{90}-80059468446622758198796944179244414275773216304886340967166998450686606958848716898739757721312778409184831024s^{89}+1004138763079804275697043916259570490495614067448884056635127265173895325706519305292407648765439933024547087131s^{88}-10573856961381771762913436897417012389280833263671551808955616987901353562852378162633710797010666619591470102324s^{87}+89592018192513343122608501728306468444080160150570013764942281507872879104525037337900571968575192851768883617072s^{86}-646389054343915846566881857708086454471929561189842086401340461316692430061948367583499858586482550023315255624968s^{85}+6456718039810642099097859415092985378580884294959550540027954767107746733301451637312622694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Found by Károly Hajba
in October 2024.
Improved by David Ellsworth
in November 2024.
Extends the $s(11)$ found by Walter Trump in 1979.

$s = \Nn{10.99683777797875}$
Found by David W. Cantrell
in February 2025.
Bounds the $s(n^2-n)=n$ conjecture to $n \lt 11$.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = 11$
Proved by Hiroshi Nagamochi
in 2005.

$s = 8 + {5\over 2}\sqrt 2 = \Nn{11.53553390593273}$
Adds three "L"s to the $s(65)$ found
by Frits Göbel in early 1979.

$s = {}^{12}🔒 = \Nn{11.60139979378801}$ $196s^{12}-26488s^{11}+1651440s^{10}-62766716s^9+1618595997s^8-29815750560s^7+402059235258s^6-3996945599928s^5+29059125001479s^4-150620866376828s^3+528127756018414s^2-1124357849826920s+1098765914618225=0$
Found and improved by David Ellsworth in December 2024, by extending the $s(102)$ found by Károly Hajba in September 2024 and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Improved by David W. Cantrell
in January 2025.
This new technique, which will need to be applied to about 20 additional packings previously thought to be finished, independently found by both David W. Cantrell and David Ellsworth in January 2025.

$s = {}^{59}🔒 = \Nn{11.77617894651987}$ $103727375923893369330209296s^{59}-66459520111369498843387398048s^{58}+20984329777930027658962486463944s^{57}-4351984463195143798965010730184216s^{56}+666687597816139249593148784451810689s^{55}-80437601187195761145119711505429371872s^{54}+7958851957658374209199569980591594814596s^{53}-663970184693824249587042395525558012303624s^{52}+47656218937812393681498721763550477091389692s^{51}-2988182303866249333810381839547924169756480176s^{50}+165654430357465514192443072787971581861849937032s^{49}-8197124382146014341147836594827868966674867968640s^{48}+364894874922743341692547213648315564575454126270264s^{47}-14706889078955847476754230966169066617800637472971200s^{46}+539587226078608281436460559285520668919736259659279616s^{45}-18103632741105160954207882647980809072466693499707260480s^{44}+557585345861528934437763964865981218857543916416201779248s^{43}-15817262947846928633382408483301037584257527948351500326912s^{42}+414431320216485974680637436345665342303734151662538663236704s^{41}-10053720706600678001976425846862433637452130993583771993589760s^{40}+226284569647540177334064359603411274762166518842198552714789200s^{39}-4733759152517412405655786220989299972751266809408192403262333952s^{38}+92179303857101690590943463901100886562361422006241927175500288192s^{37}-1672960880610675687561599553613456404353499890450249160517926849152s^{36}+28328155965071830750649114297222577573737263041261615183323834187456s^{35}-447919533736344610230445733776110830046882845020254305668342798162176s^{34}+6617899464680638705931910964826771950378130327080957719202859033233408s^{33}-91409906287902988999275876436122248409672350798530677686891645769950208s^{32}+1180762818586624946451420285182606032560987685384480825492161016877865728s^{31}-14266035758552976658869792803314865573221372493976743630736058442671376384s^{30}+161221645021528693481600237931024869745295739849234342018943309786459454464s^{29}-1703981869218340104518205680834638338093845417894482077716237293956551886848s^{28}+16838533218881790259245529284324114444679442816984936069651292036750863016960s^{27}-155507096005148562687360644397398099803664438720968695334519092641261137022976s^{26}+1341352207710179476924096858857901812229831030179784620098151738073888619790336s^{25}-10798171511029759592830752371596870956414204206741621062504417333405685159886848s^{24}+81052170118119047525304211048570354210314271885674274312331023002336366706393088s^{23}-566627077124483750134722356793280904558159317087277285384745996320116490200023040s^{22}+3684442919811890828463311011580309432000999223341205429086714471473151446435495936s^{21}-22249274396690493322552083876993000099081166687059558607644004406914233102047379456s^{20}+124552050122653107194057662720770265183766531498214018152534598663746678264316035072s^{19}-645029361671781299599296494361306865872819321944021774312439765584515217495396712448s^{18}+3082971153505608860762253281131318317152697408173385645293686789199851377994502242304s^{17}-13562323036880913892655449542328326722966446788097216930557569549306753603730726191104s^{16}+54740314326884958907759200874363245537500276472385155660821578429631150980435833520128s^{15}-201981782442364856363618820987120260153075419046198318261369689063877557069555018235904s^{14}+678454620619581023501502437862982780835374672007458222016728490158146019922598795673600s^{13}-2064436047222561869480414730777110273397659211723851059092229157758529074339050325803008s^{12}+5657822692000862951856502458470034351072956359376976212464506914883038250278596768694272s^{11}-13870526150873954065582070659971088194709355013075244287120899726399280664638800391045120s^{10}+30169237202060583946794207604883117238012283183329248484260467345739886960763300156014592s^9-57638821146611472988664430990500393055759650090200986958760477435746210148744278438838272s^8+95531208027292543052966021814412362080198304100349288922028438203568859274024955521531904s^7-135203344761287990105966704145072151250652183716897669760569045084963600901666937270960128s^6+160041814214673878562014258437703372584550901902947932679906068352094502186763966417469440s^5-154016683298825557276438882149337620795791870685777351688672777397378372447441511881113600s^4+115651046550947142095553287942602374474866214498930249064530890702755579234634640130048000s^3-63504407975087039964592928651896133586911076368041736628455664161047965497809019338752000s^2+22656559274302339443118663917820262176712502445618848038990511314458453343489281228800000s-3936908799940438911888396204756925415661006746097453884053057654812145020629208268800000=0$
Found by David Ellsworth in December 2024, based on the
$s(39)$ found by David W. Cantrell in August 2002.
Improved by David W. Cantrell in December 2024.
Improved by Thomas Schadt in December 2025, using a simulated annealing program he wrote.
Optimized by David Ellsworth
in January 2026.

