Squares in Squares: Older / Alternative Packings

This is a list of obsoleted (but record-setting at the time) and/or alternative packings, shown alongside the corresponding best known. For the main list, see Squares in Squares.

Where the word "alternative" is used, this designates an alternative packing, which is enclosed within the same-sized bounding square as another packing, but cannot be reached by continuously translating and/or rotating the squares in that packing. When the packing can be reached by such continuous transformations, the word "rearrangement" is used.

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 $s$. For more information on each packing, view its SVG's source code.

10.
$s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$ Found by Frits Göbel in early 1979.		$s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$ Alternative with minimal rotated squares. Adds an "L" to $s(5)$.

11.
$s = {5\over 2}+\sqrt 2 = \Nn{3.91421356237309}$ Side length found by Frits Göbel in early 1979.		$s = {5\over 2}+\sqrt 2 = \Nn{3.91421356237309}$ Side length found by Frits Göbel in early 1979. Alternative with minimal rotated squares documented by Martin Gardner in October 1979.		$s = 2 + {4\over 3}\sqrt 2 = \Nn{3.88561808316412}$ Found by Pertti Hämäläinen in 1980. Didn't set an overall record, but proved by Walter R. Stromquist in 2002 to be the optimal 45° packing.		$s = {}^{8}🔒 = \Nn{3.87708359002281}$ $s^8 - 20s^7 + 178s^6 - 842s^5 + 1923s^4 - 496s^3 - 6754s^2 + 12420s -6865 = 0$ Rigid. Found by Walter Trump in 1979.

17.
$s = 4 + {1\over 2}\sqrt 2 = \Nn{4.70710678118654}$ Found by Frits Göbel in early 1979. Adds two "L"s to $s(5)$.		$s = {7\over 3} + {5\over 3}\sqrt 2 = \Nn{4.69035593728849}$ Found by Pertti Hämäläinen in 1980.		$s = {}^{16}🔒 = \Nn{4.68012531131999}$ $2401s^{16}-100156s^{15}+1913156s^{14}-22179724s^{13}+174450876s^{12}-986379724s^{11}+4148847308s^{10}-13271035292s^9+32788723886s^8-63314062708s^7+96314291996s^6-115448677092s^5+107387254380s^4-74501848708s^3+35776185556s^2-10308876020s+1301040841=0$ Symmetric version found by David W. Cantrell in September 2023. Possibly found by others previously. Didn't set an overall record, but is the best known symmetric packing.		$s = {}^{18}🔒 = \Nn{4.67553009360455}$ $4775s^{18}-190430s^{17}+3501307s^{16}-39318012s^{15}+300416928s^{14}-1640654808s^{13}+6502333062s^{12}-18310153596s^{11}+32970034584s^{10}-18522084588s^9-93528282146s^8+350268230564s^7-662986732745s^6+808819596154s^5-660388959899s^4+358189195800s^3-126167814419s^2+26662976550s-2631254953=0$ Improved by John Bidwell in 1998.

18.
$s = 2 + 2 \sqrt 2 = \Nn{4.82842712474619}$ Found by Frits Göbel in early 1979.		$s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$ Found by Pertti Hämäläinen in 1980.		$s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$ Alternative with minimal rotated squares found by Mats Gustafsson in 1981.		$s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$ Alternative found by David W. Cantrell in September 2002.		$s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$ Alternative found by Thierry Gensane and Philippe Ryckelynck in 2004.

19.

$s = 4 +{2\over 3}\sqrt 2 = \Nn{4.94280904158206}$
Found by Frits Göbel
in early 1979.

$s = {7\over 2}+ \sqrt 2 = \Nn{4.91421356237309}$
Found by Charles Cottingham
in early 1979.

$s = {}^{4}🔒 = \Nn{4.88810889245683}$ $s^4-12s^3+51s^2-72s-36=0$
Found by Walter R. Stromquist
in 1984. Didn't set a record.
Fits the $s(n^2\!-\!n\!-\!1)$ pattern found by Joe DeVincentis in April 2014.
Explore group

$s = 3 +{4\over 3}\sqrt 2 = \Nn{4.88561808316412}$
Improved by Robert Wainwright
in late 1979.
(The first of many others to independently rediscover it that same year.)

$s = 3 +{4\over 3}\sqrt 2 = \Nn{4.88561808316412}$
Alternative packing found by
David W. Cantrell (see also min/max)
in 2002.

$s = 3 +{4\over 3}\sqrt 2 = \Nn{4.88561808316412}$
Alternative with minimal rotated squares found by David W. Cantrell
in 2002.

$s = 3 +{4\over 3}\sqrt 2 = \Nn{4.88561808316412}$
Alternative with minimal rotated squares found by David W. Cantrell (see also min)
in 2002.

26.
$s = 5 + {1\over 2}\sqrt 2 = \Nn{5.70710678118654}$ Found by Frits Göbel in early 1979. Adds two "L"s to $s(10)$.		$s = 5 + {1\over 2}\sqrt 2 = \Nn{5.70710678118654}$ Alternative with minimal rotated squares. Adds three "L"s to $s(5)$.		$s = {}^{3}🔒 = \Nn{5.65062919143938}$ $s^3-14s^2+67s-112=0$ Found by Walter R. Stromquist in 1984.		$s = {7\over 2} + {3\over 2}\sqrt 2 = \Nn{5.62132034355964}$ Found by Erich Friedman in 1997.

27.
$s = 5 + {1\over 2}\sqrt 2 = \Nn{5.70710678118654}$ Found by Frits Göbel in early 1979.		$s = 5 + {1\over 2}\sqrt 2 = \Nn{5.70710678118654}$ Alternative with minimal rotated squares found by David W. Cantrell in 2005.

28.
$s = 3 + 2 \sqrt 2 = \Nn{5.82842712474619}$ Found by Frits Göbel in early 1979. Explore group		$s = 3 + 2 \sqrt 2 = \Nn{5.82842712474619}$ Rigid alternative with minimal rotated squares found by David Ellsworth in June 2023.

29.
$s = 6$ No nontrivial packing previously known.		$s = {8\over 3} + {7\over 3}\sqrt 2 = \Nn{5.96649831220388}$ Found by Erich Friedman in 1997.		$s = {}^{6}🔒 = \Nn{5.96480246752266}$ $9s^6-156s^5+653s^4+1084s^3-8100s^2+7344s-39717=0$ Improved by John Bidwell in 1998.		$s = \Nn{5.93434180499654}$ Found by Thierry Gensane and Philippe Ryckelynck in April 2004.

37.
$s = 6 + {1\over 2}\sqrt 2 = \Nn{6.70710678118654}$ Found by Frits Göbel in early 1979. Adds four "L"s to $s(5)$.		$s = {9\over 2} + {3\over 2}\sqrt 2 = \Nn{6.62132034355964}$ Adds an "L" to the $s(26)$ found by Erich Friedman in 1997.		$s = {9\over 2} + {3\over 2}\sqrt 2 = \Nn{6.62132034355964}$ Alternative repacking the "L"-augmented form into a primitive packing.		$s = {}^{5}🔒 = \Nn{6.62123046462703}$ $9s^5-123s^4+404s^3+146s^2-303s+205=0$ Found by Erich Friedman in 1997.		$s = \Nn{6.60323376318593}$ Improved by Thierry Gensane and Philippe Ryckelynck in April 2004, but didn't set a record.		$s = {}^{8}🔒 = \Nn{6.59861960924436}$ $6s^4-(208+64\sqrt{2})s^3+(2058+850\sqrt{2})s^2-(7936+3658\sqrt{2})s+11163+5502\sqrt{2}=0$ $36s^8-2496s^7+59768s^6-733760s^5+5289248s^4-23462672s^3+63458276s^2-96673872s+64068561=0$ Found by David W. Cantrell in September 2002.

38.
$s = 6 + {1\over 2}\sqrt 2 = \Nn{6.70710678118654}$ Found by Frits Göbel in early 1979.		$s = 6 + {1\over 2}\sqrt 2 = \Nn{6.70710678118654}$ Alternative with minimal rotated squares found by David W. Cantrell in 2005.

