The following pictures show $n$ unit squares packed inside the smallest known square (of side length $s$). For the $n$ not shown, the closest packing shown to the right of it is the best known packing (to have the appropriate number of arbitrarily chosen squares removed). Where a polynomial root is known for $s$ of degree $3$ or higher (and has no concise closed-form expression), a 🔒 icon is shown; click this to see the polynomial root form of $s$.
See also flat grid view and older records and/or alternative packings. For more information on each packing, view its SVG's source code. Zoom:|
10.
$s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$ Found by Frits Göbel in early 1979. Proved by Walter Stromquist in 2003. Explore group |
11.
$s = {}^{8}🔒 = \Nn{3.87708359002281}$
$s^8 - 20s^7 + 178s^6 - 842s^5 + 1923s^4 - 496s^3 - 6754s^2 + 12420s - 6865 = 0$
Rigid. Found by Walter Trump in 1979. | |||||||||||||||||||||||||||||||||
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17.
$s = {}^{18}🔒 = \Nn{4.67553009360455}$
$4775s^{18}-190430s^{17}+3501307s^{16}-39318012s^{15}+300416928s^{14}-1640654808s^{13}+6502333062s^{12}-18310153596s^{11}+32970034584s^{10}-18522084588s^9-93528282146s^8+350268230564s^7-662986732745s^6+808819596154s^5-660388959899s^4+358189195800s^3-126167814419s^2+26662976550s-2631254953=0$
Found by John Bidwell in 1998. Based on packing found by Pertti Hämäläinen in 1980. | ||||||||||||||||||||||||||||||||||
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29.
$s = \Nn{5.93434180499654}$ Found by Thierry Gensane and Philippe Ryckelynck in April 2004, using a computer program they wrote. Fits an $s(n^2\!-\!n\!-\!1)$ pattern found by Joe DeVincentis in April 2014. Explore group | ||||||||||||||||||||||||||||||||||
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37.
$s = {}^{8}🔒 = \Nn{6.59861960924436}$
$6s^4-(208+64\sqrt{2})s^3+(2058+850\sqrt{2})s^2-(7936+3658\sqrt{2})s+11163+5502\sqrt{2}=0$
$36s^8-2496s^7+59768s^6-733760s^5+5289248s^4-23462672s^3+63458276s^2-96673872s+64068561=0$
Found by David W. Cantrell in September 2002. Improves upon the $s(37)$ found by Evert Stenlund in early 1980. |
41.
$s = {}^{4}🔒 = \Nn{6.93786550630255}$
$s^4-16s^3+95s^2-218s-34=0$
Found by Joe DeVincentis in April 2014. Fits an $s(n^2\!-\!n\!-\!1)$ pattern. Explore group | |||||||||||||||||||||||||||||||||
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50.
$s = {}^{8}🔒 = \Nn{7.59861960924436}$
$6s^4-(232+64\sqrt{2})s^3+(2718+1042\sqrt{2})s^2-(12700+5550\sqrt{2})s+21371+10074\sqrt{2}=0$
$36s^8-2784s^7+78248s^6-1146800s^5+9944448s^4-53242000s^3+173869324s^2-319180600s+253748689=0$
Found by David W. Cantrell in September 2002, by adding an "L" to the $s(37)$ he found. Improves upon the $s(50)$ found by Evert Stenlund in early 1980. |
51.
$s = {}^{28}🔒 = \Nn{7.70435372947124}$
$36864s^{28}-6340608s^{27}+502050816s^{26}-24636665856s^{25}+847410746368s^{24}-21857473382400s^{23}+441080063406080s^{22}-7168010813250560s^{21}+95780257115813376s^{20}-1068807979173627904s^{19}+10079527432131681024s^{18}-81076261200222141184s^{17}+560144016315152943424s^{16}-3340596154679285521280s^{15}+17248154822575215485952s^{14}-77154000981112955287360s^{13}+298459379274993606556192s^{12}-993878287428748511469056s^{11}+2827560064086331516654992s^{10}-6798876862709272559608016s^9+13620557443692132080422196s^8-22318847802748398169997192s^7+29208515137302727559556744s^6-29572821499263810227200404s^5+22139938904533199326391397s^4-11407627350518593079154528s^3+3525180882798952592954446s^2-436679755165931930913236s-28766318325274882531199=0$
Found by Károly Hajba in July 2009. |
53.
$s = {}^{32}🔒 = \Nn{7.82290804413284}$
$2401s^{32}-470596s^{31}+43819034s^{30}-2575626200s^{29}+107064827921s^{28}-3339730095676s^{27}+80881055947292s^{26}-1549634326032740s^{25}+23649840659276976s^{24}-285549917423038976s^{23}+2644342639858467944s^{22}-16975793431501140604s^{21}+42160111960108077186s^{20}+604801559847940668640s^{19}-10323577086818326059762s^{18}+91423093604241671442960s^{17}-549498547429972204556388s^{16}+2140033946880564583409424s^{15}-2116245094452089256368748s^{14}-47398133328019683350564492s^{13}+513623762890535143202861016s^{12}-3539168668169055445821007956s^{11}+19554643118966991967193841680s^{10}-91052982556166904610991305520s^9+358000595889063667996962518449s^8-1171244223040008871491078909092s^7+3123815163273613989845145831648s^6-6628626979480975566515883381512s^5+10851380919900152459339530602076s^4-13122619975009229974564386454832s^3+10946326483876900501448971940288s^2-5556443722823912253342620368384s+1267690936215846749823290569552=0$
Found by David W. Cantrell in September 2002. Improved by David W. Cantrell in December 2024. |
55.
$s = \Nn{7.95419161110664}$ Found by Joe DeVincentis in April 2014, using a computer program he wrote. Improved by David Ellsworth and David W. Cantrell in June & August 2023 and David Ellsworth in November 2024. Fits an $s(n^2\!-\!n\!-\!1)$ pattern. Explore group | |||||||||||||||||||||||||||||||
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67.
$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$ Found by Evert Stenlund in early 1980, extending the $s(52)$ found by Frits Göbel in early 1979. Explore group |
69.
$s = {}^{82}🔒 = \Nn{8.82721205592900}$
$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. |
71.
$s = {}^{4}🔒 = \Nn{8.96028765944389}$
$s^4-20s^3+151s^2-468s+12=0$
Found by Joe DeVincentis in April 2014. Fits an $s(n^2\!-\!n\!-\!1)$ pattern. Explore group | ||||||||||||||||||||||||||||||||
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83.
$s = {}^{24}🔒 = \Nn{9.63482562092335}$
$46438209s^{24}+1718447880s^{23}-1304818741864s^{22}+154362940868008s^{21}-10223870917986092s^{20}+463012769729234068s^{19}-15608677475881443482s^{18}+410530364971106359132s^{17}-8675319117762080311978s^{16}+150196459602374087471728s^{15}-2158879193002672091253360s^{14}+25993038455067669296355532s^{13}-263613888105247221344935027s^{12}+2258335015809616506745502008s^{11}-16347943921555337654669478150s^{10}+99786776593815833271369617220s^9-511154425074511891757096094175s^8+2180187656593439512672814134216s^7-7652314463979073976449593048904s^6+21727853135387976484209118127392s^5-48671720700899577518293563957136s^4+82801528406446840092722047620736s^3-100540002112755895115349929950336s^2+77621257841393908308227797286912s-28634116465193128516311336597248=0$
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. |
84.
$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$ Found by Evert Stenlund in early 1980, extending the $s(52)$ found by Frits Göbel in early 1979. Explore group |
87.
