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. Explore group		$s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$ 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 found by Charles F. Cottingham in 1979, 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. Explore group		$s = 4 + {1\over 2}\sqrt 2 = \Nn{4.70710678118654}$ Alternative with minimal rotated squares. 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. Explore group		$s = 2 + 2 \sqrt 2 = \Nn{4.82842712474619}$ Rigid alternative found by David Ellsworth in December 2024.		$s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$ 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 F. 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 $s(5)$.		$s = {5\over 2} + {9\over 4}\sqrt 2 = \Nn{5.68198051533946}$ Found by Evert Stenlund in early 1980.		$s = {}^{3}🔒 = \Nn{5.65062919143938}$ $s^3-14s^2+67s-112=0$ 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. Unextends the $s(37)$ found by Evert Stenlund in early 1980.

27.
$s = 5 + {1\over 2}\sqrt 2 = \Nn{5.70710678118654}$ Found by Frits Göbel in early 1979. Explore group		$s = 5 + {1\over 2}\sqrt 2 = \Nn{5.70710678118654}$ 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 $s(5)$.		$s = {9\over 2} + {3\over 2}\sqrt 2 = \Nn{6.62132034355964}$ Found by Evert Stenlund in early 1980.		$s = {9\over 2} + {3\over 2}\sqrt 2 = \Nn{6.62132034355964}$ Alternative constructed by adding an "L" to the $s(26)$ found by Erich Friedman in 1997.		$s = {}^{5}🔒 = \Nn{6.62123046462703}$ $9s^5-123s^4+404s^3+146s^2-303s+205=0$ 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.		$s = {}^{8}🔒 = \Nn{6.59861960924436}$ $6s^4-(208+64\sqrt{2})s^3+(2058+850\sqrt{2})s^2-(7936+3658\sqrt{2})s+11163+5502\sqrt{2}=0$ $36s^8-2496s^7+59768s^6-733760s^5+5289248s^4-23462672s^3+63458276s^2-96673872s+64068561=0$ Improved 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. Explore group		$s = 6 + {1\over 2}\sqrt 2 = \Nn{6.70710678118654}$ 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. Extends the alternative packing of the $s(18)$ found by Pertti Hämäläinen in 1980 found by Mats Gustafsson in 1981.		$s = {}^{8}🔒 = \Nn{6.81880916998841}$ $5184s^8-197568s^7+3200144s^6-28651016s^5+154197141s^4-506760114s^3+981374688s^2-1005617394s+408278853=0$ Found by David W. Cantrell in August 2002.

41.
$s = {11\over 3} + {7\over 3}\sqrt 2 = \Nn{6.96649831220388}$ Found by Evert Stenlund in early 1980. Didn't set a record.		$s = 2 + {7\over 2}\sqrt 2 = \Nn{6.94974746830583}$ Found by Charles F. 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 Evert Stenlund in early 1980, by adding an "L" to the $s(37)$ he found.		$s = {11\over 2} + {3\over 2}\sqrt 2 = \Nn{7.62132034355964}$ Alternative found by Erich Friedman in 1997, by adding two "L"s to the $s(26)$ he found.		$s = {}^{5}🔒 = \Nn{7.62123046462703}$ $9s^5-168s^4+986s^3-1894s^2+1154s+118=0$ Found by Erich Friedman in 1997, by adding an "L" to the $s(37)$ he found.		$s = {}^{8}🔒 = \Nn{7.59861960924436}$ $6s^4-(232+64\sqrt{2})s^3+(2718+1042\sqrt{2})s^2-(12700+5550\sqrt{2})s+21371+10074\sqrt{2}=0$ $36s^8-2784s^7+78248s^6-1146800s^5+9944448s^4-53242000s^3+173869324s^2-319180600s+253748689=0$ Improved by David W. Cantrell in September 2002, along with the base $s(37)$ to which this adds an "L".