$s = {21\over 2} + {1\over 2}\sqrt 7 = \Nn{11.82287565553229}$
Extends the $s(86)$ found by
Erich Friedman in 1997.

$s = {}^{40}🔒 = \Nn{11.82509196821368}$ $6765201s^{40}-2845098648s^{39}+580218189444s^{38}-76444672534332s^{37}+7313700827489408s^{36}-541478198967270496s^{35}+32282461276159060998s^{34}-1592401650332728651212s^{33}+66264078654438595433370s^{32}-2360056578357132809629748s^{31}+72739322148379315505619294s^{30}-1956653332398832550681084364s^{29}+46240556849238042597672084213s^{28}-964953122823176982192672366676s^{27}+17849625763191276058228284471146s^{26}-293485855211047587757484370460572s^{25}+4296830596115061642805514450068373s^{24}-56062026207183424768909339152161084s^{23}+651753580603034236066787916795623964s^{22}-6743132871649823544913588155645442656s^{21}+61932541245281919280766873166108057833s^{20}-502864388097908644186877008296638914520s^{19}+3586164305875893817104519150255679222830s^{18}-22233865523258917656574596200774765950936s^{17}+117836973554812356574280571124747241305248s^{16}-517669550834165752088850944695261707233564s^{15}+1760771073593491738800908345083273891209716s^{14}-3688451224681060758343247748983014976447272s^{13}-3038946790141535208034747244682683990472658s^{12}+71735791138334020614589956733651574834185892s^{11}-381139020218879456095819418101710611273627456s^{10}+1267518021830549968006069595148713683301836788s^9-2664006348624383405491809555245054393205103691s^8+1793492920161456183420660943446317121911023248s^7+10946284003184957815907569175715318463243584102s^6-52105146308542520023979647560364421311730107464s^5+129236789770564934710391555764344704402747982753s^4-211311873022032856645395618371922980818558351272s^3+230091479010044538699295485926277667974383569488s^2-153189145317923709976188931598288062177638461184s+47601488782509082091988946077133401472942285824=0$
Found by David Ellsworth in November 2024, based on the $s(69)$ found by Maurizio Morandi in June 2010. Improved by David Ellsworth in December 2024, including using the technique from the $s(53)$ improved by David W. Cantrell in December 2024.
Improved by David W. Cantrell and David Ellsworth in January 2025.

$s = {}^{20}🔒 = \Nn{11.88130621809000}$ $512s^{20}-102912s^{19}+9795328s^{18}-586977280s^{17}+24833686144s^{16}-788425574784s^{15}+19488070063424s^{14}-383997272257792s^{13}+6125462923505760s^{12}-79879749327995680s^{11}+856196296518997712s^{10}-7556258973091735104s^9+54814239105846342840s^8-325093619883814992744s^7+1561244435270252784700s^6-5979678235068566487856s^5+17845354889140483660018s^4-40018900764845892469246s^3+63498519812052276506109s^2-63640403941763504944184s+30347196858954912683392=0$
Found and improved by David Ellsworth in December 2024, adapting/extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Unimproved form similar to the $s(70)$ found by Erich Friedman in 1997.
Improved by David Ellsworth in December 2025, using his modified version of Thomas Schadt's simulated annealing program.
Optimized by David Ellsworth in January 2026.

$s = {}^{8}🔒 = \Nn{11.91119052015898}$ $5617s^4-(259504+12612\sqrt{2})s^3+(4502202+422478\sqrt{2})s^2-(34773984+4731588\sqrt{2})s+100926915+17726970\sqrt{2}=0$ $5617s^8-519008s^7+20936788s^6-481754784s^5+6917560482s^4-63487737120s^3+363767813964s^2-1189915431840s+1701575795025=0$
Found by David Ellsworth
in November 2024.
Improved by David W. Cantrell
in November 2024.
Improved by David Ellsworth
in November 2024.
Extends the $s(88)$ found by Erich Friedman in 1997; adapts and extends the $s(37)$ improvement found by David W. Cantrell in September 2002.
Further improvement pending.

$s = \Nn{11.95654869347733}$
Found by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(131)$ found by Károly Hajba in November 2024.
Not yet analytically optimized.

$s = \Nn{11.99143643966336}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Bounded the $s(n^2-n)=n$ conjecture to $n \lt 12$.
Improved by David W. Cantrell
in March 2025.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = 12$
Proved by Hiroshi Nagamochi
in 2005.

$s = 9 + {5\over 2}\sqrt 2 = \Nn{12.53553390593273}$
Adds four "L"s to the $s(65)$ found by Frits Göbel in early 1979.

$s = {}^{16}🔒 = \Nn{12.60090777851301}$ $4s^{16}-664s^{15}+51488s^{14}-2476628s^{13}+82758797s^{12}-2038683388s^{11}+38335080074s^{10}-561971068668s^9+6500206337793s^8-59622322943944s^7+433020664362230s^6-2468567342450368s^5+10847755732088938s^4-35571513994776316s^3+82161582314776868s^2-119456945705052640s+82323170428761425=0$
Found by David W. Cantrell in January 2025 by switching to the rotationally symmetric form of adding an "L" to the $s(123)$ found and improved by David Ellsworth in December 2024 and improved by David W. Cantrell in January 2025 with a technique independently found by both David W. Cantrell and David Ellsworth.
This is the first record-setting packing found that is doubly semi-primitive.

$s = 5 + {11\over 2}\sqrt 2 = \Nn{12.77817459305202}$
Found by David Ellsworth
in January 2025.
Extends the $s(85)$ found by Erich Friedman in 1997 how the $s(66)$ found by Evert Stenlund in early 1980 extends $s(26)$.

$s = {23\over 2} + {1\over 2}\sqrt 7 = \Nn{12.82287565553229}$
Found by David Ellsworth
in November 2024.
Improved by David W. Cantrell
in December 2024.