39.
$s = 4 + 2 \sqrt 2 = \Nn{6.82842712474619}$ Found by Frits Göbel in early 1979. Adds an "L" to $s(28)$.		$s = {11\over 2}+{1\over 2}\sqrt 7 = \Nn{6.82287565553229}$ Found by Erich Friedman in 1997.		$s = {}^{8}🔒 = \Nn{6.81880916998841}$ $5184s^8-197568s^7+3200144s^6-28651016s^5+154197141s^4-506760114s^3+981374688s^2-1005617394s+408278853=0$ Found by David W. Cantrell in August 2002.

41.
$s = 2 + {7\over 2}\sqrt 2 = \Nn{6.94974746830583}$ Found by Charles Cottingham in 1979.		$\begin{aligned}s &= 3 + {1\over 4}({7\sqrt 2 + \sqrt{46-8\sqrt 2}}) \\ &= \Nn{6.94725045864072}\end{aligned}$ $16s^4-192s^3+576s^2+112s-1471=0$ Found by David Cantrell in October 2005.		$s = {}^{16}🔒 = \Nn{6.94614417648499}$ $128s^8-(4608-208\sqrt{2})s^7+(71872-6636\sqrt{2})s^6-(638064-89464\sqrt{2})s^5+(3555432-667296\sqrt{2})s^4-(12870448-3018620\sqrt{2})s^3+(29967224-8441484\sqrt{2})s^2-(41793736-13833116\sqrt{2})s+27410100-10616645\sqrt{2}=0$ $8192s^{16}-589824s^{15}+19773184s^{14}-410097792s^{13}+5896832880s^{12}-62424822464s^{11}+504125766080s^{10}-3174083848032s^9+15779426694720s^8-62284061030400s^7+195013700484408s^6-480451336747312s^5+915434726864080s^4-1307583226605576s^3+1324167214909432s^2-851847919501960s+262943639948975=0$ Improved by David Ellsworth in November 2024. Didn't set a record.		$s = {}^{4}🔒 = \Nn{6.93786550630255}$ $s^4-16s^3+95s^2-218s-34=0$ Found by Joe DeVincentis in April 2014. Fits an $s(n^2\!-\!n\!-\!1)$ pattern. Explore group

50.
$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$ Found by Frits Göbel in early 1979. Adds four "L"s to $s(10)$.		$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$ Alternative with minimal rotated squares. Adds five "L"s to $s(5)$.		$s = {11\over 2} + {3\over 2}\sqrt 2 = \Nn{7.62132034355964}$ Found by Erich Friedman in 1997, by adding two "L"s to $s(26)$.		$s = {}^{8}🔒 = \Nn{7.59861960924436}$ $6s^4-(232+64\sqrt{2})s^3+(2718+1042\sqrt{2})s^2-(12700+5550\sqrt{2})s+21371+10074\sqrt{2}=0$ $36s^8-2784s^7+78248s^6-1146800s^5+9944448s^4-53242000s^3+173869324s^2-319180600s+253748689=0$ Found by David W. Cantrell in September 2002, by adding an "L" to $s(37)$.

51.
$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$ Found by Frits Göbel in early 1979.		$s = {}^{28}🔒 = \Nn{7.70435372947124}$ $36864s^{28}-6340608s^{27}+502050816s^{26}-24636665856s^{25}+847410746368s^{24}-21857473382400s^{23}+441080063406080s^{22}-7168010813250560s^{21}+95780257115813376s^{20}-1068807979173627904s^{19}+10079527432131681024s^{18}-81076261200222141184s^{17}+560144016315152943424s^{16}-3340596154679285521280s^{15}+17248154822575215485952s^{14}-77154000981112955287360s^{13}+298459379274993606556192s^{12}-993878287428748511469056s^{11}+2827560064086331516654992s^{10}-6798876862709272559608016s^9+13620557443692132080422196s^8-22318847802748398169997192s^7+29208515137302727559556744s^6-29572821499263810227200404s^5+22139938904533199326391397s^4-11407627350518593079154528s^3+3525180882798952592954446s^2-436679755165931930913236s-28766318325274882531199=0$ Found by Károly Hajba in July 2009.

52.
$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$ Found by Frits Göbel in early 1979.		$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$ Rigid alternative with minimal rotated squares found by David W. Cantrell in 2005.

53.
$s = 5 + 2 \sqrt 2 = \Nn{7.82842712474619}$ Found by Evert Stenlund in 1980, by adding an "L"s to the $s(40)$ found by Frits Göbel in early 1979.		$s = 5 + 2 \sqrt 2 = \Nn{7.82842712474619}$ Alternative repacking the "L"-augmented form into a primitive packing.		$s = {}^{3}🔒 = \Nn{7.82303789616673}$ $s^3-17s^2+36s+280=0$ Found by David W. Cantrell in September 2002.

54.
$s = {13\over 2} + \sqrt 2 = \Nn{7.91421356237309}$ Provided for comparison.		$s = {13\over 2} + \sqrt 2 = \Nn{7.91421356237309}$ Alternative with rotational symmetry found by David W. Cantrell in November 2024.		$s = 6 + {4\over 3}\sqrt 2 = \Nn{7.88561808316412}$ Found by Erich Friedman in 1997, extending the $s(19)$ found by Robert Wainwright in late 1979.		$s = {}^{24}🔒 = \Nn{7.85109314217532}$ $18432s^{24}-3446784s^{23}+304664576s^{22}-16968519680s^{21}+669449619968s^{20}-19932214017536s^{19}+465775491192832s^{18}-8768302695054848s^{17}+135403072780311424s^{16}-1737169753235173760s^{15}+18680996284830988288s^{14}-169371496869909229568s^{13}+1299023794031261063520s^{12}-8437530238970805727264s^{11}+46364613247151797971136s^{10}-214838591130149411496032s^9+834615127459846686846440s^8-2694590714870358438730024s^7+7138799170846114834605888s^6-15241751239015139566326992s^5+25549157414710670461692094s^4-32328821957836025603803370s^3+28978412289382495954074792s^2-16361602577473933571257634s+4362099385839709619937827=0$ Apparently found by Maurizio Morandi in June 2010 or slightly earlier, but didn't set a record.		$s = {}^{26}🔒 = \Nn{7.84878975975240}$ $4096s^{26}-696320s^{25}+56739840s^{24}-2949122048s^{23}+109766839552s^{22}-3113698237440s^{21}+69948465260800s^{20}-1276594350771968s^{19}+19265599452587584s^{18}-243443276207946880s^{17}+2598512723829467904s^{16}-23570528241163997888s^{15}+182367913759609388096s^{14}-1205666564490332194624s^{13}+6810634124400001375824s^{12}-32812890065132800032688s^{11}+134342994605566431921940s^{10}-464769276437493261704728s^9+1347820021349550459367488s^8-3241044285502742122448756s^7+6369772559544867940758561s^6-10037363738549635149843920s^5+12358928187941177640594208s^4-11472837613066933113730048s^3+7610215413527273079398656s^2-3277082686415784635207680s+712999405005519711846400=0$ Found by David W. Cantrell in October 2005.		$\begin{aligned}s &= 7-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{7.84666719284348}\end{aligned}$ Found by Joe DeVincentis in April 2014.

55.
$s = 8$ No nontrivial packing previously known.		$s = {}^{6}🔒 = \Nn{7.98701198965972}$ $481s^6-15166s^5+186713s^4-1132594s^3+3533776s^2-5430600s+3263162=0$ Found by David W. Cantrell in August 2002.		$s = \Nn{7.9547901}$ Found by Joe DeVincentis in April 2014. Fits an $s(n^2\!-\!n\!-\!1)$ pattern. Explore group		$s = \Nn{7.95424222760119}$ Improved by David Ellsworth in June 2023.		$s = \Nn{7.95421084599443}$ Improved by David W. Cantrell in August 2023.		$s = \Nn{7.95419161110664}$ Improved by David Ellsworth in November 2024.

65.
$s = 5 + {5\over 2}\sqrt 2 = \Nn{8.53553390593273}$ Found by Frits Göbel in early 1979. Explore group		$s = 5 + {5\over 2}\sqrt 2 = \Nn{8.53553390593273}$ Rearrangement with minimal rotated squares found by David Ellsworth in June 2023.

66.
$s = 5 + {5\over 2}\sqrt 2 = \Nn{8.65685424949238}$ Found by Evert Stenlund in 1980.		$s = 5 + {5\over 2}\sqrt 2 = \Nn{8.65685424949238}$ Alternative with rotational symmetry found by David W. Cantrell in 2023.