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. Explore group		$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$ 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 early 1980, by adding an "L" to the $s(40)$ found by Frits Göbel in early 1979.		$s = 5 + 2 \sqrt 2 = \Nn{7.82842712474619}$ Alternative converting 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.		$s = {}^{32}🔒 = \Nn{7.82290804413284}$ $2401s^{32}-470596s^{31}+43819034s^{30}-2575626200s^{29}+107064827921s^{28}-3339730095676s^{27}+80881055947292s^{26}-1549634326032740s^{25}+23649840659276976s^{24}-285549917423038976s^{23}+2644342639858467944s^{22}-16975793431501140604s^{21}+42160111960108077186s^{20}+604801559847940668640s^{19}-10323577086818326059762s^{18}+91423093604241671442960s^{17}-549498547429972204556388s^{16}+2140033946880564583409424s^{15}-2116245094452089256368748s^{14}-47398133328019683350564492s^{13}+513623762890535143202861016s^{12}-3539168668169055445821007956s^{11}+19554643118966991967193841680s^{10}-91052982556166904610991305520s^9+358000595889063667996962518449s^8-1171244223040008871491078909092s^7+3123815163273613989845145831648s^6-6628626979480975566515883381512s^5+10851380919900152459339530602076s^4-13122619975009229974564386454832s^3+10946326483876900501448971940288s^2-5556443722823912253342620368384s+1267690936215846749823290569552=0$ Improved by David W. Cantrell in December 2024.

54.
$s = {13\over 2} + \sqrt 2 = \Nn{7.91421356237309}$ Found by Evert Stenlund in early 1980.	$s = 6 + {4\over 3}\sqrt 2 = \Nn{7.88561808316412}$ 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.

$s = {13\over 2} + \sqrt 2 = \Nn{7.91421356237309}$ Alternative of the above with rotational symmetry, found by David W. Cantrell in November 2024.	$s = 6 + {4\over 3}\sqrt 2 = \Nn{7.88561808316412}$ Alternative of the above with rotational symmetry.	$s = {}^{24}🔒 = \Nn{7.85109314217532}$ $18432s^{24}-3446784s^{23}+304664576s^{22}-16968519680s^{21}+669449619968s^{20}-19932214017536s^{19}+465775491192832s^{18}-8768302695054848s^{17}+135403072780311424s^{16}-1737169753235173760s^{15}+18680996284830988288s^{14}-169371496869909229568s^{13}+1299023794031261063520s^{12}-8437530238970805727264s^{11}+46364613247151797971136s^{10}-214838591130149411496032s^9+834615127459846686846440s^8-2694590714870358438730024s^7+7138799170846114834605888s^6-15241751239015139566326992s^5+25549157414710670461692094s^4-32328821957836025603803370s^3+28978412289382495954074792s^2-16361602577473933571257634s+4362099385839709619937827=0$ Alternative of the above with rotational symmetry.	$s = {}^{26}🔒 = \Nn{7.84878975975240}$ $4096s^{26}-696320s^{25}+56739840s^{24}-2949122048s^{23}+109766839552s^{22}-3113698237440s^{21}+69948465260800s^{20}-1276594350771968s^{19}+19265599452587584s^{18}-243443276207946880s^{17}+2598512723829467904s^{16}-23570528241163997888s^{15}+182367913759609388096s^{14}-1205666564490332194624s^{13}+6810634124400001375824s^{12}-32812890065132800032688s^{11}+134342994605566431921940s^{10}-464769276437493261704728s^9+1347820021349550459367488s^8-3241044285502742122448756s^7+6369772559544867940758561s^6-10037363738549635149843920s^5+12358928187941177640594208s^4-11472837613066933113730048s^3+7610215413527273079398656s^2-3277082686415784635207680s+712999405005519711846400=0$ 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.
$s = 3 + 4 \sqrt 2 = \Nn{8.65685424949238}$ Found by Evert Stenlund in early 1980.		$s = 3 + 4 \sqrt 2 = \Nn{8.65685424949238}$ Alternative with rotational symmetry found by David W. Cantrell in 2023.

67.
$s = 8 + {5\over 7} = \Nn{8.71428571428571}$ Found by David Ellsworth in December 2024. Didn't set a record, but along with $s(104)$ is the first competitively-intended packing found with a rational side length. Its tilt angle is determined by the Pythagorean triple $\{3, 4, 5\}$.		$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$ Found by Evert Stenlund in early 1980, extending the $s(52)$ found by Frits Göbel in early 1979. Explore group		$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$ Alternative constructed by 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 early 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 converting 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, by combining two copies of his $s(18)$ alternative packing.