$s = {}^{23}🔒 = \Nn{9.83892657002494}$
$s^{23}-138s^{22}+8984s^{21}-366792s^{20}+10538108s^{19}-226899940s^{18}+3814912554s^{17}-51682985704s^{16}+579852353410s^{15}-5521708432172s^{14}+45434118479338s^{13}-324120092543232s^{12}+1970502666465045s^{11}-9840546468521178s^{10}+38505534857507358s^9-116163600689532188s^8+342445835317445719s^7-1749940540506500506s^6+11136111412137553730s^5-53096842537804640028s^4+169960636556199528165s^3-350586565237438644834s^2+425167991192928955284s-231741242909814395880=0$
Found by David W. Cantrell in January 2025. |
88.
$s = {}^{20}🔒 = \Nn{9.88815305375857}$
$3528s^{10}-(300552+15456\sqrt{2})s^9+(11614660+1180832\sqrt{2})s^8-(268405824+40209136\sqrt{2})s^7+(4111948776+801750848\sqrt{2})s^6-(43682208312+10328732976\sqrt{2})s^5+(326223055436+89277369408\sqrt{2})s^4-(1692962073984+518553084040\sqrt{2})s^3+(5849274524474+1954912407552\sqrt{2})s^2-(12163170266098+4347871933856\sqrt{2})s+11572065260145+4353477802040\sqrt{2}=0$
$254016s^{20}-43279488s^{19}+3506260608s^{18}-179642577984s^{17}+6530192527760s^{16}-179088304328704s^{15}+3846118270819200s^{14}-66261902137415296s^{13}+930479746642904384s^{12}-10759858027891736896s^{11}+103070340120029179008s^{10}-819709665351861223904s^9+5405590814889373243192s^8-29412949608198679086720s^7+130831566348158107359392s^6-468664620024162429231904s^5+1321046745485882223459068s^4-2825402176181244872057384s^3+4315682289270565775115128s^2-4199847844458434080013540s+1959329723251932809573425=0$
Found by Erich Friedman in 1997, by extending the $s(41)$ found by Charles F. Cottingham in 1979. Improved by David Ellsworth in November 2024, by adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002. 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. |
89.
$s = 5 + {7\over 2}\sqrt 2 = \Nn{9.94974746830583}$ Found by Evert Stenlund in early 1980, by continuing a pattern found by Frits Göbel in early 1979. Explore group | ||||||||||||||||||||||||||||||
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102.
$s = {}^{8}🔒 = \Nn{10.61138823373863}$
$24s^4-(1400+352\sqrt{2})s^3+(27061+10102\sqrt{2})s^2-(218629+97462\sqrt{2})s+641430+317240\sqrt{2}=0$
$576s^8-67200s^7+3011120s^6-72041376s^5+1033920257s^4-9243724322s^3+50697397293s^2-156795019420s+210150009700=0$
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. |
104.
$s = 10 + {1\over 2}\sqrt 2 = \Nn{10.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
106.
$s = {}^{32}🔒 = \Nn{10.82297973416944}$
$2401s^{32}-701092s^{31}+96532058s^{30}-8317521660s^{29}+501462496833s^{28}-22376320364004s^{27}+760407348527794s^{26}-19848555869936524s^{25}+391901782318184024s^{24}-5471444559723346548s^{23}+39629249963743971218s^{22}+353696512578770314160s^{21}-16715802829777049603255s^{20}+292780863461637719269068s^{19}-3146950297418725382386108s^{18}+17996731060753457271434416s^{17}+58068494391477930003466013s^{16}-2710032149344351718373370304s^{15}+35897708426171881544444261010s^{14}-321777454480517334707593455472s^{13}+2318288875965343221612347387046s^{12}-15552954072951813301922897418028s^{11}+110324583017828739076547861980256s^{10}-814314195444111054964406531808040s^9+5535691601850017528577776913992458s^8-31551366481166818299554205010301876s^7+144126073330457054877480503221027356s^6-514883701839008702934553147517634876s^5+1404879821405123399570947930851054697s^4-2828245713344639081326540531975430356s^3+3961424449372054137804580730689208222s^2-3448216591537867586914544066365388952s+1404226509074988020217588819457924033=0$
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. |
108.
$s = {}^{144}🔒 = \Nn{10.92591939016138}$
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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. |
109.
$s = 6 + {7\over 2}\sqrt 2 = \Nn{10.94974746830583}$ Continues a pattern found by Frits Göbel in early 1979. Explore group | ||||||||||||||||||||||||||||||
|
123.
$s = {}^{12}🔒 = \Nn{11.60139979378801}$
$196s^{12}-26488s^{11}+1651440s^{10}-62766716s^9+1618595997s^8-29815750560s^7+402059235258s^6-3996945599928s^5+29059125001479s^4-150620866376828s^3+528127756018414s^2-1124357849826920s+1098765914618225=0$
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. |
124.
$s = 6 + 4 \sqrt 2 = \Nn{11.65685424949238}$ Continues a pattern found by Frits Göbel in early 1979. Explore group |
125.
$s = 11 + {1\over 2}\sqrt 2 = \Nn{11.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
126.
$s = {}^{9}🔒 = \Nn{11.77652079061690}$
$185761s^9-17452914s^8+733790386s^7-18120925928s^6+289457197920s^5-3096648733600s^4+22129577852576s^3-101471852915328s^2+269345322921472s-312506709170176=0$
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. |
128.
$s = {}^{40}🔒 = \Nn{11.82509196821368}$
$6765201s^{40}-2845098648s^{39}+580218189444s^{38}-76444672534332s^{37}+7313700827489408s^{36}-541478198967270496s^{35}+32282461276159060998s^{34}-1592401650332728651212s^{33}+66264078654438595433370s^{32}-2360056578357132809629748s^{31}+72739322148379315505619294s^{30}-1956653332398832550681084364s^{29}+46240556849238042597672084213s^{28}-964953122823176982192672366676s^{27}+17849625763191276058228284471146s^{26}-293485855211047587757484370460572s^{25}+4296830596115061642805514450068373s^{24}-56062026207183424768909339152161084s^{23}+651753580603034236066787916795623964s^{22}-6743132871649823544913588155645442656s^{21}+61932541245281919280766873166108057833s^{20}-502864388097908644186877008296638914520s^{19}+3586164305875893817104519150255679222830s^{18}-22233865523258917656574596200774765950936s^{17}+117836973554812356574280571124747241305248s^{16}-517669550834165752088850944695261707233564s^{15}+1760771073593491738800908345083273891209716s^{14}-3688451224681060758343247748983014976447272s^{13}-3038946790141535208034747244682683990472658s^{12}+71735791138334020614589956733651574834185892s^{11}-381139020218879456095819418101710611273627456s^{10}+1267518021830549968006069595148713683301836788s^9-2664006348624383405491809555245054393205103691s^8+1793492920161456183420660943446317121911023248s^7+10946284003184957815907569175715318463243584102s^6-52105146308542520023979647560364421311730107464s^5+129236789770564934710391555764344704402747982753s^4-211311873022032856645395618371922980818558351272s^3+230091479010044538699295485926277667974383569488s^2-153189145317923709976188931598288062177638461184s+47601488782509082091988946077133401472942285824=0$
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. |
129.
$s = 10 + {4\over 3}\sqrt 2 = \Nn{11.88561808316412}$ Found and improved by David Ellsworth in December 2024, adapting/extending the $s(37)$ improvement found by David W. Cantrell in September 2002. Similar to the $s(70)$ found by Erich Friedman in 1997. Quasi-extends the $s(19)$ found by Robert Wainwright in late 1979, resembling one of its alternative packings found by found by David W. Cantrell in 2002. |
130.