67.
$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$ Found by Erich Friedman in 1997, extending the $s(52)$ found by Frits Göbel in early 1979.		$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$ Alternative by Erich Friedman in 1997, adding an "L" to the $s(52)$ found by Frits Göbel in early 1979.		$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$ Alternative with minimal rotated squares found by David W. Cantrell in June 2023.

68.
$s = 5 + 2 \sqrt 2 = \Nn{8.82842712474619}$ Found by Evert Stenlund in 1980, by adding two "L"s to the $s(40)$ found by Frits Göbel in early 1979.		$s = 5 + 2 \sqrt 2 = \Nn{8.82842712474619}$ Alternative repacking the "L"-augmented form into a primitive packing.		$s = {15\over 2} + {1\over 2}\sqrt 7 = \Nn{8.82287565553229}$ Found by David W. Cantrell in September 2002.		$s = {13\over 3} + 2 \sqrt 5 = \Nn{8.80546928833291}$ Found by Sigvart Brendberg in June 2023.		$s = {13\over 3} + 2 \sqrt 5 = \Nn{8.80546928833291}$ Rearrangement with minimal rotated squares found by Sigvart Brendberg in June 2023.

69.

$s = {5\over 2} + {9\over 2}\sqrt 2 = \Nn{8.86396103067892}$
Found by Erich Friedman
in 1997.

$s = {}^{6}🔒 = \Nn{8.85617974640243}$ $16s^6-504s^5+6117s^4-34854s^3+86322s^2-53182s+19237=0$
Found by David W. Cantrell
in September 2002.

$s = {}^{24}🔒 = \Nn{8.85109314217532}$ $18432s^{24}-3889152s^{23}+389027840s^{22}-24580483072s^{21}+1102466163200s^{20}-37385083702784s^{19}+996598199576064s^{18}-21433776649410560s^{17}+378654550444888448s^{16}-5564791746074648960s^{15}+68634423179436341888s^{14}-714589619551925523840s^{13}+6301668522983200715232s^{12}-47124317021386371701792s^{11}+298546848767309267273056s^{10}-1597315404347371966113984s^9+7177015917694030181782984s^8-26850335202074919621125800s^7+82608242230494967829610872s^6-205341156374029643891989208s^5+401949708478623153033594262s^4-596112355282004302680524122s^3+629100458661410266649761474s^2-420590216248553598525473360s+133760979570165275895770709=0$
Apparently found by Maurizio Morandi
in June 2010 or slightly earlier.

$s = {}^{26}🔒 = \Nn{8.84878975975240}$ $4096s^{26}-802816s^{25}+75479040s^{24}-4530423808s^{23}+194919613696s^{22}-6398615904768s^{21}+166556019374080s^{20}-3527133377275648s^{19}+61862476687325248s^{18}-910107143467106304s^{17}+11332922227072242880s^{16}-120197511537209237184s^{15}+1090196653398278259840s^{14}-8474103290188848197504s^{13}+56471863481873946657168s^{12}-322223865161764417071024s^{11}+1569458277137839730778564s^{10}-6493082769370743372961904s^9+22652899039672064985204476s^8-65980365270454114217274836s^7+158260034464024031096444789s^6-306765404716748327800087354s^5+467930910665959192575853315s^4-540089919969690029016072512s^3+443048213754547127055903719s^2-230214712619268014090364786s+57068214173713936231485937=0$
Found by Maurizio Morandi (by adding an "L" to the $s(54)$ found by David W. Cantrell)
in June 2010 or slightly earlier.

$s = {}^{8}🔒 = \Nn{8.84116469390844}$ $2686s^4-(75940-5664\sqrt{2})s^3+(790046-138378\sqrt{2})s^2-(3559776-1127334\sqrt{2})s-3066408\sqrt{2}+5789327=0$ $5372s^8-303760s^7+7406440s^6-101251088s^5+842958976s^4-4326575672s^3+13090556164s^2-20394602112s+10953530207=0$
Experiment by David Ellsworth
in November 2024, by adapting the $s(37)$ improvement by David W. Cantrell in September 2002.
Didn't set a record.

$s = {}^{6}🔒 = \Nn{8.82868705685148}$ $s^6-32s^5+423s^4-3296s^3+18806s^2-75584s+142565=0$
Found by Maurizio Morandi
in June 2010.

$s = {}^{82}🔒 = \Nn{8.82721205592900}$ $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$
Improved by David W. Cantrell
in August 2023.

70.
$s = 4 +{7\over 2}\sqrt 2 = \Nn{8.94974746830583}$ Unpublished, but was likely found before the improvements that followed.		$s = {15\over 2}+ \sqrt 2 = \Nn{8.91421356237309}$ Found by Erich Friedman in 1997.		$s = {}^{7}🔒 = \Nn{8.91209113581151}$ $16s^7-960s^6+24610s^5-346304s^4+2850209s^3-13406809s^2+31814883s-25624307=0$ Improved by David W. Cantrell in August 2002.		$s = {}^{8}🔒 = \Nn{8.90476635933426}$ $8s^4-(160+48\sqrt{2})s^3+(1162+840\sqrt{2})s^2-(4138+4368\sqrt{2})s+7961+5772\sqrt{2}=0$ $64s^8-2560s^7+39584s^6-276768s^5+551924s^4+3620472s^3-21928360s^2+34963148s-3254447=0$ Improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement by David W. Cantrell in November 2024. Didn't set a record.		$s = {}^{4}🔒 = \Nn{8.88166675700900}$ $23s^4-742s^3+8848s^2-45876s+86229=0$ Found by Joe DeVincentis in April 2014.

71.		71.
$s = 9$ No nontrivial packing previously known.		$s = {}^{8}🔒 = \Nn{8.96326750850139}$ $s^4-(18+10\sqrt{2})s^3+(157+152\sqrt{2})s^2-(746+742\sqrt{2})s+1353+1094\sqrt{2}=0$ $s^8-36s^7+438s^6-1064s^5-21677s^4+211944s^3-784922s^2+1228316s-563063=0$ Found by David Cantrell in October 2005.		$s = {}^{4}🔒 = \Nn{8.96028765944389}$ $s^4-20s^3+151s^2-468s+12=0$ Found by Joe DeVincentis in April 2014. Fits an $s(n^2\!-\!n\!-\!1)$ pattern. Explore group

83.
$s = 4 + 4 \sqrt 2 = \Nn{9.65685424949238}$ Found by Erich Friedman in 1997 (or Evert Stenlund in 1980) by adding an "L" to the $s(66)$ found by Evert Stenlund in 1980.		$s = 4 + 4 \sqrt 2 = \Nn{9.65685424949238}$ Alternative without diagonal mirror symmetry, as originally presented.		$s = {}^{26}🔒 = \Nn{9.63681171725651}$ $3301165359s^{26}-655973563500s^{25}+50204111425925s^{24}-2016728082067340s^{23}+42539291051118064s^{22}-155936223658816784s^{21}-18310692686166193312s^{20}+591490273594615246296s^{19}-8704776447911059155340s^{18}+29105577044266076411528s^{17}+1565268183079984272144364s^{16}-38531265127295718043578176s^{15}+457091443662013291766365392s^{14}-2222742682755255371679742512s^{13}-24936806055484971890366802672s^{12}+698108382960198940777683768000s^{11}-8883512560110334504239194416640s^{10}+78358434090848901490183344375936s^9-523167090946476999256538020431344s^8+2718783627950104467777347864419584s^7-11055205056181316074113447108321152s^6+34889337532013365693949953532068352s^5-83794842528024412407912862460950272s^4+148113646516510229311356493201954816s^3-182221209141584860870287890177921024s^2+140464471690452627000592739126640640s-51694146925208010027347936896811008=0$ Improved by Károly Hajba in September 2024.		$s = {}^{24}🔒 = \Nn{9.63482562092335}$ $46438209s^{24}+1718447880s^{23}-1304818741864s^{22}+154362940868008s^{21}-10223870917986092s^{20}+463012769729234068s^{19}-15608677475881443482s^{18}+410530364971106359132s^{17}-8675319117762080311978s^{16}+150196459602374087471728s^{15}-2158879193002672091253360s^{14}+25993038455067669296355532s^{13}-263613888105247221344935027s^{12}+2258335015809616506745502008s^{11}-16347943921555337654669478150s^{10}+99786776593815833271369617220s^9-511154425074511891757096094175s^8+2180187656593439512672814134216s^7-7652314463979073976449593048904s^6+21727853135387976484209118127392s^5-48671720700899577518293563957136s^4+82801528406446840092722047620736s^3-100540002112755895115349929950336s^2+77621257841393908308227797286912s-28634116465193128516311336597248=0$ Improved by David W. Cantrell in November 2024.