$s = {15\over 2} + {1\over 2}\sqrt 7 = \Nn{8.82287565553229}$
Alternative found by David Ellsworth in January 2025, based on the $s(86)$ found by Erich Friedman in 1997 extending the $s(18)$ found by Mats Gustafsson in 1981.

$s = {15\over 2} + {1\over 2}\sqrt 7 = \Nn{8.82287565553229}$
Alternative with diagonally symmetric background of unrotated squares found by David Ellsworth in January 2025, based on the $s(86)$ found by Erich Friedman in 1997 extending the $s(18)$ found by Mats Gustafsson in 1981.

$s = {15\over 2} + {1\over 2}\sqrt 7 = \Nn{8.82287565553229}$
Alternative with minimal rotated squares found by David Ellsworth in December 2024, by using the $s(18)$ found by Mats Gustafsson in 1981.

$s = {13\over 3} + 2 \sqrt 5 = \Nn{8.80546928833291}$
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 found 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.
$s = 4 +{7\over 2}\sqrt 2 = \Nn{8.94974746830583}$ Found by Evert Stenlund in early 1980.		$s = {15\over 2}+ \sqrt 2 = \Nn{8.91421356237309}$ Found by Erich Friedman in 1997, by extending and adapting the $s(41)$ found by Charles F. Cottingham in 1979. Extends a version of the $s(11)$ whose side length was found by Frits Göbel in early 1979.		$s = {}^{7}🔒 = \Nn{8.91209113581151}$ $16s^7-960s^6+24610s^5-346304s^4+2850209s^3-13406809s^2+31814883s-25624307=0$ 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 found 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 Evert Stenlund in early 1980, by adding an "L" to the $s(66)$ he found.		$s = 4 + 4 \sqrt 2 = \Nn{9.65685424949238}$ 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. With these improvements, this now 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		$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$ Rearrangement. Can also be constructed by adding an "L" to the $s(67)$ that extends the $s(52)$ found by Frits Göbel in early 1979.		$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$ Alternative constructed by 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.
$s = 2 + {11\over 2}\sqrt 2 = \Nn{9.77817459305202}$ Found by Evert Stenlund in early 1980. Extends the $s(18)$ found by Frits Göbel in early 1979. Explore group		$s = {11\over 2} + 3 \sqrt 2 = \Nn{9.74264068711928}$ Found by Erich Friedman in 1997.		$s = {11\over 2} + 3 \sqrt 2 = \Nn{9.74264068711928}$ Alternative with minimal rotated squares.

86.
$s = 7 + 2 \sqrt 2 = \Nn{9.82842712474619}$ Found by Evert Stenlund in early 1980.		$s = {17\over 2} + {1\over 2}\sqrt 7 = \Nn{9.82287565553229}$ Found by Erich Friedman in 1997. Extends the alternative packing of the $s(18)$ found by Pertti Hämäläinen in 1980 found by Mats Gustafsson in 1981.		$s = {17\over 2} + {1\over 2}\sqrt 7 = \Nn{9.82287565553229}$ Alternative packing with a diagonally symmetric background of unrotated squares.

87.
$s = { 7\over 2}+{ 9\over 2}\sqrt 2 = \Nn{9.86396103067892}$ Provided for comparison.		$s = {14\over 3}+{11\over 3}\sqrt 2 = \Nn{9.85211639536801}$ Found by Evert Stenlund in early 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.		$s = {}^{4}🔒 = \Nn{9.84666719284348}$ $4s^4-144s^3+1932s^2-11432s+25121=0$ Found by David Ellsworth in December 2024, based on the $s(107)$ found by Károly Hajba in November 2024 and the $s(54)$ found by Joe DeVincentis in April 2014.		$s = {}^{23}🔒 = \Nn{9.83892657002494}$ $s^{23}-138s^{22}+8984s^{21}-366792s^{20}+10538108s^{19}-226899940s^{18}+3814912554s^{17}-51682985704s^{16}+579852353410s^{15}-5521708432172s^{14}+45434118479338s^{13}-324120092543232s^{12}+1970502666465045s^{11}-9840546468521178s^{10}+38505534857507358s^9-116163600689532188s^8+342445835317445719s^7-1749940540506500506s^6+11136111412137553730s^5-53096842537804640028s^4+169960636556199528165s^3-350586565237438644834s^2+425167991192928955284s-231741242909814395880=0$ Found by David W. Cantrell in January 2025.