$s = {}^{8}🔒 = \Nn{11.91119052015898}$
$5617s^4-(259504+12612\sqrt{2})s^3+(4502202+422478\sqrt{2})s^2-(34773984+4731588\sqrt{2})s+100926915+17726970\sqrt{2}=0$
$5617s^8-519008s^7+20936788s^6-481754784s^5+6917560482s^4-63487737120s^3+363767813964s^2-1189915431840s+1701575795025=0$
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. |
131.
$s = \Nn{11.97350182495032}$ Found by David Ellsworth in November 2024. Fits an $s(n^2\!-\!n\!-\!1)$ pattern found by Joe DeVincentis in April 2014. Explore group | |||||||||||||||||||||||||||
|
146.
$s = {}^{16}🔒 = \Nn{12.60090777851301}$
$4s^{16}-664s^{15}+51488s^{14}-2476628s^{13}+82758797s^{12}-2038683388s^{11}+38335080074s^{10}-561971068668s^9+6500206337793s^8-59622322943944s^7+433020664362230s^6-2468567342450368s^5+10847755732088938s^4-35571513994776316s^3+82161582314776868s^2-119456945705052640s+82323170428761425=0$
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 best known packing found that is doubly semi-primitive. |
148.
$s = 7 + 4 \sqrt 2 = \Nn{12.65685424949238}$ Continues a pattern found by Frits Göbel in early 1979. Explore group |
149.
$s = 12 + {1\over 2}\sqrt 2 = \Nn{12.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
152.
$s = {}^{84}🔒 = \Nn{12.83100282216725}$
$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}-559755021749304754461642532733408124950018980895075585999339578832360136778923426307553163340902938629628s^{21}+1924575434803718372008270042897715699164358183917612079945858652072146738653167981508516336683663859061599s^{20}-6191354447775618609836439234314634277309179603333612701552040527406315440826478331716065451128871996534670s^{19}+18595526459860121841165340574719138895821326772209658546058960991768583728860803392397281925779742313102481s^{18}-52017877357788563455349165010925854251068156661668439408177886419332674892024542973348383001969034647203800s^{17}+135157501868692938192366500327069236098923290104122566272833032709737094637647438296644161923551840062478047s^{16}-325194345405021237494009612258086437235941593129725154570748027251348017358438973985857832818544091573466890s^{15}+722027337085845458293091421627894365163696335388182663348454109292879460761558869358682885259404296739097605s^{14}-1473490515445565711145580914132319537412939207599828450536062468067653025235732598973121176466499270242277300s^{13}+2751263110572258451275788319799047606274296134838009812884081346353904687815554600474281715625532788431857500s^{12}-4675004465981033004681427309219791294001064124937869088877001968417898773888626422941020050924220157893424000s^{11}+7183663179262068580520363661753568724897226652642552988417888182903983412500744464905921688273238836620034000s^{10}-9906503785168325598131719638667321753622423751680769939351178446313898592975073862661742325684296198395800000s^9+12146944042621050502155105222176120253288451344101362562568904240946898405180038535485859612413971290688600000s^8-13090089533045063828551500666675702412696959576885680069538751766268511536417556308969034651288724717832000000s^7+12214708422409722054795434067910885386456675508392415014637505990855532116097110284775444747515260212130400000s^6-9676741582044228838733364917804914719594151753924645775721115515700953860674142281619525345349780994608000000s^5+6333689917226114819893572648325901156172023728019394254226767788794528703035140896587138185275926267216000000s^4-3291213620680948097795240144789699630837845626207176614176801595943955328302192020992095429065402069760000000s^3+1274139733072931259621243156734865080962818490295409701367431118912261529814385684303575896038896856000000000s^2-326961724135799194553082826640306328403985650565477078726674395723377915426470127881664726913341548800000000s+41749176420191321903553288073015514269407918838782863228971623490248342604054718097305744082801772800000000=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. |
155.
$s = {}^{4}🔒 = \Nn{12.97970624703929}$
$s^4-28s^3+299s^2-1376s+332=0$
Continues the $s(n^2\!-\!n\!-\!1)$ pattern found by Joe DeVincentis in April 2014. Explore group | ||||||||||||||||||||||||||||||
|
171.
$s = {}^{8}🔒 = \Nn{13.59569998314336}$
$96s^4-(5968+928\sqrt{2})s^3+(133541+34462\sqrt{2})s^2-(1291883+427186\sqrt{2})s+4598280+1769696\sqrt{2}=0$
$9216s^8-1145856s^7+59534528s^6-1714063968s^5+30175010609s^4-334467918206s^3+2288154693249s^2-8856922100656s+14880531093568=0$
Found and improved by David Ellsworth in November and 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. Further improvement pending. |
172.
$s = {}^{8}🔒 = \Nn{13.61898898660160}$
$54s^4-(3936+864\sqrt{2})s^3+(98658+33466\sqrt{2})s^2-(1047336+434398\sqrt{2})s+4049739+1891234\sqrt{2}=0$
$2916s^8-425088s^7+24654168s^6-774089568s^5+14674175968s^4-173849337008s^3+1265419604436s^2-5196681822080s+9246853882609=0$
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. |
174.
$s = 13 + {1\over 2}\sqrt 2 = \Nn{13.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
177.
$s = {}^{32}🔒 = \Nn{13.82302875075647}$
$2401s^{32}-931588s^{31}+169658874s^{30}-19231837912s^{29}+1515206475113s^{28}-87554613482844s^{27}+3800602014647796s^{26}-123749571598485028s^{25}+2895774212866682688s^{24}-40503170127651197920s^{23}-85797782465034115616s^{22}+22358489056150565928884s^{21}-670371983793922205889766s^{20}+11585516453329663611601440s^{19}-107946265554474207035275274s^{18}-325416702082583543878844088s^{17}+31678001381593454789856242308s^{16}-627531463707625262161828471384s^{15}+7738051841906036676459384893372s^{14}-65825971528188631650991553430380s^{13}+390859965437244296776723867974104s^{12}-1981263185614787515387345660456708s^{11}+20697660792797450992561898119685608s^{10}-341458570720939415832614447938585072s^9+4331777593881436240458282284914909233s^8-39425956807257141453494313251818730308s^7+265332233224572263186725591178500406376s^6-1341713349412185465951061216890867086592s^5+5078477010422928472451310243101300746580s^4-14041240100101609173845558593200611383568s^3+26892648324465918884312724426364749628480s^2-31980711702815550619890642368200074811200s+17821422876028786503270705680802036472000=0$
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. |
182.