$s = {}^{84}🔒 = \Nn{12.83100282216725}$ $15197358585941502961s^{84}-14964876021186235227298s^{83}+7272595222812913346894003s^{82}-2325304163405568837985711872s^{81}+550199014011902976417247729787s^{80}-102744907802613577303858434436162s^{79}+15770698108244892301462590278910373s^{78}-2046188803886728893706559592846166940s^{77}+229043365901827791271124033850409760904s^{76}-22465861798723536733581343377318969312880s^{75}+1954663063042408634836851737534542472988316s^{74}-152349951255709169941326823887066293346193624s^{73}+10723829667261655134952521210622665945643784968s^{72}-686326223380322052977518105478585500875790507640s^{71}+40166977336061074256998638756973363193038422762460s^{70}-2160190971013429916138221567315873809290109191227128s^{69}+107210371553542169414048201729669662441830314197024226s^{68}-4928367697648782759676964001463807238961374648608591036s^{67}+210520282427005491506665365123774698375707578804045991166s^{66}-8380048643861127985429658865920308979713621168800528636888s^{65}+311642979584387473990467919553614272825113220053872893167686s^{64}-10851807764621222577237269161423835499294065873227886719724876s^{63}+354529200442735318891489329639289130837080983716609686105844682s^{62}-10886472654344044554989820001276664684754352275834326626073676952s^{61}+314707830740258412075262705001018514633277114685570369659228230886s^{60}-8577131365576184973326383639361903919077060020501484349486798462044s^{59}+220676659864024529834222826550027169451751436159279143212643304100750s^{58}-5366108560854773099323062782542881689148622349128272868132841503304592s^{57}+123455035085747414638373053783111838415779807885794670228653620606756656s^{56}-2689771757042576617382266796760465801989789910549251473683839639432673824s^{55}+55545344660722669971771487790593823603421739027747117357084896573560148132s^{54}-1088014439077084998878807873686497414222608555337013172037085789785737027128s^{53}+20228760646280093419477169754640747236777672568049390223893488286064642987733s^{52}-357199319350132942687604624139422297924839641572382189922202437482576762356002s^{51}+5993578492259294457183648659781874954535555138522882079496926437004560861449475s^{50}-95607888374734158621452463758465962667527766889151119870233929745815458197870464s^{49}+1450449889734808340651325100183427866248693256913059099309145940155004921623663049s^{48}-20934174377970907243549394685205967930306737373954662348822439943609230423042271998s^{47}+287521214531466154844493934641558475944025274728304204144868014752283863593850841071s^{46}-3758693037139824522492196965950225707948971038134502497610180651441239588793706153212s^{45}+46776156166368620484370637225042926774946526631884418623684508412824264790410695322622s^{44}-554213498552523069017939780278197270272937025886205771218096298876624373232634030307268s^{43}+6251950732475223994427997225766260343105672057938693543298710223719438546214455983422174s^{42}-67148743958288503662262110524028512420572740818205612649788782970039163837877122850002224s^{41}+686624504976838308996418090639964921283797740974390943510927597462462930108635451388956672s^{40}-6683624210795197215545578694769827454378481281503786991205805051093122539577000607518492112s^{39}+61922038451063372274525115664711614781745351857035675107097735737909228871958371309771201252s^{38}-545914626236899516397451030889695660434986026741727925639735898323422010790358226899471142192s^{37}+4578596394614695256434177101284715486045596223316501591974779425322438106683698243945133312657s^{36}-36519557390596147521078972913970989194289986313602140505586138261168362886227188120613809866274s^{35}+276908452084060463131678792053749272309227620650243616383774652462199807972600272617027105313715s^{34}-1995125618573564239441569968811948768877908975264805329921557835273555692406902924736432832072656s^{33}+13652302794929101077954196396097847995447641729570461853834329101037063785834852682999232389861183s^{32}-88673260640330913380557229503418078388290914029958962882747927428883329136607168216172985057506786s^{31}+546323078740762777551499851151749299500660383154079131025685851053212263193896570529297338202937057s^{30}-3190532419019051137207920527310810985363082905053938554402580063971581164010258646801817355978728172s^{29}+17647597956569345600264156757559024534779396494222803173516820647844865659043181903919579547455993405s^{28}-92370185046480996167766003031798069104355227778256861958172638341670501748986559657158143287384201746s^{27}+457063596008246343843246947652497192740604188316726144667941077879214729203288916667939807194976754475s^{26}-2135754321030318339803946148640432723925062036840170746475770465651607920747496502359936750118636877936s^{25}+9413245107777963706229663058434971410824508757077884118452667020417078901491483099639931354293862639251s^{24}-39081567129013269416348118385885630160005422872090989882489179092615712699596245833330552131314742641914s^{23}+152623748753623567775238681507133061671421661548557325363509181608802271715863896420772452309156580126893s^{22}-5597550217493047544616425327334081249500189808950755859993395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Found by David Ellsworth and David W. Cantrell in January 2025, based on the $s(53)$ found by David W. Cantrell in September 2002 and the $s(69)$ found by Maurizio Morandi in June 2010, adapting and extending the $s(69)$ improvement found by David W. Cantrell in August 2023.

$s = {}^{4}🔒 = \Nn{12.88166675700900}$ $23s^4-1110s^3+19960s^2-158164s+464677=0$
Found by David Ellsworth in November 2024, based on the $s(70)$ found by Joe DeVincentis in April 2014.

$s = \Nn{12.93171183926903}$
Originally found by David Ellsworth in December 2024, by combining two slightly modified copies of the $s(41)$ found by Joe DeVincentis in April 2014.
Improved by David W. Cantrell
in February 2025.
Improved independently by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program. Resembles the $s(179)$ found by David Ellsworth in December 2024.
Not yet analytically optimized.

$s = \Nn{12.95851388606690}$
Found by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(182)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024 and January 2026, with 27 squares removed.
Not yet analytically optimized.

$s = \Nn{12.98219172354800}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n)<n$ for $n=13$.
Improved by David Ellsworth in December 2024.
Improved by David W. Cantrell
in March 2025.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = 13$
Proved by Hiroshi Nagamochi
in 2005.

$s = 10 + {5\over 2}\sqrt 2 = \Nn{13.53553390593273}$
Extends the $s(65)$ found by Frits Göbel in early 1979. This alternative, converting the $s(65)$ augmented by five "L"s into a primitive packing, found by Károly Hajba in November 2024.

$s = 13 + {4\over 7} = \Nn{13.57142857142857}$
Found by David Ellsworth in December 2025, by combining two copies of the $s(50)$ found by Thomas Schadt in December 2025 (using a simulated annealing program he wrote, starting from randomness) and optimized by David Ellsworth.

$s = {}^{8}🔒 = \Nn{13.61898898660160}$ $54s^4-(3936+864\sqrt{2})s^3+(98658+33466\sqrt{2})s^2-(1047336+434398\sqrt{2})s+4049739+1891234\sqrt{2}=0$ $2916s^8-425088s^7+24654168s^6-774089568s^5+14674175968s^4-173849337008s^3+1265419604436s^2-5196681822080s+9246853882609=0$
Found by Károly Hajba in November 2024, extending the $s(102)$ he found in September 2024. Improved by David Ellsworth in November 2024 by adapting the $s(37)$ improvement found by David W. Cantrell in September 2002. Improved by David Ellsworth in December 2024.
Further improvement pending.