84.
$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$ Found by Erich Friedman in 1997, extending the $s(52)$ found by Frits Göbel in early 1979.		$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$ Alternative by Erich Friedman in 1997, extending and adding an "L" to the $s(52)$ found by Frits Göbel in early 1979.		$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$ Alternative by Erich Friedman in 1997, adding two "L"s to the $s(52)$ found by Frits Göbel in early 1979.		$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$ Alternative with minimal rotated squares found by David W. Cantrell in 2005.

85.
$s = 2 + {11\over 2}\sqrt 2 = \Nn{9.77817459305202}$ Extends the $s(18)$ found by Frits Göbel in early 1979.		$s = {11\over 2} + 3 \sqrt 2 = \Nn{9.74264068711928}$ Found by Erich Friedman in 1997.		$s = {11\over 2} + 3 \sqrt 2 = \Nn{9.74264068711928}$ Alternative with minimal rotated squares.

86.
$s = 7 + 2 \sqrt 2 = \Nn{9.82842712474619}$ Unpublished, but was likely found before the improvements that followed.		$s = {17\over 2} + {1\over 2}\sqrt 7 = \Nn{9.82287565553229}$ Found by Erich Friedman in 1997.

87.
$s = { 7\over 2}+{ 9\over 2}\sqrt 2 = \Nn{9.86396103067892}$ Unpublished, but was likely found before the improvements that followed.		$s = {14\over 3}+{11\over 3}\sqrt 2 = \Nn{9.85211639536801}$ Found by Evert Stenlund in 1980. Similar to the $s(19)$ found by Robert Wainwright in late 1979. Note that Evert Stenlund was one of the 11 independent rediscoverers of that $s(19)$ before March 1980, as listed by Martin Gardner.		$s = {}^{6}🔒 = \Nn{9.85197533993158}$ $9s^6-282s^5+2045s^4+5870s^3-89663s^2+554156s-3691596=0$ Improved by David W. Cantrell in August 2002.

88.
$s = {17\over 2}+\sqrt 2 = \Nn{9.91421356237309}$ Found by Erich Friedman in 1997.		$s = {}^{12}🔒 = \Nn{9.90177651254408}$ $4s^6-(320+56\sqrt{2})s^5+(9952+2828\sqrt{2})s^4-(55956\sqrt{2}+156632)s^3+(543586\sqrt{2}+1329945)s^2-(5815938+2595866\sqrt{2})s+10276983+4876156\sqrt{2}=0$ $16s^{12}-2560s^{11}+175744s^{10}-6988864s^9+181397032s^8-3260558096s^7+41816558200s^6-386837301552s^5+2568083107241s^4-11953457726884s^3+37081298010138s^2-68909201625724s+58062584909617=0$ Improved by David W. Cantrell in August 2002.		$s = {}^{8}🔒 = \Nn{9.89444803576420}$ $4s^4-(84+28\sqrt{2})s^3+(612+532\sqrt{2})s^2-(2202+2870\sqrt{2})s+5289+3164\sqrt{2}=0$ $16s^8-672s^7+10384s^6-60848s^5-100696s^4+2877928s^3-11884252s^2+13029964s+7951729=0$ Improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{9.888160100452437}$ $4898s^4-(186856+8508\sqrt{2})s^3+(2676964+234258\sqrt{2})s^2-(17078860+2159580\sqrt{2})s+40971345+6672600\sqrt{2}=0$ $9796s^8-747424s^7+24905648s^6-473558928s^5+5621403852s^4-42670393040s^3+202313221240s^2-547916568600s+649082862225=0$ Improved by David Ellsworth in November 2024, by adapting and extending the technique from the $s(37)$ found by David W. Cantrell in September 2002.

89.
$s = 5 + {7\over 2}\sqrt 2 = \Nn{9.94974746830583}$ Found by Evert Stenlund in 1980, by extending a pattern found by Frits Göbel in early 1979. Explore group		$s = 5 + {7\over 2}\sqrt 2 = \Nn{9.94974746830583}$ Rearrangement with minimal rotated squares found by David Ellsworth in June 2023.

102.
$s = {17\over 2} + {3\over 2}\sqrt 2 = \Nn{10.62132034355964}$ Found by Károly Hajba in September 2024. Extends the $s(37)$ that results from adding an "L" to the $s(26)$ found by Erich Friedman in 1997 and converting that to a primitive packing. Unlike the $s(66)$ found by Evert Stenlund in 1980, is clamped on its narrow side while having play along its long side.		$s = {}^{8}🔒 = \Nn{10.61393128238062}$ $8s^4-(192+48\sqrt{2})s^3+(1614+984\sqrt{2})s^2-(5918+6048\sqrt{2})s+10249+9876\sqrt{2}=0$ $64s^8-3072s^7+58080s^6-525536s^5+1943764s^4+2662200s^3-43922048s^2+117613028s-90028751=0$ Improved by David W. Cantrell in November 2024. This improvement is very similar to that of $s(37)$.		$s = {}^{8}🔒 = \Nn{10.61350257784129}$ $392s^4-(33432+13888\sqrt{2})s^3+(706840+371152\sqrt{2})s^2-(5716880+3287856\sqrt{2})s+16051345+9689424\sqrt{2}=0$ $3136s^8-534912s^7+26247104s^6-635223424s^5+8904202320s^4-76237792816s^3+395289628464s^2-1144847001376s+1426036763377=0$ Improved by David Ellsworth in November 2024, by adapting the technique from the $s(37)$ found by David W. Cantrell in September 2002.

106.
$s = 8 + 2 \sqrt 2 = \Nn{10.82842712474619}$ Extends the s(28) found by Frits Göbel in early 1979.		$s = 8 + 2 \sqrt 2 = \Nn{10.82842712474619}$ Alternative combining the $s(40)$ found by Frits Göbel in early 1979, and the $s(18)$ alternative packing found by Mats Gustafsson in 1981.		$s = {}^{4}🔒 = \Nn{10.82308816660919}$ $s^4-38s^3+355s^2+714s-14857=0$ Found by David Ellsworth in November 2024, based on the $s(53)$ found by David W. Cantrell in September 2002.

107.

$s = {19\over 2} + \sqrt 2 = \Nn{10.91421356237309}$
Provided for comparison.

$s = {}^{6}🔒 = \Nn{10.90177651254408}$ $4s^6-(344+56\sqrt{2})s^5+(11612+3108\sqrt{2})s^4-(199720+67828\sqrt{2})s^3+(1862813+728982\sqrt{2})s^2-(8987156+3862498\sqrt{2})s17589774+8074448\sqrt{2}=0$ $16s^{12}-2752s^{11}+204960s^{10}-8890624s^9+252635608s^8-4985280208s^7+70344750088s^6-717292684304s^5+5257253669425s^4-27055380682296s^3+92919601709108s^2-191413928881072s+179006728361668=0$
Adds an "L" to the $s(88)$ found by
David W. Cantrell in August 2002.

$s = 9 + {4\over 3}\sqrt 2 = \Nn{10.88561808316412}$
Extends the $s(54)$ found by Erich Friedman in 1997.

$s = {}^{24}🔒 = \Nn{10.85109314217532}$ $18432s^{24}-4773888s^{23}+588277760s^{22}-45931980800s^{21}+2552543416832s^{20}-107500243934720s^{19}+3566520312916480s^{18}-95644799494952960s^{17}+2110582635042826624s^{16}-38807300890882986368s^{15}+599770576843313706880s^{14}-7836620660791447974272s^{13}+86855117839205769003744s^{12}-817505803876276332946720s^{11}+6528487118774181169397536s^{10}-44097570952979852814707648s^9+250550088290813287983018440s^8-1187342150090217786690276136s^7+4635908831980002371952919080s^6-14654160330940064886216876440s^5+36560932559920850709129952486s^4-69286092260078962945268233802s^3+93708449241192303154538746350s^2-80560987713141442549975798736s+33076489286419613987330352941=0$
Extends the $s(54)$ apparently found by Maurizio Morandi in June 2010 or slightly earlier.