$s = {}^{66}🔒 = \Nn{13.98264294789927}$
$1713850177070388831714241s^{66}-923663845843030533029467964s^{65}+244356526675459843171256658812s^{64}-42301149163464927098830722044712s^{63}+5389637680061232065659444372903884s^{62}-538990344040466547260756897354234800s^{61}+44060430495129161810848298418942728768s^{60}-3027597018002804743276382310802134926912s^{59}+178476725866222249279843650493042891414174s^{58}-9167125837455703044437102107796599060405072s^{57}+415277865358575078055753826628996914179647584s^{56}-16755219749633813087434458219665047861571005696s^{55}+606951588795740859789587928711977301863618562444s^{54}-19872617896564458386479749674438181110594571783440s^{53}+591430450596695946776559043737897061925817091883896s^{52}-16076403382596088035670122941655178909884706102259360s^{51}+400780790911848317908826372889780636781368953842134545s^{50}-9196292015835695622526434508695387052579786474968530764s^{49}+194831950657120243957930949495672896263326187482945017636s^{48}-3821488517527886494201924904813013938910357960132992875112s^{47}+69561259675051249201492552497458936188600783387550785194512s^{46}-1177541097875302951212548144402084258394462103241418878091904s^{45}+18572022118855878686507143350115259541331165972203297602035016s^{44}-273350438905317141283665359004770430754125267452024933490938688s^{43}+3759875778293105916627657039864999051693799407851712882736113208s^{42}-48390183666058035250293916819527929345794012592770993441540548200s^{41}+583359495992769368180893839833879135938973736557186058881098499424s^{40}-6593404023892398478887639745053689330072639258527396503597740822704s^{39}+69922254584057803430223785412744087886168418412416437299744434763920s^{38}-696199801753965501285725548974198631608011266843672974698091490432160s^{37}+6511650429478482167950610493981767940220071476762446758258297829362464s^{36}-57234890555844218227733552885344124231228022552073037991205841355754784s^{35}+472897245542288005015724033828084119792469894384102779109267997968480096s^{34}-3673527397539404818977007733654565672521620497606425069188768378430791392s^{33}+26830967247160316756709981320435477112019319138841629014760220636370474816s^{32}-184248363314616695085417127987503784871627748574838179238693969981716890752s^{31}+1189362835357759435647680170936163404089852639152762150758843550851500029312s^{30}-7215198553515381078457422107373261572387237955159216402793836179726031898368s^{29}+41118237990763833027717928080095280450530652728789049212131791394290259869888s^{28}-220013586830299858346544628394840804319064288631784649846740295442808772668160s^{27}+1104624680933328901706883047576578922329679532689002041544873488282388304886016s^{26}-5199881073657329493566722212089680068736612815975219398932756978414737762064896s^{25}+22929072742861089738590065098448598702132646005025936214483918913337650353069824s^{24}-94608008640637296299957783267672779275948371642923398380136113133454728406140416s^{23}+364819925689129881972579265818449668453976797240020471862122424163060417660220928s^{22}-1312862171184299432821030617749268935136513935049032859605965972625387377719365120s^{21}+4401912097721267325119409154299200433015462297229066215750087595154529481296731136s^{20}-13725788367315319506315635643739707608001009560356143859397586956998117539703622656s^{19}+39717900153429166873869781810993511500356117237564376538643805576134915824234523648s^{18}-106399332388497647615120413383665293526457978124604308307567938191333861608693833728s^{17}+263146214957976354333140167648787212861652057702279153736060651315259635818278014976s^{16}-598948423050050797962529487281503994476619892474808687630034845197011146318218905600s^{15}+1250084500161595434165407078512122830805401588230922174957693497088606844442474217472s^{14}-2382458928043568775151925441566626859323222839887782280481843477807775050701314756608s^{13}+4125994826986881535378038074633666561772029457781662960093867812493907378777176526848s^{12}-6456003870217606716135295888226937623998759028219550792812724057683607497186984583168s^{11}+9065372956178732821827510381589091533382082905889742319984764707994104046708843806720s^{10}-11330803663084115039545790076138860721501300902182053330467921505356766127529332162560s^9+12481974955426380241965795213639280113780047248776326121594789973305180053750293086208s^8-11970325319419753577043006848391910395086707654239813304187044220003154526297518047232s^7+9838402392037009878211651167751684462461327699694726737875855220251210658084493393920s^6-6789017987208597545069669454098514525060401317357596876693764512756899474772483768320s^5+3824088158296275561318991814461422505555214264984822149493993857503600579756629164032s^4-1687927982575032415641479063772412094421237083850884119669396632875631662093020692480s^3+547314545007055176499085872328704154179592545295820829835019843713968328240803086336s^2-115875113076825478429958642319077276340668682757823451130118465600944508946094227456s+12012260314488957307342435041239878634388761717780836316682092965799623955237044224=0$
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. | ||||||||||||||||||||||||||||||
|
198.
$s = {}^{8}🔒 = \Nn{14.59153604974246}$
$7889s^4-(496196+66432\sqrt{2})s^3+(11353289+2588872\sqrt{2})s^2-(112836070+33552520\sqrt{2})s+413156140+144732976\sqrt{2}=0$
$7889s^8-992392s^7+52797090s^6-1566650388s^5+28530003977s^4-327826283468s^3+2327673598588s^2-9356471793680s+16326876326032=0$
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. Further improvement pending. |
199.
$s = {}^{8}🔒 = \Nn{14.61898898660160}$
$54s^4-(4152+864\sqrt{2})s^3+(110790+36058\sqrt{2})s^2-(1256676+503922\sqrt{2})s+5199723+2359962\sqrt{2}=0$
$2916s^8-448416s^7+27711432s^6-931104720s^5+18929518528s^4-240795061296s^3+1883132387964s^2-8311787117640s+15898277993841=0$
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. |
201.
$s = 14 + {1\over 2}\sqrt 2 = \Nn{14.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
202.
$s = 2 + 9 \sqrt 2 = \Nn{14.72792206135785}$ Extends the $s(18)$ found by Frits Göbel in early 1979. Explore group |
205.
$s = {}^{40}🔒 = \Nn{14.82445114612408}$
$6765201s^{40}-3656922768s^{39}+958435862652s^{38}-162258762163956s^{37}+19944495118895680s^{36}-1896817614997924732s^{35}+145244607337067116310s^{34}-9200231755130148537584s^{33}+491532552940826676421598s^{32}-22471266124455622725351620s^{31}+888775367178021320983990528s^{30}-30670374642432504931208393536s^{29}+929494097506723220926745638527s^{28}-24862568075564651560878275579812s^{27}+589158528037990588430776578667278s^{26}-12400310845434669601706739856078348s^{25}+232178750736807601084495428509773447s^{24}-3869279274647141302600286447060842028s^{23}+57359828211423634121225283370282978472s^{22}-755020486735335189866154838905984601284s^{21}+8794114159490040759213225924279409129174s^{20}-90126731671943479290165241436087360926864s^{19}+805381183069421598181526905692816890290502s^{18}-6181562346162193722542638238059633411805108s^{17}+39656170956144484959733239448754004343189341s^{16}-200557184022962853819962178120475887680432104s^{15}+668299828757988666056469128930860611880478162s^{14}+47787548247188382784775375655144589392168024s^{13}-20274687460176342070123115001595419340631113986s^{12}+171806640590364466782586102889249290168160938968s^{11}-887645374096288133345472520371583078066772417438s^{10}+2995081869452847889056141420143309758645762048480s^9-4490901292932914262604076259256067123710050815195s^8-18138284909745955939878418372807115317752603411176s^7+165869320264318841451647718375161237581492991097810s^6-695527958949195756061996697585676748194320009594040s^5+1948735509481744597514888658808885791132097191929625s^4-3838064311683628303762259793886602147590339042385000s^3+5169742707762800848775847149992971475362078199650000s^2-4315559969700226767159393981391684637349630444000000s+1694040395896101772224866098180283214283057728000000=0$
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. |
206.
$s = 13 + {4\over 3}\sqrt 2 = \Nn{14.88561808316412}$ Found and improved by David Ellsworth in December 2024, adapting/extending the $s(37)$ improvement found by David W. Cantrell in September 2002. Similar to the $s(70)$ found by Erich Friedman in 1997. Quasi-extends the $s(19)$ found by Robert Wainwright in late 1979, resembling one of its alternative packings found by found by David W. Cantrell in 2002. |
207.