$s = 8 + 4 \sqrt 2 = \Nn{13.65685424949238}$
Adds an "L" to the $s(148)$ that continues a pattern found by Frits Göbel in early 1979.

$s = 6 + {11\over 2}\sqrt 2 = \Nn{13.77817459305202}$
Found by David Ellsworth
in December 2024.
Based on the $s(233)$ that continues a pattern found by Frits Göbel in early 1979.

$s = {25\over 2} + {1\over 2}\sqrt 7 = \Nn{13.82287565553229}$
Extends the $s(86)$ found by
Erich Friedman in 1997.

$s = {}^{32}🔒 = \Nn{13.82302875075647}$ $2401s^{32}-931588s^{31}+169658874s^{30}-19231837912s^{29}+1515206475113s^{28}-87554613482844s^{27}+3800602014647796s^{26}-123749571598485028s^{25}+2895774212866682688s^{24}-40503170127651197920s^{23}-85797782465034115616s^{22}+22358489056150565928884s^{21}-670371983793922205889766s^{20}+11585516453329663611601440s^{19}-107946265554474207035275274s^{18}-325416702082583543878844088s^{17}+31678001381593454789856242308s^{16}-627531463707625262161828471384s^{15}+7738051841906036676459384893372s^{14}-65825971528188631650991553430380s^{13}+390859965437244296776723867974104s^{12}-1981263185614787515387345660456708s^{11}+20697660792797450992561898119685608s^{10}-341458570720939415832614447938585072s^9+4331777593881436240458282284914909233s^8-39425956807257141453494313251818730308s^7+265332233224572263186725591178500406376s^6-1341713349412185465951061216890867086592s^5+5078477010422928472451310243101300746580s^4-14041240100101609173845558593200611383568s^3+26892648324465918884312724426364749628480s^2-31980711702815550619890642368200074811200s+17821422876028786503270705680802036472000=0$
Found and improved by David Ellsworth in November 2024 and December 2024, based on the $s(53)$ found/improved by David W. Cantrell in September 2002 and December 2024, respectively.

$\begin{aligned}s &= 13-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{13.84666719284348}\end{aligned}$
Extends the $s(54)$ found by Joe DeVincentis in April 2014.

$s = {25\over 2} + \sqrt 2 = \Nn{13.89540982243640}$
Found by David Ellsworth in January 2025, using a computer program he wrote.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = \Nn{13.93513929847193}$
Found by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(209)$ found by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(240)$ converted in December 2024 from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024 and January 2026, with 31 squares removed, with 29 squares removed.
Not yet analytically optimized.

$s = \Nn{13.95698416446504}$
Found by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019, improved, fixed, and improved by David Ellsworth in December 2024, December 2025, and January 2026, with 29 squares removed.
Not yet analytically optimized.

$s = \Nn{13.97442960739443}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n)<n$ for $n=14$.
Improved by David Ellsworth in December 2024.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = 14$
Proved by Hiroshi Nagamochi
in 2005.

$s = 11 + {5\over 2}\sqrt 2 = \Nn{14.53553390593273}$
Adds six "L"s to the $s(65)$ found by Frits Göbel in early 1979.

$s = 14 + {4\over 7} = \Nn{14.57142857142857}$
Adds an "L" to the $s(171)$ found by David Ellsworth in December 2025 by combining two copies of the $s(50)$ found by Thomas Schadt in December 2025 (using a simulated annealing program he wrote, starting from randomness) and optimized by David Ellsworth.

$s = {}^{8}🔒 = \Nn{14.61898898660160}$ $54s^4-(4152+864\sqrt{2})s^3+(110790+36058\sqrt{2})s^2-(1256676+503922\sqrt{2})s+5199723+2359962\sqrt{2}=0$ $2916s^8-448416s^7+27711432s^6-931104720s^5+18929518528s^4-240795061296s^3+1883132387964s^2-8311787117640s+15898277993841=0$
Adds an "L" to the $s(172)$ found by Károly Hajba in November 2024, improved by David Ellsworth in November 2024 by adapting the $s(37)$ improvement found by David W. Cantrell in September 2002, and improved by David Ellsworth in December 2024.
Further improvement pending.

$s = 9 + 4 \sqrt 2 = \Nn{14.65685424949238}$
Adds two "L"s to the $s(148)$ that continues a pattern found by Frits Göbel in early 1979.

$s = 7 + {11\over 2}\sqrt 2 = \Nn{14.77817459305202}$
Found by David Ellsworth
in December 2024.
Based on the $s(233)$ that continues a pattern found by Frits Göbel in early 1979.

$s = {27\over 2} + {1\over 2}\sqrt 7 = \Nn{14.82287565553229}$
Extends the $s(86)$ found by
Erich Friedman in 1997.

$s = {}^{40}🔒 = \Nn{14.82445114612408}$ $6765201s^{40}-3656922768s^{39}+958435862652s^{38}-162258762163956s^{37}+19944495118895680s^{36}-1896817614997924732s^{35}+145244607337067116310s^{34}-9200231755130148537584s^{33}+491532552940826676421598s^{32}-22471266124455622725351620s^{31}+888775367178021320983990528s^{30}-30670374642432504931208393536s^{29}+929494097506723220926745638527s^{28}-24862568075564651560878275579812s^{27}+589158528037990588430776578667278s^{26}-12400310845434669601706739856078348s^{25}+232178750736807601084495428509773447s^{24}-3869279274647141302600286447060842028s^{23}+57359828211423634121225283370282978472s^{22}-755020486735335189866154838905984601284s^{21}+8794114159490040759213225924279409129174s^{20}-90126731671943479290165241436087360926864s^{19}+805381183069421598181526905692816890290502s^{18}-6181562346162193722542638238059633411805108s^{17}+39656170956144484959733239448754004343189341s^{16}-200557184022962853819962178120475887680432104s^{15}+668299828757988666056469128930860611880478162s^{14}+47787548247188382784775375655144589392168024s^{13}-20274687460176342070123115001595419340631113986s^{12}+171806640590364466782586102889249290168160938968s^{11}-887645374096288133345472520371583078066772417438s^{10}+2995081869452847889056141420143309758645762048480s^9-4490901292932914262604076259256067123710050815195s^8-18138284909745955939878418372807115317752603411176s^7+165869320264318841451647718375161237581492991097810s^6-695527958949195756061996697585676748194320009594040s^5+1948735509481744597514888658808885791132097191929625s^4-3838064311683628303762259793886602147590339042385000s^3+5169742707762800848775847149992971475362078199650000s^2-4315559969700226767159393981391684637349630444000000s+1694040395896101772224866098180283214283057728000000=0$
Found by David Ellsworth in December 2024, by extending the $s(128)$ he found/improved in November/December 2024, based on the $s(69)$ found by Maurizio Morandi in June 2010, and using the technique from the $s(53)$ improved by David W. Cantrell in December 2024.
Improved by David Ellsworth in January 2025, using the technique from the $s(128)$ improved by David W. Cantrell and David Ellsworth in January 2025.