$s = {}^{26}🔒 = \Nn{10.84878975975240}$ $4096s^26-1015808s^25+120944640s^24-9201993728s^23+502399546624s^22-20953136291328s^21+693848568469504s^20-18719597068130048s^19+418950222375670336s^18-7878536261143035136s^17+125644760096620381632s^16-1710258057372252411584s^15+19954539058257375074048s^14-200038166892087062773888s^13+1724103508752218328450192s^12-12763352735201165788216624s^11+80937126937487507548487524s^10-437648167759224144329248000s^9+2004234645426817826431993644s^8-7699646718693974721330444692s^7+24488166278454036208661847501s^6-63303938085773821186731652934s^5+129578943655421572443592443667s^4-201986692988967941421692106568s^3+225151551461908324070320627711s^2-159766200573552065970752291662s+54201975899799278005783078177=0$
Extends the $s(54)$ found by
David W. Cantrell in October 2005.

$\begin{aligned}s &= 10-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{10.84666719284348}\end{aligned}$
Found by Károly Hajba
in November 2024.

$\begin{aligned}s &= 10-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{10.84666719284348}\end{aligned}$
Alternative packing extending the $s(54)$ found by Joe DeVincentis in April 2014.

108.
$s = {}^{48}🔒 = \Nn{10.92610452776589}$ $s^{48}-296s^{47}+37526s^{46}-2457236s^{45}+60844617s^{44}+3122963280s^{43}-331745446062s^{42}+10983160663460s^{41}+62452173904874s^{40}-19446218282632488s^{39}+679354503491649660s^{38}-306347914574021296s^{37}-824192700536782793245s^{36}+30911983353740820507596s^{35}-262121177343638691053278s^{34}-20657417122552298628935512s^{33}+971695388124063306053267226s^{32}-16884352389057164640444940516s^{31}-166659722723317426893191108440s^{30}+17691521821468611178286376431996s^{29}-468921640726619185864899949407898s^{28}+3438326638194136822140720890342188s^{27}+217961407836578733314911846614259564s^{26}-11248600516826176233529679347044225712s^{25}+323920805642357351326081112233623957573s^{24}-6887388276136117234932790212364322541116s^{23}+115882135156598208423601413146959010390534s^{22}-1567334776845205905421716254395582625876388s^{21}+16682906372542830966045907261996287027604830s^{20}-127034908598646065907756912304581873823655412s^{19}+394733542489197810146567993704557834850847040s^{18}+6705651563244549820196163041336291829216481116s^{17}-149808196690779798092249384238501456462483764077s^{16}+1765316723831751122567779994720019780273788899744s^{15}-14414520548700152134078391432348665546702557725700s^{14}+77782340758050929719490920739995509475465836939316s^{13}-122708778574798015128015461066223615343279092778398s^{12}-2859086568813124377779141812318153754010977141736244s^{11}+40645452384731308590882059237227780403713187160844132s^{10}-338255715966963536305264184392547735966960852835234560s^9+2097972040533665959486451035925303239919231900516664905s^8-10276265765870648891612144344761814178341387935812072112s^7+40341689255275766626501645859028711393105697769978383064s^6-126359457459344553289312516972810042412505269755888407136s^5+309947750328179116200888948608223797814828405187929737584s^4-574944902893431216495052138731525783921261450808775002240s^3+758896117426851846626726372201059822573691790491620054400s^2-634934004875500649691379467228925311855209793769543360000s+252784775851498947264108627439343213239274355138539040000=0$ Found by Károly Hajba in October 2024.		$s = \Nn{10.92591939016138}$ Improved by David Ellsworth in November 2024, using a similar technique to that of the $s(87)$ improvement by David W. Cantrell in August 2002.

109.
$s = {}^{24}🔒 = \Nn{10.97240394480333}$ $s^4-24s^3+219s^2-846s+126=0$ Continues the $s(n^2\!-\!n\!-\!1)$ pattern found by Joe DeVincentis in April 2014, but is not optimal. Explore group		$s = 6 + {7\over 2}\sqrt 2 = \Nn{10.94974746830583}$ Extends a pattern found by Frits Göbel in early 1979. Explore group

123.
$s = {19\over 2} + {3\over 2}\sqrt 2 = \Nn{11.62132034355964}$ Adds an "L" to the $s(102)$ found by Károly Hajba in September 2024.		$s = {19\over 2} + {3\over 2}\sqrt 2 = \Nn{11.62132034355964}$ Alternative with rotational symmetry found by David Ellsworth in November 2024 by combining two copies of the $s(26)$ found by Erich Friedman in 1997.		$s = {}^{8}🔒 = \Nn{11.61393128238062}$ $8s^4-(224+48\sqrt{2})s^3+(2238+1128\sqrt{2})s^2-(9754+8160\sqrt{2})s+17981+16956\sqrt{2}=0$ $64s^8-3584s^7+81376s^6-942112s^5+5554644s^4-11640920s^3-34053200s^2+202670492s-251695511=0$ Improved by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{11.61350257784129}$ $392s^4-(35000+13888\sqrt{2})s^3+(809488+412816\sqrt{2})s^2-(7232424+4071824\sqrt{2})s+22508889+13362320\sqrt{2}=0$ $3136s^8-560000s^7+30079296s^6-804114816s^5+12492967440s^4-118750675952s^3+684185482592s^2-2203093815248s+3051977396929=0$ Improved by David Ellsworth in November 2024, by adapting the technique from the $s(37)$ found by David W. Cantrell in September 2002.

128.
$s = 9 + 2 \sqrt 2 = \Nn{11.82842712474619}$ Extends the s(40) found by Frits Göbel in early 1979.		$s = {}^{4}🔒 = \Nn{11.82629667551039}$ $2s^4-82s^3+1248s^2-8354s+20759=0$ Found by David Ellsworth in November 2024, based on the $s(69)$ found by Maurizio Morandi in June 2010.

130.
$s = {21\over 2}+\sqrt 2 = \Nn{11.91421356237309}$ Extends the $s(88)$ found by Erich Friedman in 1997.		$s = {}^{8}🔒 = \Nn{11.91266577563213}$ $2s^4-(50+16\sqrt{2})s^3+(401+368\sqrt{2})s^2-(1316+2344\sqrt{2})s+3264+2576\sqrt{2}=0$ $4s^8-200s^7+3592s^6-21812s^5-115407s^4+2233400s^3-10430960s^2+15561728s-2617856=0$ Improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{11.91251680114691}$ $18s^4-(654+84\sqrt{2})s^3+(8975+2408\sqrt{2})s^2-(55632+22694\sqrt{2})s+133015+70042\sqrt{2}=0$ $324s^8-23544s^7+736704s^6-12932964s^5+138883709s^4-930455300s^3+3777858858s^2-8441648368s+7881226697=0$ Improved by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{11.91119052015898}$ $5617s^4-(259504+12612\sqrt{2})s^3+(4502202+422478\sqrt{2})s^2-(34773984+4731588\sqrt{2})s+100926915+17726970\sqrt{2}=0$ $5617s^8-519008s^7+20936788s^6-481754784s^5+6917560482s^4-63487737120s^3+363767813964s^2-1189915431840s+1701575795025=0$ Improved by David Ellsworth in November 2024, by adapting and extending the technique from the $s(37)$ found by David W. Cantrell in September 2002.		$s = {}^{8}🔒 = \Nn{11.91119052015898}$ $5617s^4-(259504+12612\sqrt{2})s^3+(4502202+422478\sqrt{2})s^2-(34773984+4731588\sqrt{2})s+100926915+17726970\sqrt{2}=0$ $5617s^8-519008s^7+20936788s^6-481754784s^5+6917560482s^4-63487737120s^3+363767813964s^2-1189915431840s+1701575795025=0$ Alternative with rotational symmetry found by David W. Cantrell in November 2024.