$s = {}^{12}🔒 = \Nn{14.89564154083791}$
$16s^6-(1144+96\sqrt{2})s^5+(33985+6024\sqrt{2})s^4-(537992+150024\sqrt{2})s^3+(4801816+1850340\sqrt{2})s^2-(23022312+11275344\sqrt{2})s+27067200\sqrt{2}+46636900=0$
$256s^{12}-36608s^{11}+2377824s^{10}-92660208s^9+2409377665s^8-43965076368s^7+576052330352s^6-5445742834320s^5+36732970030440s^4-171582571927744s^3+523309619017472s^2-926610560558400s+709733809930000=0$
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 and the $s(130)$ improvement found by David W. Cantrell in November 2024. Further improvement pending. |
208.
$s = 10 + {7\over 2}\sqrt 2 = \Nn{14.94974746830583}$ Originally found by David Ellsworth in December 2024. Alternative found by David Ellsworth in January 2025, using a computer program he wrote. Extends the $s(89)$ found by Evert Stenlund in early 1980, which continues a pattern found by Frits Göbel in early 1979. |
210.
$s = {}^{66}🔒 = \Nn{14.98264294789927}$
$1713850177070388831714241s^{66}-1036777957529676195922607870s^{65}+308070885285072811862199123417s^{64}-59939613454150601563431685241960s^{63}+8588813678023291261040553534326772s^{62}-966583568773741127569549391339079520s^{61}+88973510619748651618414859862026730876s^{60}-6888549315978477501144600760825939298456s^{59}+457813058489637019372876565898875762561962s^{58}-26525924226609242481987455544648574755740060s^{57}+1356297999041102109336083334683724973134359610s^{56}-61799914557147552310687442895919575623521766840s^{55}+2529586572366492011492859050137381026019860276228s^{54}-93635049365714878103413599709118844924967191715568s^{53}+3152090707330317380212505369273541211594578660668828s^{52}-96964556710709267169581794961613905049794026876686584s^{51}+2736977327538183203599671149196691182209406477480426999s^{50}-71141341918871267999443622612794042553629895963414923554s^{49}+1708099525395866416691912276683615496482061285927149098767s^{48}-37985865900286624496819504298337826100292397570681660970624s^{47}+784296971154264745366602352186813069922416329570936529960120s^{46}-15065885306353527422811618762658136922751746094197077043607808s^{45}+269749236797310845207931452385959355761508726043413698724913752s^{44}-4508953939245292278547355693056226796525821395294109220913219760s^{43}+70461706096440343504339359622207886316103484007784533453402451564s^{42}-1030684846592380569306994628265946835298274035988655898515756518536s^{41}+14127213285791424916401341828825119374306871082227591450747564708428s^{40}-181611012865775124333222503250168271689890578495810300493810490865200s^{39}+2191389437227659168703888781115852382017210534407970049772895556135144s^{38}-24835252575107480313771638509524748878675003716854534342036362523912096s^{37}+264493817830994381831108752939852505433266002142853663596745464038105112s^{36}-2648106506791820424837514470078773670571332661118776845107705934142835760s^{35}+24931817840154101718618675302330328982025705361265314835179979339556004095s^{34}-220774232592783180535637731350600608019775280385941635854721272379834520226s^{33}+1838859783489983364796248010427404448952293664994580697429468489690817448391s^{32}-14405793276814386692861915917557813798574320839280787544681770605702716243496s^{31}+106132652991766969969790599313795738256162256172065649640195937493783785627012s^{30}-735141103620029965659773734189613183265405874983552791121595763935147775210784s^{29}+4785629939469775130214760518204470967279654407141021153131677479220692014706252s^{28}-29264396180717787590970697217467423077531807932966908615153820237900145532452792s^{27}+167997715129752800946484031210802390736511802747887658272905155103288606648529418s^{26}-904700154474475752208581796921737926689902979381082659180178913034340167719783580s^{25}+4566216117618812719619282953953565095633492925168145038704985120782850946704225818s^{24}-21577735591142156619368360651289725338415069076557628274035844600998082681026841624s^{23}+95351638011084223101955380792874473279508189366232625045913569058491535016623736724s^{22}-393477264331394419045246438963291356261882874125326625922439475690536334420518557680s^{21}+1513871326020646900995786895290089433666501518786395002635104303984667876045713548492s^{20}-5420589319604280486120080489351603688574661016232355428256686514216840385980014845144s^{19}+18025671901154216272856257915455921861722569907273247804240829505876591992203422527785s^{18}-55538726517617123321306980401388416698101944528011997777048904298548850658621880672382s^{17}+158119388173030536226827428726717197786937874399240446800307626931688853687437117891505s^{16}-414679766978138921331059461345942561913790083283653894727325899938505427858892965125872s^{15}+998228943688800328823203749490058494562116495933106174279441691815211346156075844428112s^{14}-2196572339864925849617348696491086772493215350378298856026223386996595289374884651752448s^{13}+4397165113585946993607002307549324635278671461053934532945038939803841295777334091620192s^{12}-7962739349659864081806356456004225116652369166110620089333680683881143975151724678690048s^{11}+12957137888727586584941034560243767737163859971700725359751881246527790566047851156973568s^{10}-18794133216992559109409136095289044747144304027555381568607235733444082266478813293896192s^9+24062707252931451750084904210379578567640678822720136431418887027294986269907390293532928s^8-26864643906507885513435835499475292614570149380412717542767363827389611363117057750564864s^7+25750555425531123093287716207567520607853292058501569356839679301847154190381313748566016s^6-20763167580966447999929009292788490161886160932306769296176687436564687869505524614365184s^5+13694576920953853335160219343243224031929703522356095792630808590038480485959704606801920s^4-7094101031359142126885338736264230663456345418028425142721873216376995826523389107896320s^3+2706320051143905447935556673344831161461373449684695751208915990592616770507314550538240s^2-675931907390084983300050927121665770468081954992407462935298185140040600624082498617344s+82905798822484597710700208785408047898314652058813563312936555293758738816713843802112=0$
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. | ||||||||||||||||||||||||||
|
228.
$s = {}^{12}🔒 = \Nn{15.60902282132495}$
$24s^6-(3008+632\sqrt{2})s^5+(143990+46836\sqrt{2})s^4-(3498934+1393560\sqrt{2})s^3+(46342192+20798868\sqrt{2})s^2-(320302448+155625644\sqrt{2})s+907787225+466763736\sqrt{2}=0$
$576s^{12}-144384s^{11}+15160736s^{10}-915791264s^9+36096990788s^8-988137514952s^7+19384693200660s^6-275725098032832s^5+2830331680270636s^4-20490408263563724s^3+99460046477154240s^2-290971312817869664s+388340875383845233=0$
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. |
230.
$s = 15 + {28\over 41} = \Nn{15.68292682926829}$
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. |
231.
$s = 15 + {1\over 2}\sqrt 2 = \Nn{15.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
233.
$s = 8 + {11\over 2}\sqrt 2 = \Nn{15.77817459305202}$ Continues a pattern found by Frits Göbel in early 1979. Explore group |
234.
$s = {29\over 2} + {1\over 2}\sqrt 7 = \Nn{15.82287565553229}$ 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. |
235.
$s = {}^{83}🔒 = \Nn{15.82660563342856}$
$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. |
236.
$s = {}^{12}🔒 = \Nn{15.87607676541001}$
$6596s^6-(619856+19000\sqrt{2})s^5+(24226944+1444540\sqrt{2})s^4-(504036792+43818500\sqrt{2})s^3+(5886496273+662906450\sqrt{2})s^2-(36585732598+5001593450\sqrt{2})s+94529982167+15055629320\sqrt{2}=0$
$26384s^{12}-4958848s^{11}+426380416s^{10}-22179443008s^9+777414394344s^8-19344554351312s^7+350411708641272s^6-4655972006018640s^5+45039272938487593s^4-309348405383979564s^3+1432057511182322314s^2-4011939813305628468s+5144071303851334561=0$
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. Further improvement pending. |
240.