$s = \Nn{14.87253189075152}$
Found and improved by David Ellsworth in December 2024, adapting/extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Unimproved form similar to the $s(70)$ found by Erich Friedman in 1997.
Improved by David Ellsworth in December 2025, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = \Nn{14.89397859563780}$
Found by David Ellsworth
in November 2024, by extending the $s(88)$ found by Erich Friedman in 1997, and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002 and the $s(130)$ improvement found by David W. Cantrell in November 2024.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = \Nn{14.93783044811097}$
Found by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(239)$ found by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(272)$ converted in December 2024 from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024 and January 2026, with 33 squares removed, with 31 squares removed.
Not yet analytically optimized.

$s = \Nn{14.95868244078260}$
Found by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(240)$ converted in December 2024 from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024 and January 2026, with 31 squares removed.
Not yet analytically optimized.

$s = \Nn{14.97421396826961}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n)<n$ for $n=15$.
Improved by David Ellsworth in December 2024.
Fixed by David Ellsworth in December 2025.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = 15$
Proved by Hiroshi Nagamochi
in 2005.

$s = 12 + {5\over 2}\sqrt 2 = \Nn{15.53553390593273}$
Adds seven "L"s to the $s(65)$ found by Frits Göbel in early 1979.

$s = {17\over 2} + 5 \sqrt 2 = \Nn{15.57106781186547}$
Found by David Ellsworth in January 2025, using a computer program he wrote.
Continues the $s(26)$, $s(85)$ series found by Erich Friedman in 1997.

$s = {}^{12}🔒 = \Nn{15.60902282132495}$ $24s^6-(3008+632\sqrt{2})s^5+(143990+46836\sqrt{2})s^4-(3498934+1393560\sqrt{2})s^3+(46342192+20798868\sqrt{2})s^2-(320302448+155625644\sqrt{2})s+907787225+466763736\sqrt{2}=0$ $576s^{12}-144384s^{11}+15160736s^{10}-915791264s^9+36096990788s^8-988137514952s^7+19384693200660s^6-275725098032832s^5+2830331680270636s^4-20490408263563724s^3+99460046477154240s^2-290971312817869664s+388340875383845233=0$
Found by David Ellsworth in November 2024, by extending the $s(102)$ found by Károly Hajba in September 2024 and improved by David W. Cantrell and David Ellsworth in November 2024.
Improved by David Ellsworth in December 2024.
Further improvement pending.

$s = 10 + 4 \sqrt 2 = \Nn{15.65685424949238}$
Adds three "L"s to the $s(148)$ that continues a pattern found by Frits Göbel in early 1979.

$s = 15 + {28\over 41} = \Nn{15.68292682926829}$
Found and improved by David Ellsworth in January 2025. Uses a rotational symmetry technique found by David W. Cantrell in January 2025, and the technique from the $s(293)$ improved by David W. Cantrell in January 2025.
This is the second record-setting packing found with a rational side length, thanks to the Pythagorean triple $\{20, 21, 29\}$ determining its tilt angle.

$s = {29\over 2} + {1\over 2}\sqrt 7 = \Nn{15.82287565553229}$
Found by David Ellsworth in December 2024, by extending the $s(128)$ and $s(205)$ he found/improved in November/December 2024, based on the $s(69)$ found by Maurizio Morandi in June 2010, and using the technique from the $s(53)$ improved by David W. Cantrell in December 2024.
Improved by David Ellsworth in January 2025, adapting the technique from the $s(128)$ improved by David W. Cantrell and David Ellsworth in January 2025.