131.
$s = \Nn{11.97614140898726}$ Found by Károly Hajba in November 2024.		$s = \Nn{11.97350182495032}$ Found by David Ellsworth in November 2024. Fits an $s(n^2\!-\!n\!-\!1)$ pattern found by Joe DeVincentis in April 2014. Explore group

132.
$s = 12$ No nontrivial packing previously known.		$s = \Nn{11.99790201730589}$ Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019. Bounds the $s(n^2-n)=n$ conjecture to $n<12$.

146.
$s = {21\over 2} + {3\over 2}\sqrt 2 = \Nn{12.62132034355964}$ Adds two "L"s to the $s(102)$ found by Károly Hajba in September 2024.		$s = {}^{8}🔒 = \Nn{12.61393128238062}$ $8s^4-(256+48\sqrt{2})s^3+(2958+1272\sqrt{2})s^2-(14934+10560\sqrt{2})s+30205+26292\sqrt{2}=0$ $64s^8-4096s^7+108256s^6-1509216s^5+11615764s^4-45037160s^3+44916240s^2+208411140s-470196503=0$ Improved by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{12.61350257784129}$ $392s^4-(36568+13888\sqrt{2})s^3+(916840+454480\sqrt{2})s^2-(8957968+4939120\sqrt{2})s+30586193+17860848\sqrt{2}=0$ $3136s^8-585088s^7+34087104s^6-996526208s^5+16984550480s^4-177385055408s^3+1123899500496s^2-3981893672992s+6071335095409=0$ Improved by David Ellsworth in November 2024, by adapting the technique from the $s(37)$ found by David W. Cantrell in September 2002.

149.
$s = 12 + {1\over 2}\sqrt 2 = \Nn{12.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979.		$s = 12 + {1\over 2}\sqrt 2 = \Nn{12.70710678118654}$ Rigid alternative with minimal rotated squares found by David Ellsworth in November 2024, based on the rigid $s(52)$ found by David W. Cantrell in 2005.

150.
$s = {11\over 2}+{ 9\over 2}\sqrt 2 = \Nn{12.86396103067892}$ Provided for comparison.		$s = {23\over 3}+{11\over 3}\sqrt 2 = \Nn{12.85211639536801}$ Extends the $s(87)$ found by Evert Stenlund in 1980.		$s = {}^{6}🔒 = \Nn{12.85197533993158}$ $9s^6-444s^5+7490s^4-48910s^3+55012s^2+902432s-6078789=0$ Extends the $s(87)$ improved by David W. Cantrell in August 2002.		$s = {23\over 2} + {1\over 2}\sqrt 7 = \Nn{12.82287565553229}$ Extends and adds an "L" to the $s(86)$ found by Erich Friedman in 1997.

152.		153.
$s = 3 + 7 \sqrt 2 = \Nn{12.89949493661166}$ Extends the $s(28)$ found by Frits Göbel in early 1979.		$s = {}^{4}🔒 = \Nn{12.88166675700900}$ $23s^4-1110s^3+19960s^2-158164s+464677=0$ Found by David Ellsworth in November 2024, based on the $s(70)$ found by Joe DeVincentis in April 2014.

153.
$s = {23\over 2}+\sqrt 2 = \Nn{12.91421356237309}$ Extends the $s(88)$ found by Erich Friedman in 1997.		$s = {}^{4}🔒 = \Nn{12.88166675700900}$ $23s^4-1110s^3+19960s^2-158164s+464677=0$ Found by David Ellsworth in November 2024, based on the $s(70)$ found by Joe DeVincentis in April 2014.

154.
$s = \Nn{12.97614140898726}$ Adds an "L" to the $s(131)$ found by Károly Hajba in November 2024.		$s = \Nn{12.97350182495032}$ Adds an "L" to the $s(131)$ found by David Ellsworth in November 2024.

156.
$s = 13$ No nontrivial packing previously known.		$s = \Nn{12.99404229036268}$ Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019. Shows $s(n^2-n)<n$ for $n=13$.

170.
$s = 10 + {5\over 2}\sqrt 2 = \Nn{13.53553390593273}$ Adds five "L"s to the $s(65)$ found by Frits Göbel in early 1979.		$s = 10 + {5\over 2}\sqrt 2 = \Nn{13.53553390593273}$ Alternative repacking the "L"-augmented form into a primitive packing.

172.
$s = {23\over 2} + {3\over 2}\sqrt 2 = \Nn{13.62132034355964}$ Extends the $s(102)$ found by Károly Hajba in September 2024.		$s = {}^{8}🔒 = \Nn{13.61977561767287}$ $4s^4-(116+36\sqrt{2})s^3+(1136+936\sqrt{2})s^2-(4998+6930\sqrt{2})s+14139+10746\sqrt{2}=0$ $16s^8-928s^7+19952s^6-168752s^5-186968s^4+12857640s^3-79179012s^2+156545676s-31041711=0$ Improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{13.61970688383567}$ $562s^4-(166824+99232\sqrt{2})s^3+(5529694+3579002\sqrt{2})s^2-(64343912+42896630\sqrt{2})s+252737371+171045700\sqrt{2}=0$ $1124s^8-667296s^7+51073464s^6-1767572224s^5+34464149128s^4-405438721504s^3+2869488680332s^2-11299238055184s+19085109334561=0$ Improved by David Ellsworth in November 2024, by adapting the technique from the $s(37)$ found by David W. Cantrell in September 2002.

177.
$s = 11 + 2 \sqrt 2 = \Nn{13.82842712474619}$ Combines two copies of the $s(40)$ found by Frits Göbel in early 1979.		$s = {}^{4}🔒 = \Nn{13.82311724757295}$ $s^4-50s^3+565s^2+2676s-49396=0$ Found by David Ellsworth in November 2024, based on the $s(53)$ found by David W. Cantrell in September 2002.

178.
$s = 4 + 7 \sqrt 2 = \Nn{13.89949493661166}$ Extends the $s(40)$ found by Frits Göbel in early 1979.		$s = {15\over 2} + {9\over 2}\sqrt 2 = \Nn{13.86396103067892}$ Extends the $s(26)$, $s(85)$ series found by Erich Friedman in 1997.		$\begin{aligned}s &= 13-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{13.84666719284348}\end{aligned}$ Extends the $s(54)$ found by Joe DeVincentis in April 2014.		$\begin{aligned}s &= 13-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{13.84666719284348}\end{aligned}$ Alternative without rotational symmetry.

179.
$s = 9 + {7\over 2}\sqrt 2 = \Nn{13.94974746830583}$ Extends $s(109)$, which continues a pattern found by Frits Göbel in early 1979.		$s = {}^{4}🔒 = \Nn{13.93786550630255}$ $s^4-44s^3+725s^2-5272s+14036=0$ Combines two copies of the $s(41)$ found by Joe DeVincentis in April 2014, which fits an $s(n^2\!-\!n\!-\!1)$ pattern.

182.
$s = 14$ No nontrivial packing previously known.		$s = \Nn{13.98318264138415}$ Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019. Shows $s(n^2-n)<n$ for $n=14$.

198.
$s = {25\over 2} + {3\over 2}\sqrt 2 = \Nn{14.62132034355964}$ Extends the $s(102)$ found by Károly Hajba in September 2024.		$s = {}^{8}🔒 = \Nn{14.59861960924436}$ $6s^4-(400+64\sqrt2)s^3+(9354+2386\sqrt2)s^2-(93088+29546\sqrt2)s+337435+121934\sqrt2=0$ $36s^8-4800s^7+264056s^6-7989440s^5+147067168s^4-1698236176s^3+12068439396s^2-48411650704s+84126578513=0$ Adds an "L" to two combined copies of the $s(37)$ found by David W. Cantrell in September 2002.		$s = {}^{8}🔒 = \Nn{14.59153604974246}$ $7889s^4-(496196+66432\sqrt{2})s^3+(11353289+2588872\sqrt{2})s^2-(112836070+33552520\sqrt{2})s+413156140+144732976\sqrt{2}=0$ $7889s^8-992392s^7+52797090s^6-1566650388s^5+28530003977s^4-327826283468s^3+2327673598588s^2-9356471793680s+16326876326032=0$ Improved by David Ellsworth in December 2024, by adapting and extending the technique from the $s(37)$ found by David W. Cantrell in September 2002.