88.
$s = {17\over 2}+\sqrt 2 = \Nn{9.91421356237309}$ Found by Erich Friedman in 1997, by extending the $s(41)$ found by Charles F. Cottingham in 1979.		$s = {}^{12}🔒 = \Nn{9.90177651254408}$ $4s^6-(320+56\sqrt{2})s^5+(9952+2828\sqrt{2})s^4-(55956\sqrt{2}+156632)s^3+(543586\sqrt{2}+1329945)s^2-(5815938+2595866\sqrt{2})s+10276983+4876156\sqrt{2}=0$ $16s^{12}-2560s^{11}+175744s^{10}-6988864s^9+181397032s^8-3260558096s^7+41816558200s^6-386837301552s^5+2568083107241s^4-11953457726884s^3+37081298010138s^2-68909201625724s+58062584909617=0$ 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 found 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 $s(37)$ improvement found by David W. Cantrell in September 2002.		$s = {}^{20}🔒 = \Nn{9.888153053758572}$ $3528s^{10}-(300552+15456\sqrt{2})s^9+(11614660+1180832\sqrt{2})s^8-(268405824+40209136\sqrt{2})s^7+(4111948776+801750848\sqrt{2})s^6-(43682208312+10328732976\sqrt{2})s^5+(326223055436+89277369408\sqrt{2})s^4-(1692962073984+518553084040\sqrt{2})s^3+(5849274524474+1954912407552\sqrt{2})s^2-(12163170266098+4347871933856\sqrt{2})s+11572065260145+4353477802040\sqrt{2}=0$ $254016s^{20}-43279488s^{19}+3506260608s^{18}-179642577984s^{17}+6530192527760s^{16}-179088304328704s^{15}+3846118270819200s^{14}-66261902137415296s^{13}+930479746642904384s^{12}-10759858027891736896s^{11}+103070340120029179008s^{10}-819709665351861223904s^9+5405590814889373243192s^8-29412949608198679086720s^7+130831566348158107359392s^6-468664620024162429231904s^5+1321046745485882223459068s^4-2825402176181244872057384s^3+4315682289270565775115128s^2-4199847844458434080013540s+1959329723251932809573425=0$ Improved by David Ellsworth in January 2025. This new technique also independently found by David W. Cantrell around the same time.

89.
$s = 5 + {7\over 2}\sqrt 2 = \Nn{9.94974746830583}$ Found by Evert Stenlund in early 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 $s(37)$ found by Evert Stenlund in early 1980. Unlike the $s(66)$ found by Evert Stenlund, is clamped on its narrow side while having play along its long side.		$s = {17\over 2} + {3\over 2}\sqrt 2 = \Nn{10.62132034355964}$ Alternative with rotational symmetry found by David Ellsworth in December 2024 by combining two copies of the $s(26)$ found by Erich Friedman in 1997.		$s = {17\over 2} + {3\over 2}\sqrt 2 = \Nn{10.62132034355964}$ Alternative with mirror symmetry found by David Ellsworth in December 2024 by combining two copies of the $s(26)$ found by Erich Friedman in 1997.		$s = {}^{8}🔒 = \Nn{10.61393128238062}$ $8s^4-(192+48\sqrt{2})s^3+(1614+984\sqrt{2})s^2-(5918+6048\sqrt{2})s+10249+9876\sqrt{2}=0$ $64s^8-3072s^7+58080s^6-525536s^5+1943764s^4+2662200s^3-43922048s^2+117613028s-90028751=0$ 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 $s(37)$ improvement found by David W. Cantrell in September 2002.		$s = {}^{8}🔒 = \Nn{10.61138823373863}$ $24s^4-(1400+352\sqrt{2})s^3+(27061+10102\sqrt{2})s^2-(218629+97462\sqrt{2})s+641430+317240\sqrt{2}=0$ $576s^8-67200s^7+3011120s^6-72041376s^5+1033920257s^4-9243724322s^3+50697397293s^2-156795019420s+210150009700=0$ Improved by David Ellsworth in December 2024.