$s = {}^{66}🔒 = \Nn{15.98264294789927}$
$1713850177070388831714241s^{66}-1149892069216321858815747776s^{65}+379137661154317748641195681912s^{64}-81891074048159729003070697178408s^{63}+13032601897654000375127635736914044s^{62}-1629704803955659741528480932743629600s^{61}+166761277204901256215813369062945614448s^{60}-14358776608758577299719024788683561291200s^{59}+1061744286766486909447032265740305813536222s^{58}-68474327034163224324683196882521071680487400s^{57}+3898680014040310378546424786982864974691587032s^{56}-197893479763233829851953449206470703678382928256s^{55}+9027058972990094483551124675146985797402686345676s^{54}-372525707164344655382696831764231550931861454524704s^{53}+13986254561093171332246318032879802874415524621400936s^{52}-480021129624779505518793563427218381659637764171707744s^{51}+15122329258714384953594013673578917012766694983733804881s^{50}-438855029419115553162393305979405927798765869173444577904s^{49}+11768226418611255713545406862693986409497534694395932490528s^{48}-292389611865138370972616101847672913969792672213028811518184s^{47}+6746862076174715126011529791080074820752050442542414504370560s^{46}-144888150698948224043410517981223612722489152566776581298794208s^{45}+2900996894928148652325377674448953709338108189036698898804529608s^{44}-54242556288254952045704327412146319996536264265171980175919237760s^{43}+948462355395453479168602938292256298703061926500378052173528780792s^{42}-15528080343980828527604766556375005181552902663086802848142696321224s^{41}+238282964650245168300609610231809513127916963399478723451985201685904s^{40}-3430367236060179440070732218664376912627362282540244578551905978189104s^{39}+46365532467244547446360743216142017095989148306459526039623515846047792s^{38}-588755762949838005565806711016983484397068959152714283441733945344802848s^{37}+7027249712549848127431398466867724250461309354203649153646258736184974752s^{36}-78871493856005778995281528812829746466339050516177676306484461418414160416s^{35}+832653557924952811014500182820751911383034261282044436164334245143830456352s^{34}-8269810361173954931078914169946911637391596761131737508844305945597487794144s^{33}+77275880265085319916055821144235805497262635145714014287742658219786671877504s^{32}-679349084345154135379649690958316686561437063246031516423762907925538312289408s^{31}+5617964001822564205130667238529394632220025771132883175722208150517560549227648s^{30}-43690824350734077930796967911338231155740749007361979222513577735261402688590592s^{29}+319422913565064992742490317937286445800967315990365504365040042142459301153592512s^{28}-2194292900885520709718961937109526038033002419198710908780183901772626370705331456s^{27}+14155019838518846812037482168797435038238593984348815057869321629794699529203855616s^{26}-85682417222445926795565973572329420277366975422152054005379234967639143358197815808s^{25}+486245607801686576677241553060465905470056389505993650262344015281342122162160575232s^{24}-2584368483713735021047686463913913640136052794643430856590242035633830225424725988864s^{23}+12848980595866502714148654836356043391589862545760566610324380590058932864681066715648s^{22}-59676137078910398640252148320907121417908906930343892171666836705986652465992239468032s^{21}+258503073948652279571122377723799998795218261647593540516168322657994235939676499460096s^{20}-1042512327688912764291535477319351885866939675291965746264513281156719001472917009644544s^{19}+3906196167448619261818676733246713157402634575850720548507828059644983940218233231610880s^{18}-13566400556764072761657734569104915733372068338209863555592826673337289078412816532291584s^{17}+43555912849470441098868802000969681226084842115362413993508512687803571381115014140829696s^{16}-128873854467322076750495059627569588067583487401622277235861210691551499032718516262586368s^{15}+350170085209767527530821777446058436886835554296012946187110660487759175384250607478042624s^{14}-870180189847506227220609230477835558202475594829635350625252034102067479987046653583601664s^{13}+1968252700727960939216791826879252136521049007411500029477449576057390685915953096594960384s^{12}-4029555319153116978053280766009889585799476763772621260420608643127295976220742724892581888s^{11}+7417299288136909883224183664841980102926026482386873232431581240176756976673835812755652608s^{10}-12177862422215869003301230978568940238954188047906715505518821155614152635401370021975375872s^9+17659998109675690653749221764562831419249106682956363249429630212164222854382762578641272832s^8-22347378432178696652088994803216423167402716129816166966820581562620109814623380610932342784s^7+24296916754024602064906452756050319984681043769067192580115816651405229556935324316920250368s^6-22239046622093583453544338095037087546806112181982198692585543077930687627404701078594387968s^5+16664365087788761186074795638291069350851003090434486799819196634836845588756587148389384192s^4-9816056457754695102925029410241846551198661538424486286292785393106187869550293083972698112s^3+4262107623285338836956260105382884155349919936507235251826717599203646332610595428224729088s^2-1212790846648326411539652251907964256150144830955555521253734442565303245011213397155381248s+169654700822801341540046526070943346738341569829349407278942246963305979508724394755096576=0$
Originally found by Károly Hajba in September 2015. Bounded the $s(n^2-n)=n$ conjecture to $n<16$. Improvement based on 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. |
241.
$s = \Nn{15.99379865105946}$ 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. | ||||||||||||||||||||||||||
|
259.
$s = {}^{12}🔒 = \Nn{16.60255251726339}$
$64s^6-(4352+288\sqrt{2})s^5+(112386+18636\sqrt{2})s^4-(1295184+466536\sqrt{2})s^3+(4902306+5553822\sqrt{2})s^2+(19879740-30390606\sqrt{2})s-143635023+56063664\sqrt{2}=0$
$4096s^{12}-557056s^{11}+33159424s^{10}-1122522624s^9+23299339236s^8-290070700224s^7+1703658414312s^6+5714085847824s^5-182330704044288s^4+1346739639595488s^3-4105729915848180s^2+1104381069893496s+14344750990000737=0$
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. Further improvement pending. |
261.
$s = 16 + {28\over 41} = \Nn{16.68292682926829}$ 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. |
262.
$s = 16 + {1\over 2}\sqrt 2 = \Nn{16.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
265.
$s = 9 + {11\over 2}\sqrt 2 = \Nn{16.77817459305202}$ Continues a pattern found by Frits Göbel in early 1979. Explore group |
266.
$s = {}^{32}🔒 = \Nn{16.82306208283780}$
$2401s^{32}-1162084s^{31}+263199482s^{30}-36971795756s^{29}+3593304360857s^{28}-254523535144036s^{27}+13408548240776810s^{26}-519901936408593644s^{25}+13805430523268790240s^{24}-171760035794017558004s^{23}-4426818799267672804198s^{22}+326745541238344524670064s^{21}-10052823560617182774582207s^{20}+181074087488548519346667052s^{19}-1180165534713863709787052484s^{18}-39814767996810201813233201880s^{17}+1608559573416436468095371051805s^{16}-32123383071505854643466513964696s^{15}+412614513240021198536387912093202s^{14}-3166929683248511957762311522115552s^{13}+4864413227687588362660593637481982s^{12}+178773674416065677195226856423404204s^{11}-1078437021027255344046160578887057944s^{10}-30113420634289372798330615891075770272s^9+791230647693671970529285145442783714082s^8-10342818482011428376427887130138935213892s^7+91890490062472897914498395018088233984940s^6-595883639785360944056309451122760067196636s^5+2858083210066295032341927117330041150532457s^4-9961121615142990990156112009477337084333564s^3+23996879793794528190240708983850819688279982s^2-35874639749283350605126520242841070953510704s+25142156270060598069723106950936800398659193=0$
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. |
269.