199.
$s = {25\over 2} + {3\over 2}\sqrt 2 = \Nn{14.62132034355964}$ Adds an "L" to $s(172)$, which extends the $s(102)$ found by Károly Hajba in September 2024.		$s = {}^{8}🔒 = \Nn{14.61977561767287}$ $4s^4-(132+36\sqrt{2})s^3+(1508+1044\sqrt{2})s^2-(7634+8910\sqrt{2})s+20393+18648\sqrt{2}=0$ $16s^8-1056s^7+26896s^6-308848s^5+989672s^4+11485576s^3-116867004s^2+353254396s-279621359=0$ Improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{14.61970688383567}$ $562s^4-(169072+99232\sqrt{2})s^3+(6033538+3876698\sqrt{2})s^2-(75906020+50352330\sqrt{2})s+322778363+217620564\sqrt{2}=0$ $1124s^8-676288s^7+55776008s^6-2088089168s^5+44091546248s^4-562015927840s^3+4311045608500s^2-18401537158840s+33695558254817=0$ Improved by David Ellsworth in November 2024, by adapting the technique from the $s(37)$ found by David W. Cantrell in September 2002.

201.
$s = 14 + {1\over 2}\sqrt 2 = \Nn{14.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979.		$s = 14 + {1\over 2}\sqrt 2 = \Nn{14.70710678118654}$ Alternative combining the $s(52)$ rigid alternative with minimal rotated squares found by David W. Cantrell in 2005, and the $s(51)$ found by Károly Hajba in July 2009.

202.
$s = {21\over 2} + 3 \sqrt 2 = \Nn{14.74264068711928}$ Found by David Ellsworth in November 2024, by extending the $s(85)$ found by Erich Friedman in 1997.		$s = {}^{8}🔒 = \Nn{14.73657855744445}$ $776s^4-(33320+336\sqrt{2})s^3+(528568+7752\sqrt{2})s^2-(3662244+43296\sqrt{2})s+9301325-28968\sqrt{2}=0$ $6208s^8-533120s^7+19900352s^6-421620160s^5+5543230416s^4-46288934864s^3+239607688384s^2-702396496296s+891886272841=0$ Improved by David Ellsworth in November 2024, by adapting the technique from the $s(37)$ found by David W. Cantrell in September 2002. Didn't set a record.		$s = 2 + 9 \sqrt 2 = \Nn{14.72792206135785}$ Extends the $s(18)$ found by Frits Göbel in early 1979.

205.
$\begin{aligned}s &= 14-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{14.84666719284348}\end{aligned}$ Combines the $s(40)$ found by Frits Göbel in early 1979, and the $s(54)$ found by Joe DeVincentis in April 2014.		$s = 12 + 2 \sqrt 2 = \Nn{14.82842712474619}$ Found by Károly Hajba in November 2024, by extending the $s(28)$ and $s(40)$ found by Frits Göbel in early 1979.

207.
$s = {27\over 2} + \sqrt 2 = \Nn{14.91421356237309}$ Extends the $s(88)$ found by Erich Friedman in 1997.		$s = {}^{8}🔒 = \Nn{14.89947689335427}$ $2s^4-(62+20\sqrt{2})s^3+(593+580\sqrt{2})s^2-(2104+4570\sqrt{2})s+6590+5220\sqrt{2}=0$ $4s^8-248s^7+5416s^6-35548s^5-399495s^4+7707496s^3-41637644s^2+67690880s-11068700=0$ Improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement by David W. Cantrell in November 2024.		$s = {}^{12}🔒 = \Nn{14.89564154083791}$ $16s^6-(1144+96\sqrt{2})s^5+(33985+6024\sqrt{2})s^4-(537992+150024\sqrt{2})s^3+(4801816+1850340\sqrt{2})s^2-(23022312+11275344\sqrt{2})s+27067200\sqrt{2}+46636900=0$ $256s^{12}-36608s^{11}+2377824s^{10}-92660208s^9+2409377665s^8-43965076368s^7+576052330352s^6-5445742834320s^5+36732970030440s^4-171582571927744s^3+523309619017472s^2-926610560558400s+709733809930000=0$ Improved by David Ellsworth in November 2024, by adapting and extending the technique from the $s(37)$ found by David W. Cantrell in September 2002, and the $s(130)$ improvement by David W. Cantrell in November 2024.

209.		210.
$s = {}^{4}🔒 = \Nn{14.98444916984191}$ $s^4-32s^3+391s^2-2082s+654=0$ Continues the $s(n^2\!-\!n\!-\!1)$ pattern found by Joe DeVincentis in April 2014, but is not optimal. Explore group		$s = \Nn{14.98318264138415}$ Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019. Shows $s(n^2-n)<n$ for $n=15$.

210.
$s = 15$ No nontrivial packing previously known.		$s = \Nn{14.98318264138415}$ Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019. Shows $s(n^2-n)<n$ for $n=15$.

227.
$s = 🔒 = \Nn{15.59861960924436}$ $6s^4-(424+64\sqrt2)s^3+(10590+2578\sqrt2)s^2-(113020+34510\sqrt2)s+440283+153930\sqrt2=0$ $36s^8-5088s^7+298664s^6-9676592s^5+191145728s^4-2371850384s^3+18129507980s^2-78273072120s+146460230289=0$ Combines two copies of the $s(50)$ found by David W. Cantrell in September 2002.		$s = {}^{8}🔒 = \Nn{15.59153604974246}$ $7889s^4-(527752+66432\sqrt{2})s^3+(12889211+2788168\sqrt{2})s^2-(137062792+38929560\sqrt{2})s+537849584+180940800\sqrt{2}=0$ $7889s^8-1055504s^7+59964726s^6-1904714944s^5+37190498217s^4-458703920560s^3+3498811994208s^2-15117574901504s+28368998453504=0$ Adds an "L" to the $s(198)$ found by David Ellsworth in December 2024, by extending the $s(102)$ found by Károly Hajba in September 2024, and adapting and extending the technique from the $s(37)$ found by David W. Cantrell in September 2002.

228.
$s = {27\over 2} + {3\over 2}\sqrt 2 = \Nn{15.62132034355964}$ Extends the $s(102)$ found by Károly Hajba in September 2024.		$s = {}^{8}🔒 = \Nn{15.61235446152284}$ $8s^4-(272+80\sqrt{2})s^3+(3106+2440\sqrt{2})s^2-(15406+21320\sqrt{2})s+46589+40180\sqrt{2}=0$ $64s^8-4352s^7+110880s^6-1155360s^5+43924s^4+99894312s^3-774485896s^2+1991050132s-1058329879=0$ Improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{15.61205342220945}$ $1568s^4-(152672+46784\sqrt{2})s^3+(4577260+1970944\sqrt{2})s^2-(55519100+27618080\sqrt{2})s+238931729+128802168\sqrt{2}=0$ $50176s^8-9771008s^7+679296768s^6-24549228544s^5+524805011408s^4-6925893926752s^3+55687992448488s^2-251050505939960s+487926003263857=0$ Improved by David Ellsworth in November 2024, by adapting the technique from the $s(37)$ found by David W. Cantrell in September 2002.

234.
$s = 13 + 2 \sqrt 2 = \Nn{15.82842712474619}$ Extends the $s(40)$ found by Frits Göbel in early 1979.		$s = 13 + 2 \sqrt 2 = \Nn{15.82842712474619}$ Alternative combining two copies of the $s(40)$ with the $s(5)$ all found by Frits Göbel in early 1979.

236.
$s = {29\over 2}+\sqrt 2 = \Nn{15.91421356237309}$ Extends the $s(88)$ found by Erich Friedman in 1997.		$s = {}^{8}🔒 = \Nn{15.87893084682989}$ $16s^4-(752+104\sqrt{2})s^3+(13186+3952\sqrt{2})s^2-(103642+48724\sqrt{2})s+315949+192660\sqrt{2}=0$ $256s^8-24064s^7+965824s^6-21504256s^5+288352740s^4-2358058568s^3+11280245560s^2-27942509156s+25588019401=0$ Improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement by David W. Cantrell in November 2024.

240.
$s = 16$ No nontrivial packing previously known.		$s = \Nn{15.98470216018343}$ Found by Károly Hajba in September 2015. Bounded the $s(n^2-n)=n$ conjecture to $n \lt 16$.		$s = \Nn{15.98318264138415}$ Based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.