$s = {}^{83}🔒 = \Nn{15.82660563342856}$ $15197358585941502961s^{83}-19054029395700781380870s^{82}+11796924216629042649354579s^{81}-4808108614769876120043468180s^{80}+1451030856404178122570781430789s^{79}-345799767275884788306353984453266s^{78}+67774498813564804634645374167057335s^{77}-11234508000945633407554425804213908384s^{76}+1607532344825153174801872971520978793189s^{75}-201668659118682967539387133057113450996510s^{74}+22454296891522560058771023859593458711158383s^{73}-2240889622579706580689718577985790904704763388s^{72}+202076916434677863681321953179975198173682882349s^{71}-16577690712578589166013893731214562354253387341274s^{70}+1244306501679531491606992301185434118328699167129267s^{69}-85872410283218882421430889383517071885115187773104056s^{68}+5471931933759042230241340087362460711039430660491160647s^{67}-323138905823957289824620179077427648515971541643473332874s^{66}+17742026808852576915081042170482522649963659530339729297241s^{65}-908282679855850349803743965878897364257354889699334907097884s^{64}+43465080817124537825282652035527867092782838207883438915180955s^{63}-1948672651714009470065517948906139063310384408556369781794367782s^{62}+82014228252452305308486172380001566000442685829960411231106701149s^{61}-3246186057482410594749950508613065532781882056581161088848208781448s^{60}+121030129123359970782879324721925653713348068894577966308153780376769s^{59}-4256774383890838346683359771242014391309685520375828879287765027421558s^{58}+141417395891604974478673542444206052527574423688874097457316926184111667s^{57}-4442950325166710767652318494176477005346284036871039689545149858425019188s^{56}+132143761554097136892656409949813501644806291750873673851743056646810847307s^{55}-3724278121653142074595785794031359776637918996855481246500264411247627583870s^{54}+99547162759226989443922275915301342990394148481013546431915854939056693130149s^{53}-2525451166316636919882169794346033300450573261760214407544561168810183261623712s^{52}+60851054822925884393362872088231784921476592249704525359339306693211978978637574s^{51}-1393407983201463082750103812667205031231534119903960161407503385026149533812640388s^{50}+30338905815230015350337507443629547938133322986372599251613731735431147148807587298s^{49}-628396204847897965060061083685759495211044889335692127791727849670706521484518435856s^{48}+12386583193977551660461241876390719826604576164873235895959785437607584760441587730192s^{47}-232433724444205471983096462597776075369365835330568989261391110164425311728212391035632s^{46}+4153323734217430140145070137613411920743170003732119525339537946385437046481746033523872s^{45}-70686380467717783902448903568646078963423628086622300763415121270383317250590261969131336s^{44}+1146011275499749445486974909121594149473099806147735475740109750306258973873693081841338630s^{43}-17701136698331714953844021503637463123650314086731790857767064280112784186506532020777615276s^{42}+260492277459372683985340515012561460001230750454462824800331349575718851511822995239394574714s^{41}-3652320068116991521691903234094675077664783199210342925565230028260558957233172085378428693648s^{40}+48786365568085939831648565613081789083584480449521988080326278540514634832953535303132015187720s^{39}-620776840486987337546656613456515581406101803594950169186131730819070047268868384631491241542280s^{38}+7523288801124157164696453440451668126537802179655932385275366967957982556047497857359919983352588s^{37}-86819738273128010242526385979967991470567862056150689620462183702554367768947413111616038233016416s^{36}+953774954925025648253304474733609592484747994431472390301868754639755982678716269084529930641443023s^{35}-9971081293651892380457737953978637874233272651346699205006125992828217866601482502757629011345866570s^{34}+99159113935527095843791193320314995744620431737333925602216138254954437828859473360567133620147394445s^{33}-937591882456243116942831217151994956616595933093755511723294563972551195048414611565248985607152821236s^{32}+8424667140126948167633953265193969962679276499025327494839034104790577492192909219817624591386758730279s^{31}-71892659494830178227616871456048478350079377496302951700870381917053589308972575388983039596535479420278s^{30}+582251421494115025057046976615977029976720394400712813199098003690123303907349827395875756580289035731733s^{29}-4471937667537034121585882112126200967254037591271575148365797097846048547850509627026833861786334101222136s^{28}+32543614650180016207874244187164028503629361855235911764700956406683943044733068359145297584031071397155994s^{27}-224183523933854882502864682078640188285826762871716486751957673723635189271192115484721934301078509843540820s^{26}+1460312333249097739412739848153412311136035088888884922686424538738960526601793346522957489286185673762602902s^{25}-8984148217876593612167877570397820635649870435669726928588777366245474742877218816201439836639845502892953560s^{24}+52134501077125370653590265849501908514957010403776225078820415168994018070692909692704843148671899014807836538s^{23}-284941297083362220078712187212822093342654560298933781771361807935255252780142074341846765908056419933065354916s^{22}+1464404571156043076914637252120097347255977567642248877431122565189765063022644175773142656036823662408630919410s^{21}-7064080779433205483997411627458329256081928935674885720573960927817446941144054662390438254917264293853571028424s^{20}+31919881457624033726193536951838036401111886302890312451699329606929040067283389942070240214272698940267541956493s^{19}-134801722329859078816742606094646334228067206837309021726684217834848607820988762454709381034223562103046457675998s^{18}+530707012396024913429576740380121941086737225273466472672403349930854389215481675641529836588022052334151834279003s^{17}-1942205476550803484882315247161577147422888428395169781637047795709098197490266699760943181834041758117906367397268s^{16}+6585767384476720675187977121595259536735035857996075281802198937005198489937485851461054520341053364378881846325263s^{15}-20614892940117095331097736607876870976314189060882921080291907574510740821764737500291921914248821586266401315666942s^{14}+59316353011229384909666880954476222015479423693489595098884075044567497728679992849404330173399583652230381335502305s^{13}-156117670782437502269990886042735284992957692699464203987769629925609348266900609376317045942894365095896351357414520s^{12}+373698270769972479755939957244422481116455737532015818487606025512111631824244935754777237146223432771503247544608560s^{11}-808048121315624822538485230387020159879703732818682364581834764813478780649668941927219429385916131623619099840318128s^{10}+1565583417239292630942979288694634658573835168674069318551077876822765011658251873382962913367365429573513955164437064s^9-2691230551310123431856658453025381333299196967548104418289076204697729928184484894708129837595375747556018998818512192s^8+4054542066273187475893579325962915065845321084583224373643180486702374676704731329107411831884877578115801740253541904s^7-5270878662724145249388262906951781140123258747428763886318687210813398134336463475140327838867254422860035408448158432s^6+5792836201656880167367652921094881746142782983189532980946755918671194010952081295188183352763611672876752443571623568s^5-5233614558124453824846563012926178182173958634839557295689659286203290796724301748107667652717293231008062406343836800s^4+3732102862328364130585244845143109502510706215299361787475517662634187688740089578345011212002758654232473965037848832s^3-1969628432244786328367806409266293170003496844663625159204453312874075246094750590119459796978694116823499446641506304s^2+683920819913111461189983402245349089127449075173547013376466597761071475636877690918573610486424418270329963001143296s-117203977757280124647356183279343983773836101226288348510440504667964165618210763520372674407323269025478320565518336=0$
Found by David Ellsworth and David W. Cantrell in January 2025, based on the $s(53)$ found by David W. Cantrell in September 2002 and the $s(69)$ found by Maurizio Morandi in June 2010, adapting and extending the $s(69)$ improvement found by David W. Cantrell in August 2023.

$s = {}^{12}🔒 = \Nn{15.87607676541001}$ $6596s^6-(619856+19000\sqrt{2})s^5+(24226944+1444540\sqrt{2})s^4-(504036792+43818500\sqrt{2})s^3+(5886496273+662906450\sqrt{2})s^2-(36585732598+5001593450\sqrt{2})s+94529982167+15055629320\sqrt{2}=0$ $26384s^{12}-4958848s^{11}+426380416s^{10}-22179443008s^9+777414394344s^8-19344554351312s^7+350411708641272s^6-4655972006018640s^5+45039272938487593s^4-309348405383979564s^3+1432057511182322314s^2-4011939813305628468s+5144071303851334561=0$
Found by David Ellsworth
in November 2024, by extending the $s(88)$ found by Erich Friedman in 1997 and adapting the $s(102)$ improvement found by David W. Cantrell in November 2024.
Improved by David Ellsworth
in December 2024.

$s = {29\over 2} + \sqrt 2 = \Nn{15.91421356237309}$
Found by David Ellsworth in December 2024.
Similar to the $s(70)$ found by Erich Friedman in 1997.

$s = \Nn{15.93984676308969}$
Found by David Ellsworth in February 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(239)$ found by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(272)$ converted in December 2024 from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024 and January 2026, with 33 squares removed, with 1 square removed and 4 straightened.
Not yet analytically optimized.