$s = {}^{8}🔒 = \Nn{16.90596764828402}$
$1338s^4-(92136+4180\sqrt{2})s^3+(2378600+206890\sqrt{2})s^2-(27290208+3419060\sqrt{2})s+18867680\sqrt{2}+117431679=0$
$8028s^8-1105632s^7+66453952s^6-2277483296s^5+48690729660s^4-665109430256s^3+5669988144368s^2-27584892248768s+58646728859167=0$
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. |
272.
$s = {}^{66}🔒 = \Nn{16.98264294789927}$
$1713850177070388831714241s^{66}-1263006180902967521708887682s^{65}+457556854283194653508246334297s^{64}-108626085650108755375383219863016s^{63}+19007543579258154993013605104517540s^{62}-2614249620222774694075991845075567408s^{61}+294320445544799712214422954346585080428s^{60}-27891472736856262183567432641837740604344s^{59}+2270607650860185114208141699776485160991834s^{58}-161270542469096575205017999202879333703379396s^{57}+10115431614750251067098547571363673856667344506s^{56}-565808816641730403858512432034552952494802469080s^{55}+28450183984319404845113532374059123476095944389108s^{54}-1294559647452187836265838445107719695018437168558336s^{53}+53606558683243577767537060160465091965389251586245100s^{52}-2029777274949400008320972989373569605256603237798407896s^{51}+70566124724622022053181378677212143452798761833741049255s^{50}-2260494118677599144499162628207117877734197301937499627454s^{49}+66928391584632217881721045867244937443110647589022537806159s^{48}-1836487293516144718606242405600092685339691196348395055392352s^{47}+46812433957159983606351028994995744914840143394813517611414488s^{46}-1110784325503673809119053441923324973802716059719307513683034144s^{45}+24580067642598124655873296327812538035270416503768622579007969464s^{44}-508058601195291180778593698054098715271789503226858668644025004528s^{43}+9822636844453945473008973446300398331806350662367666559112039098828s^{42}-177850157754818541348101942178417197242325906137100400008326274953784s^{41}+3018908592957234274621453725329581711584511453209430776141062678498508s^{40}-48084823616106717807937793455351705691203975334728299955852985477677936s^{39}+719219081331785321193765357384648827974404178174478952936526177053613640s^{38}-10108484874977669848442591400277761623042675059618304149046369427119689280s^{37}+133569141781888512562398911029947506237982901071358900752904689441632946104s^{36}-1659947237436600687047903428474338638444186680114020006663182877218281431536s^{35}+19407721788005616167841988847134491979124178917074454947061592816288392155103s^{34}-213512563646549268743663877471869657452486579360256542887702449445267477625758s^{33}+2210398398130635465592076383059864672001655782088693299537565902349619712869319s^{32}-21532723313632709226861394724764610661805015535470154323486739829302573530834280s^{31}+197353488882411915369080553242112202824443643000941071851504553934557696342479860s^{30}-1701361607748327036898500039362885770951773534397536787240248737455726000275481776s^{29}+13790944656640719079864365329065256075236685711621827154509298129212864382169075708s^{28}-105057191253365590055038856706594446361850240187411125808555225806066647805699033432s^{27}+751670513866794149869446078936059679095796755093931999846310783101254874128343844858s^{26}-5047524051032968924904168688747563678450991211248789313855857351391502072525065835844s^{25}+31783178768787466515646386129275724683780750656572320239619905303340921614133673928026s^{24}-187472727052955447696160431653518028007026362437178333588607245141115925448495562239416s^{23}+1034626229621705034003065868277725184320524563387572538925459708898162378122695884371524s^{22}-5335045699393078934531764329130636548870723955424210927081461414593591826916976872572736s^{21}+25663737575409659302307158237342619977251070651367656518135022159442843909863332058148316s^{20}-114960649295229282113311852806854796874367423592350138710159368893751322473994310949804856s^{19}+478559753673706084748519492363770439999236334487313334845093021068854656585876271428583577s^{18}-1846990700033721376192730686948971722144142619617398680320187325046521383970748784637316834s^{17}+6591326456994561029578640106261265890282504157582308701300923406400578110653255439076001265s^{16}-21683429758094255971784837646080964873309338986609460623022430958540451763873664347265934416s^{15}+65523328529399452484083881848848270952170741513603155833639414320711386575541360843917428752s^{14}-181134395473848110073827498749917703659200447864218040383057641661298047344658812692009492864s^{13}+455904206381436590247441608773490326831830404955375957062019998388748856279690733152466535776s^{12}-1038920953582196812525580854613110667241159361649685005599580196703726210365369892962261589248s^{11}+2129321555265993058808909413739742507947670301867618671926711075672095702442458870941704884224s^{10}-3893858393082444876460677140167356642408686917745920213899715750218779803640863536637271852544s^9+6291634657144980094912045673643258772753140381469197256391134629731334665393467914754165776640s^8-8874016200990822056453884612581892989147523385532622735339375442773419838200440294959402524672s^7+10757961171340950446264308166124992160748835994596014475038180685505768156638580567927181983744s^6-10983792379878536403573363876927768332193956898933674258258398402852665358347513825072163651584s^5+9184651747912342152838771898129517117129618056068600195075502388329608767166607606008661409792s^4-6040013853226973048374847053430821660325834738741764444484502435456762249243146187494035292160s^3+2929211695042187145319679151141672508929021947850940754766129715898448812797031858745271910400s^2-931421152741943199632685430128893095342601525238972381704891083745166622082394765118220009472s+145672913717139262806845359990077919840464280273804604430606108170184726692142354311210336256=0$
Originally found by Lars Cleemann between 1991 and 1998. Bounded the $s(n^2-n)=n$ conjecture to $n<17$. Improvement based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019. Improved by David Ellsworth in December 2024. |
273.
$s = {}^{5}🔒 = \Nn{16.99295005377111}$
$10s^5-412s^4+7204s^3-73814s^2+416937s-935217=0$
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. | |||||||||||||||||||||||||||
|
(290)
$s = 14 + {5\over 2}\sqrt 2 = \Nn{17.53553390593273}$ First $s(n^2+1)$ whose best known side length is the same as $s(n^2+2)$. To construct, either add an "L" to $s(257)$ or remove a square from $s(291)$. |
292.
$s = {}^{12}🔒 = \Nn{17.60255251726339}$
$64s^6-(4736+288\sqrt{2})s^5+(135106+20076\sqrt{2})s^4-(1789528+543960\sqrt{2})s^3+(9506654+7068126\sqrt{2})s^2-(42973842\sqrt{2}-5717888)s-157200471+92493552\sqrt{2}=0$
$4096s^{12}-606208s^{11}+39557376s^{10}-1485656064s^9+34988158692s^8-521042431968s^7+4488051338520s^6-12054166969344s^5-173414241628704s^4+2087574544612928s^3-9264733241063660s^2+14101503785216640s+7601873759468433=0$
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. Further improvement pending. |
293.
$s = 17 + {26\over 41} = \Nn{17.63414634146341}$ 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. |
296.
$s = 17 + {1\over 2}\sqrt 2 = \Nn{17.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
300.