241.
$s = 16$ No nontrivial packing previously known.		$s = \Nn{15.99509176822518}$ 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.		$s = \Nn{15.99379865105946}$ Improved by David Ellsworth in November 2024, by adapting a technique from the $s(240)$ found by Károly Hajba in September 2015.

258.
$s = {29\over 2} + {3\over 2}\sqrt 2 = \Nn{16.62132034355964}$ Extends the $s(102)$ found by Károly Hajba in September 2024.		$s = {}^{8}🔒 = \Nn{16.59861960924436}$ $6s^4-(448+64\sqrt{2})s^3+(11898+2770\sqrt{2})s^2-(135496+39858\sqrt{2})s+564323+191082\sqrt{2}=0$ $36s^8-5376s^7+335288s^6-11577440s^5+244189248s^4-3239352592s^3+26493287236s^2-122462432992s+245435786881=0$ Combines two copies of the $s(50)$ found by David W. Cantrell in September 2002.		$s = {}^{8}🔒 = \Nn{16.58884686366654}$ $1864s^4-(130112+14184\sqrt{2})s^3+(3340332+638892\sqrt{2})s^2-(37526084+9576432\sqrt{2})s+156111517+47789280\sqrt{2}=0$ $14912s^8-2081792s^7+124375616s^6-4175466176s^5+86460492992s^4-1133641742816s^3+9208543852232s^2-42428775277352s+84992168129633=0$ Improved by David Ellsworth in November 2024, by adapting and extending the technique from the $s(37)$ found by David W. Cantrell in September 2002.

259.
$s = {29\over 2} + {3\over 2}\sqrt 2 = \Nn{16.62132034355964}$ Adds an "L" to $s(228)$, which extends the $s(102)$ found by Károly Hajba in September 2024.		$s = {29\over 2} + {3\over 2}\sqrt 2 = \Nn{16.62132034355964}$ Alternative repacking the "L"-augmented form into a primitive packing.		$s = {}^{8}🔒 = \Nn{16.61235446152284}$ $8s^4-(304+80\sqrt{2})s^3+(3970+2680\sqrt{2})s^2-(22466+26440\sqrt{2})s+65381+64020\sqrt{2}=0$ $64s^8-4864s^7+143136s^6-1915616s^5+7640724s^4+85791512s^3-1060595304s^2+3833056108s-3922445639=0$ Improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{16.61205342220945}$ $1568s^4-(158944+46784\sqrt{2})s^3+(5044684+2111296\sqrt{2})s^2-(65137908+31700320\sqrt{2})s+299182329+158437976\sqrt{2}=0$ $50176s^8-10172416s^7+749098752s^6-28833010176s^5+658086103248s^4-9284536988320s^3+79870392630248s^2-385430283384616s+802140438936961=0$ Improved by David Ellsworth in November 2024, by adapting the technique from the $s(37)$ found by David W. Cantrell in September 2002.		$s = {}^{8}🔒 = \Nn{16.60651495297546}$ $17714s^4-(1505768+323208\sqrt{2})s^3+(43929590+14599998\sqrt{2})s^2-(540077424+219437586\sqrt{2})s+2402899929+1097885178\sqrt{2}=0$ $35428s^8-6023072s^7+408123288s^6-14966022368s^5+330968229088s^4-4567307767008s^3+38656232063604s^2-184242397528080s+379725001766289=0$ Improved by David Ellsworth in December 2024, by using the primitive packing as a basis, and again adapting the technique from the $s(37)$ found by David W. Cantrell in September 2002.

267.
$s = {}^{12}🔒 = \Nn{16.90177651254408}$ $4s^6-(488+56\sqrt{2})s^5-(-24092-4788\sqrt{2})s^4-(619528+162580\sqrt{2})s^3-(-8786765-2742174\sqrt{2})s^2-(65359184+22983898\sqrt{2})s+199624218+76607060\sqrt{2}=0$ $16s^{12}-3904s^{11}+424608s^{10}-27397504s^9+1173110104s^8-35222114128s^7+762052962472s^6-11990755860848s^5+136356518122609s^4-1094014672541520s^3+5883126429752828s^2-19051636577756704s+28112545128424324=0$ Combines the $s(40)$ found by Frits Göbel in early 1979, and the $s(88)$ found by David W. Cantrell in August 2002.		$\begin{aligned}s &= 16-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{16.84666719284348}\end{aligned}$ Extends the $s(107)$ found by Károly Hajba in November 2024.		$\begin{aligned}s &= 16-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{16.84666719284348}\end{aligned}$ Alternative packing extending the $s(54)$ found by Joe DeVincentis in April 2014.

269.
$s = {31\over 2}+\sqrt 2 = \Nn{16.91421356237309}$ Extends the $s(88)$ found by Erich Friedman in 1997.		$s = {}^{8}🔒 = \Nn{16.90898430955091}$ $4s^4-(140+52\sqrt{2})s^3+(1620+1716\sqrt{2})s^2-(9638+15314\sqrt{2})s+50961+18772\sqrt{2}=0$ $16s^8-1120s^7+27152s^6-173776s^5-3343896s^4+63523672s^3-339883516s^2+167573396s+1892247553=0$ Improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{16.90810332579893}$ $11534s^4-(776040+34380\sqrt{2})s^3+(19549332+1653822\sqrt{2})s^2-(218566796+26528724\sqrt{2})s+915240215+141924780\sqrt{2}=0$ $23068s^8-3104160s^7+182215728s^6-6096166544s^5+127172500628s^4-1694326878096s^3+14081791966552s^2-66762945978200s+138265886040775=0$ Improved by David Ellsworth in November 2024, by adapting and extending the technique from the $s(37)$ found by David W. Cantrell in September 2002.

271.						272.
$s = {}^{4}🔒 = \Nn{16.98770404979942}$ $s^4-36s^3+495s^2-2988s+1116=0$ Continues the $s(n^2\!-\!n\!-\!1)$ pattern found by Joe DeVincentis in April 2014, but is not optimal. Explore group		$s = \Nn{16.98470216018343}$ Adds an "L" to the $s(240)$ found by Károly Hajba in September 2015.		$s = \Nn{16.98318264138415}$ Adds an "L" to the $s(240)$ based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.		$s = \Nn{16.98318264138415}$ Based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.

272.
$s = 17$ No nontrivial packing previously known.		$s < 17$ Found by Lars Cleemann between 1991 and 1998. Bounded the $s(n^2-n)=n$ conjecture to $n \lt 17$.		$s = {}^{5}🔒 = \Nn{16.99151646824600}$ $s^5-49s^4+872s^3-6894s^2+24437s-34521=0$ Local optimum squeeze found by David Ellsworth in May 2023.		$s = {}^{4}🔒 = \Nn{16.98786575951164}$ $10s^4-470s^3+7840s^2-54504s+134721=0$ Improved by Károly Hajba in September 2015. (see also max staggering angle)		$s = \Nn{16.98318264138415}$ Based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.

273.
$s = 17$ No nontrivial packing previously known.		$s = \Nn{16.99509176822518}$ Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019. Shows $s(n^2-n+1) \lt n$ for $n=17$.		$s = \Nn{16.99379865105946}$ Improved by David Ellsworth in November 2024, by adapting a technique from the $s(240)$ found by Károly Hajba in September 2015.

307.
$s = 18$ No nontrivial packing previously known.		$s = \Nn{17.99509176822518}$ Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019. Shows $s(n^2-n+1) \lt n$ for $n=18$.		$s = \Nn{17.99379865105946}$ Improved by David Ellsworth in November 2024, by adapting a technique from the $s(240)$ found by Károly Hajba in September 2015.

1765.
$s = {}^{4}🔒 = \Nn{42.49171522113325}$ $229s^4-13040s^3-12743s^2+4396114s+90109052=0$ Found by Károly Hajba in November 2024. Bounds $\{s(n^2\!+\!1)\} \ge {1\over 2}$ to $n \lt 42$. Beats the $s(1765)$ Göbel square.		$s = {}^{4}🔒 = \Nn{42.49171522113325}$ $229s^4-13040s^3-12743s^2+4396114s+90109052=0$ Rearranged for comparison with subsequent improvement.		$s = {}^{4}🔒 = \Nn{42.48797851186022}$ $2s^4-212s^3+8129s^2-148140s+1362276=0$ Improved by David Ellsworth in November 2024.