$s = \Nn{15.95643304058435}$
Found by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(272)$ converted in December 2024 from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024 and January 2026, with 33 squares removed.
Not yet analytically optimized.

$s = \Nn{15.97559404379946}$
Originally found by Károly Hajba
in September 2015.
Bounded the $s(n^2-n)=n$ conjecture to $n \lt 16$.
Converted in December 2024 from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = \Nn{15.99091684780193}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n+1) \lt n$ for $n=16$.
Improved by David Ellsworth in November 2024.
Improved again by David Ellsworth in November 2024, by adapting a technique from the $s(240)$ found by Károly Hajba in September 2015.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = 16$
Proved by Hiroshi Nagamochi
in 2005.

$s = 13 + {5\over 2}\sqrt 2 = \Nn{16.53553390593273}$
Extends the $s(65)$ found by Frits Göbel in early 1979.
This alternative, converting the $s(65)$ augmented by eight "L"s into a primitive packing, found by David Ellsworth in January 2025, using a computer program he wrote.

$s = {19\over 2} + 5 \sqrt 2 = \Nn{16.57106781186547}$
Adds an "L" to the $s(227)$ found by David Ellsworth in January 2025 (using a computer program he wrote) which continues the $s(26)$, $s(85)$ series found by Erich Friedman in 1997.

$s = {}^{8}🔒 = \Nn{16.60257141234448}$ $150s^4-(11600+1720\sqrt{2})s^3+(324554+80686\sqrt{2})s^2-(3938328+1264082\sqrt{2})s+6617328\sqrt{2}+17614577=0$ $22500s^8-3480000s^7+226009400s^6-8156031520s^5+180271536264s^4-2511556300176s^3+21612680769420s^2-105284582762928s+222695263169761=0$
Found by David Ellsworth in December 2024, by extending the $s(102)$ found by Károly Hajba in September 2024 and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Improved by David Ellsworth in December 2024.
Fixed by David Ellsworth in January 2026.
Further improvement pending.

$s = 11 + 4 \sqrt 2 = \Nn{16.65685424949238}$
Extends the $s(124)$ that continues a pattern found by Frits Göbel in early 1979. This alternative, converting the $s(148)$ augmented by four "L"s into a primitive packing, found by David Ellsworth in December 2024.

$s = 16 + {28\over 41} = \Nn{16.68292682926829}$
Adds an "L" to the $s(230)$ found and improved by David Ellsworth in January 2025, which uses a rotational symmetry technique found by David W. Cantrell in January 2025, and the technique from the $s(293)$ improved by David W. Cantrell in January 2025.

$s = {25\over 2} + 3 \sqrt 2 = \Nn{16.74264068711928}$
Found by David Ellsworth in January 2025, based on
the $s(297)$ he found.

$s = {}^{32}🔒 = \Nn{16.82306208283780}$ $2401s^{32}-1162084s^{31}+263199482s^{30}-36971795756s^{29}+3593304360857s^{28}-254523535144036s^{27}+13408548240776810s^{26}-519901936408593644s^{25}+13805430523268790240s^{24}-171760035794017558004s^{23}-4426818799267672804198s^{22}+326745541238344524670064s^{21}-10052823560617182774582207s^{20}+181074087488548519346667052s^{19}-1180165534713863709787052484s^{18}-39814767996810201813233201880s^{17}+1608559573416436468095371051805s^{16}-32123383071505854643466513964696s^{15}+412614513240021198536387912093202s^{14}-3166929683248511957762311522115552s^{13}+4864413227687588362660593637481982s^{12}+178773674416065677195226856423404204s^{11}-1078437021027255344046160578887057944s^{10}-30113420634289372798330615891075770272s^9+791230647693671970529285145442783714082s^8-10342818482011428376427887130138935213892s^7+91890490062472897914498395018088233984940s^6-595883639785360944056309451122760067196636s^5+2858083210066295032341927117330041150532457s^4-9961121615142990990156112009477337084333564s^3+23996879793794528190240708983850819688279982s^2-35874639749283350605126520242841070953510704s+25142156270060598069723106950936800398659193=0$
Found and improved by David Ellsworth in November 2024 and in December 2024, based on the $s(53)$ found/improved by David W. Cantrell in September 2002 and December 2024, respectively.

$\begin{aligned}s &= 16-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{16.84666719284348}\end{aligned}$
Extends the $s(107)$ found by
Károly Hajba in November 2024.

$s = {}^{6}🔒 = \Nn{16.87933209237563}$ $2s^6-176s^5+6280s^4-115238s^3+1129107s^2-5432722s+9316447=0$
Found by David Ellsworth in November 2024, based on the $s(70)$ found by Joe DeVincentis in April 2014.
Improved by David W. Cantrell
in February 2025.

$s = {}^{8}🔒 = \Nn{16.90596764828402}$ $1338s^4-(92136+4180\sqrt{2})s^3+(2378600+206890\sqrt{2})s^2-(27290208+3419060\sqrt{2})s+18867680\sqrt{2}+117431679=0$ $8028s^8-1105632s^7+66453952s^6-2277483296s^5+48690729660s^4-665109430256s^3+5669988144368s^2-27584892248768s+58646728859167=0$
Found by David Ellsworth in November 2024, by extending the $s(88)$ found by Erich Friedman in 1997, and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Improved by David Ellsworth in January 2025.
Further improvement pending.

$s = \Nn{16.94073593015457}$
Found by David Ellsworth in February 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the latest $s(271)$ as of February 2026, with 1 square removed and 5 straightened.
Not yet analytically optimized.

$s = \Nn{16.95509447506437}$
Found by David Ellsworth in February 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the $s(272)$ converted in December 2024 from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024 and January 2026, with 1 square removed and 4 straightened.
Not yet analytically optimized.

$s = \Nn{16.96980702828259}$
Originally found by Lars Cleemann between 1991 and 1998.
Bounded the $s(n^2-n)=n$ conjecture to $n \lt 17$.
Converted in December 2024 from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = \Nn{16.98832058897683}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n+1) \lt n$ for $n=17$.
Improved by David Ellsworth in November 2024, by adapting a technique from the $s(240)$ found by Károly Hajba in September 2015.
Improved by David Ellsworth
in December 2024.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = 17$
Proved by Hiroshi Nagamochi
in 2005.

$s = 14 + {5\over 2}\sqrt 2 = \Nn{17.53553390593273}$
Combines two copies of the $s(65)$ that continues a pattern found by Frits Göbel in early 1979.
This is the first $s(n^2+2)$ that has the same side length as the best known $s(n^2+1)$, and is likely an irreducible primitive packing.