$s = {}^{40}🔒 = \Nn{17.82412338847854}$
$6765201s^{40}-4468746888s^{39}+1431055332684s^{38}-295991953326108s^{37}+44445603736025352s^{36}-5163220519287531048s^{35}+482878985597614632366s^{34}-37353497397438488815260s^{33}+2436831832955268577880066s^{32}-136013384348919954233715980s^{31}+6566863009941672632548971230s^{30}-276576685299846988827282188436s^{29}+10227647719730769459718165452749s^{28}-333724070621087598910090605855348s^{27}+9643524705514237911246918698322714s^{26}-247404094366714093683897197709917204s^{25}+5643119674396861410915876260884036989s^{24}-114477712300768469604781612467804162660s^{23}+2063707000134940629708077092426963028132s^{22}-32985932723685807897922726019150035754248s^{21}+465583759504110006491405487605416144111553s^{20}-5764247971291523576624473899585978835759824s^{19}+61914526137823063569936032519366359362310830s^{18}-566147126795280425805190165571461577530656336s^{17}+4248816809091624736372842695988910574840203032s^{16}-23948399126640716409385371068745856498999580804s^{15}+69800847299887214247772249238723707512987918380s^{14}+394782152673201159198736947784032596106016880136s^{13}-7828936745814934561867828618761111333890039348250s^{12}+67387777391090638044324068706351575262477706086740s^{11}-381562083381768156325668597682082463480074995868360s^{10}+1336650893593888696724108652818662244745263296865588s^9-614601349329475752238933120184261110575651083354603s^8-28305247058854021955410328849850580588215220244356248s^7+224981301926271768349655651977718726650378985178322646s^6-1055948455747248748890178989019952798681204013121554736s^5+3473810396018259264801879879806884179343515176608754025s^4-8177313821139351944992551555783250987371382184946020000s^3+13278518858240943321872383441026571982205038824476375000s^2-13426397690512873291977824583646716429736153596450000000s+6401405918124622124119207874790462553248468260156250000=0$
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. |
301.
$s ≈ 17.87595$ ROUGH DRAFT, UNFINISHED. There are some micro-overlaps, and the size is approximate. Found by David Ellsworth in January 2025, including adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002. Extends the $s(11)$ found by Walter Trump in 1979. Planned to be finished using a computer program yet to be written. |
302.
$s = {}^{4}🔒 = \Nn{17.88674602860566}$
$s^4-66s^3+1631s^2-17882s+73369=0$
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. |
305.
$s = {}^{66}🔒 = \Nn{17.98264294789927}$
$1713850177070388831714241s^{66}-1376120292589613184602027588s^{65}+543328464671703526463351080572s^{64}-140615202964614126638004715304744s^{63}+26829824909531386795122409867553580s^{62}-4024798138448651535381545184824888272s^{61}+494347896425777838963701966211841094208s^{60}-51121940635378797836287264998905061474368s^{59}+4542640548237608922133896403568612165085598s^{58}-352254670142390357290980730103805413351130752s^{57}+24128195020496036167298549899657263698522042112s^{56}-1474179150135552799244494323027883402848618735872s^{55}+80985138091690144145355687368928269630738567897484s^{54}-4026981590548665867874487939909468960023196494310192s^{53}+182267200638373063745230585503289601386963559885303288s^{52}-7545112281899612510165763435068918446604608201263951392s^{51}+286834925628382194495660904989654559193002073657037773457s^{50}-10049565828457746164374711704047033495168594843678677316244s^{49}+325499384321545190494163072511087433536420191967280816128740s^{48}-9772581289162465359405567577566877285512203772853480321141608s^{47}+272613842003155736225570794797609838660943262882211714349718640s^{46}-7080493440341343834411011123586959947754962517882504495538745472s^{45}+171531474757097410249414924053861168536230110955969576731895246920s^{44}-3882214349259127305137310567801724956533274558662561393205688835264s^{43}+82200585145535219135810965054893272960077010195154270039200863749688s^{42}-1630258956865275295872337774636555763342793225726628107661326519656232s^{41}+30316602368448156150200109954721666553328238001770420734846520369155456s^{40}-529100170410512162733967021266362515006823176250235236208800862485778096s^{39}+8672812114309507583523769261574291633843885317390927008959165725551281104s^{38}-133604712858611307770326141488963196041431007457229154153884192004409913632s^{37}+1935287963280671887853021351300551803147977152959329110457304496325932053152s^{36}-26369568984008683547580706652585786757749645465630170649594991804279169086240s^{35}+338078080839919096674573098826845065127283338268117746286742900602332756029664s^{34}-4079087539789791575560050969074532457336361603032072778873961672817546356973536s^{33}+46320137173197740753817591401820230866740557176026937020010829487638066263326912s^{32}-495016582000784435413106529483575249511354685535037968758201031333595965076898944s^{31}+4977924700882517458359048737510231963396017072296127685672337183902430724422017920s^{30}-47091620430092681534926593487054276521041403353074011105123802349042439773476528896s^{29}+418933385016745211947222808620255441114380728391860527906265870911858034743285241024s^{28}-3503006886259715819412667649026476439477808686694534314551125185742311900556660285184s^{27}+27514857713737746963130211058679811954406316972651571150660641469472381982354794337536s^{26}-202863150403975114963635408680873544302505729554954613863476699751624919208660885852672s^{25}+1402709995829726084689277339133025652193049640394127529908133471556083038663917684053760s^{24}-9086899324658301009708796843048145448075340279291908987325520939240439100244688624002560s^{23}+55084584772479049650257779664648209447949122588055864610352900437017173418201795772243456s^{22}-312044443572486729316802467807724154381090514130070572414647103019127107672636876664290816s^{21}+1649277211093445693448208153574041967523560611293184471904202237863220666468961287187338240s^{20}-8118640175848026108993106281833917188694597678315910558063402475202051432613391955855641600s^{19}+37144696681372172439650831549400091380866366509181988973869339080686979212069965990823787520s^{18}-157586730651964411252322536014125965772239288740159848629723235086393681235894364201883713536s^{17}+618288927781270563652014578600966260282549165002258008110053288490780862741600781299207507968s^{16}-2236565450657513656482121703891867471823841248110283410884806433679812901105466655396471359488s^{15}+7432893071780345274664428029482179027513487039424442339358853964292563899965270358695764254720s^{14}-22601990380304840114232955816799025270171303224566573276800545756165651975962250575537444556800s^{13}+62586563073321683259360467444111903207735246150889958394168784027878147573962885126129204400128s^{12}-156939492712340281577011813896174237722581326407907915345344934496499932397944939733935650037760s^{11}+354010766890679907447918498245949709618769477101420391146521973712262419115277237946938967687168s^{10}-712634824084661388879909049188671154939579018589650125862407413872672097371655108017943957979136s^9+1267801797993963237889272907735359448760706717740504858777243285714820373043141123507911382614016s^8-1969253813038576659750419438966806080674935670243968164986133969042920568174489538190545489821696s^7+2629669713639732187218315553568862244094923460774770810827814913228347803950554622508321583857664s^6-2958102093115310789329465782332338661036926390430105544964292955398971352437590664062597062066176s^5+2725943978218028478771349856122398946519913316361589789870009883220796007931840915720001460109312s^4-1976029693513971350410426740321961150195134039008222100615918618954101379818384436418594287386624s^3+1056622505101796269393565035252896788783387631720299838353179437245837174921473700994668720816128s^2-370548971618299668657098004068243761126772930024009924355548824210287559967194161234577752850432s+63933786130214659824460347409445038461589338575833265414758677167009207593207893617517426376704=0$
Adds an "L" to the $s(272)$ originally found by Lars Cleemann between 1991 and 1998, and improved based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019, and then improved by David Ellsworth in December 2024. Improvement pending. |
307.
$s = {}^{5}🔒 = \Nn{17.99295005377111}$
$10s^5-462s^4+8952s^3-97998s^2+587875s-1433594=0$
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. |
For more details, see Erich Friedman's paper on the subject: Packing Unit Squares in Squares: A Survey and New Results (or David Ellsworth's edit).