$s = {}^{8}🔒 = \Nn{17.60257141234448}$ $150s^4-(12200+1720\sqrt{2})s^3+(360254+85846\sqrt{2})s^2-(4622836+1430614\sqrt{2})s+21889209+7963816\sqrt{2}=0$ $22500s^8-3660000s^7+250999400s^6-9586427920s^5+224565209864s^4-3318846008432s^3+30314003053732s^2-156807860101352s+352292740081969=0$
Adds an "L" to the $s(259)$ found and improved by David Ellsworth in December 2024 extending the $s(102)$ found by Károly Hajba in September 2024 and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Fixed by David Ellsworth in January 2026.
Further improvement pending.

$s = 17 + {26\over 41} = \Nn{17.63414634146341}$
Found by David Ellsworth in December 2024.
Improved by David W. Cantrell
in January 2025.
This is the first record-setting packing found with a rational side length, thanks to the Pythagorean triple $\{20, 21, 29\}$ determining its tilt angle. It was foreshadowed just 3 days earlier by David Ellsworth finding an $s(104)$ with rational side length.

$s = 12 + 4 \sqrt 2 = \Nn{17.65685424949238}$
Extends the $s(148)$ that continues a pattern found by Frits Göbel in early 1979.
This is likely an irreducible primitive packing.

$s = \Nn{17.74116992948972}$
Found by David Ellsworth
in January 2025.
Extends the $s(85)$ found by Erich Friedman in 1997.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = 10 + {11\over 2}\sqrt 2 = \Nn{17.77817459305202}$
Adds an "L" to the $s(265)$ that continues a pattern found by Frits Göbel in early 1979.

$s = {33\over 2} + {1\over 2}\sqrt 7 = \Nn{17.82287565553229}$
Extends the $s(86)$ found by
Erich Friedman in 1997.

$s = {}^{40}🔒 = \Nn{17.82412338847854}$ $6765201s^{40}-4468746888s^{39}+1431055332684s^{38}-295991953326108s^{37}+44445603736025352s^{36}-5163220519287531048s^{35}+482878985597614632366s^{34}-37353497397438488815260s^{33}+2436831832955268577880066s^{32}-136013384348919954233715980s^{31}+6566863009941672632548971230s^{30}-276576685299846988827282188436s^{29}+10227647719730769459718165452749s^{28}-333724070621087598910090605855348s^{27}+9643524705514237911246918698322714s^{26}-247404094366714093683897197709917204s^{25}+5643119674396861410915876260884036989s^{24}-114477712300768469604781612467804162660s^{23}+2063707000134940629708077092426963028132s^{22}-32985932723685807897922726019150035754248s^{21}+465583759504110006491405487605416144111553s^{20}-5764247971291523576624473899585978835759824s^{19}+61914526137823063569936032519366359362310830s^{18}-566147126795280425805190165571461577530656336s^{17}+4248816809091624736372842695988910574840203032s^{16}-23948399126640716409385371068745856498999580804s^{15}+69800847299887214247772249238723707512987918380s^{14}+394782152673201159198736947784032596106016880136s^{13}-7828936745814934561867828618761111333890039348250s^{12}+67387777391090638044324068706351575262477706086740s^{11}-381562083381768156325668597682082463480074995868360s^{10}+1336650893593888696724108652818662244745263296865588s^9-614601349329475752238933120184261110575651083354603s^8-28305247058854021955410328849850580588215220244356248s^7+224981301926271768349655651977718726650378985178322646s^6-1055948455747248748890178989019952798681204013121554736s^5+3473810396018259264801879879806884179343515176608754025s^4-8177313821139351944992551555783250987371382184946020000s^3+13278518858240943321872383441026571982205038824476375000s^2-13426397690512873291977824583646716429736153596450000000s+6401405918124622124119207874790462553248468260156250000=0$
Found by David Ellsworth in January 2025, extending the $s(128)$ and $s(205)$ he found, based on the $s(69)$ found by Maurizio Morandi in June 2010 and using the technique from the $s(53)$ improved by David W. Cantrell in December 2024.
Improved by David Ellsworth in January 2025, using the technique from the $s(128)$ improved by David W. Cantrell and David Ellsworth in January 2025.

$s = \Nn{17.86899185179999}$
Drafted by David Ellsworth in January 2025, including adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Refined by David Ellsworth in December 2025, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = {}^{4}🔒 = \Nn{17.88674602860566}$ $s^4-66s^3+1631s^2-17882s+73369=0$
Found by David Ellsworth in December 2024 (including re-adapting the techniques from the $s(130)$ and $s(129)$/$s(206)$ he improved), by extending the $s(88)$ found by Erich Friedman in 1997, and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Further improvement pending.

$s = \Nn{17.93127894394689}$
Found by David Ellsworth in April 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the latest $s(305)$ as of March 2026, with 2 squares removed and 8 straightened.
Not yet analytically optimized.

$s = \Nn{17.94917201919110}$
Found by David Ellsworth in May 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the latest $s(305)$ as of March 2026, with 1 square removed and 4 straightened.
Not yet analytically optimized.

$s = \Nn{17.96075558628675}$
Found by David Ellsworth in March 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the latest $s(306)$ as of February 2026, with 1 square removed and 4 straightened.
Not yet analytically optimized.

$s = \Nn{17.96926975248972}$
Found by David Ellsworth in February 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from the latest $s(307)$ as of February 2026, with 1 square removed and 4 straightened.
Not yet analytically optimized.

$s = \Nn{17.98281564631754}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n+1) \lt n$ for $n=18$.
Improved by David Ellsworth in November 2024, by adapting a technique from the $s(240)$ found by Károly Hajba in September 2015.
Improved by David Ellsworth and Károly Hajba in December 2024.
Improved by David Ellsworth in January 2026, using his modified version of Thomas Schadt's simulated annealing program.
Not yet analytically optimized.

$s = 18$
Proved by Hiroshi Nagamochi
in 2005.

$s = {27\over 2} + {17\over 2}\sqrt 2 = \Nn{25.52081528017130}$
Found by David Ellsworth in January 2025, using a computer program he wrote.
Continues the $s(26)$, $s(85)$ series found by Erich Friedman in 1997.

$s = {}^{4}🔒 = \Nn{43.48878088476276}$ $2s^4-224s^3+9311s^2-185004s+1705932=0$
Found by Michael J. Kearney
and Peter Shiu in June 2001.
Bounded $\{s(n^2\!+\!1)\} \ge {1\over 2}$ to $n \lt 43$.