103.
$s = {13\over 2} + {1\over 2}\sqrt{71} = \Nn{10.71307488658817}$ The smallest known size at which the attempt on the right doesn't have any overlapping squares. Doesn't set a record.		$s = {}^{6}🔒 = \Nn{10.70090498232613}$ $9s^6-342s^5+5209s^4-40336s^3+164252s^2-318864s+201276=0$ Found by David Ellsworth in December 2024 - January 2025. NOT A VALID PACKING – there are two pairs of slightly intersecting squares. This is meant to demonstrate the closest attempt so far at finding an $s(103)$ smaller than the best known $s(104)$. It might be possible to modify this to make a valid packing that is still smaller than the $s(104)$ that extends the $s(52)$ found by Frits Göbel in early 1979.

104.
$s = 10 + {5\over 7} = \Nn{10.71428571428571}$ Found by David Ellsworth in December 2024. Didn't set a record, but is the first competitively-intended packing found with a rational side length. Its tilt angle is determined by the Pythagorean triple $\{3, 4, 5\}$.		$s = 10 + {1\over 2}\sqrt 2 = \Nn{10.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group		$s = 10 + {1\over 2}\sqrt 2 = \Nn{10.70710678118654}$ simplified

105.
$s = {19\over 2} + {1\over 2}\sqrt 7 = \Nn{10.82287565553229}$ Adds an "L" to the $s(86)$ found by Erich Friedman in 1997.		$s = {19\over 2} + {1\over 2}\sqrt 7 = \Nn{10.82287565553229}$ Alternative converting the "L"-augmented form into a primitive packing with a diagonally symmetric background of unrotated squares.

106.
$s = 8 + 2 \sqrt 2 = \Nn{10.82842712474619}$ Combines the $s(40)$ found by Frits Göbel in early 1979, and the $s(18)$ alternative packing found by Mats Gustafsson in 1981.		$s = 8 + 2 \sqrt 2 = \Nn{10.82842712474619}$ Alternative found by Károly Hajba in November 2024. Extends the s(28) found by Frits Göbel in early 1979.		$s = {}^{4}🔒 = \Nn{10.82308816660919}$ $s^4-38s^3+355s^2+714s-14857=0$ Found by David Ellsworth in November 2024, based on the $s(53)$ found by David W. Cantrell in September 2002.		$s = {}^{32}🔒 = \Nn{10.82297973416944}$ $2401s^{32}-701092s^{31}+96532058s^{30}-8317521660s^{29}+501462496833s^{28}-22376320364004s^{27}+760407348527794s^{26}-19848555869936524s^{25}+391901782318184024s^{24}-5471444559723346548s^{23}+39629249963743971218s^{22}+353696512578770314160s^{21}-16715802829777049603255s^{20}+292780863461637719269068s^{19}-3146950297418725382386108s^{18}+17996731060753457271434416s^{17}+58068494391477930003466013s^{16}-2710032149344351718373370304s^{15}+35897708426171881544444261010s^{14}-321777454480517334707593455472s^{13}+2318288875965343221612347387046s^{12}-15552954072951813301922897418028s^{11}+110324583017828739076547861980256s^{10}-814314195444111054964406531808040s^9+5535691601850017528577776913992458s^8-31551366481166818299554205010301876s^7+144126073330457054877480503221027356s^6-514883701839008702934553147517634876s^5+1404879821405123399570947930851054697s^4-2828245713344639081326540531975430356s^3+3961424449372054137804580730689208222s^2-3448216591537867586914544066365388952s+1404226509074988020217588819457924033=0$ Improved by David W. Cantrell in December 2024.

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 $s(88)$ improved 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.
$s = \Nn{10.93343730798119}$ Found by David Ellsworth in December 2024, by combining two modified copies of the $s(29)$ found by Thierry Gensane and Philippe Ryckelynck in April 2004 that fits an $s(n^2\!-\!n\!-\!1)$ pattern. Didn't set a record.		$s = {}^{48}🔒 = \Nn{10.92610452776589}$ $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.		$s = {}^{144}🔒 = \Nn{10.92591939016138}$ $827055562171106304s^{144}-1028746038533781676032s^{143}+614161678485829473038592s^{142}-236434676228380170030685824s^{141}+66980317656649750618706949945s^{140}-15177024487481458274595256127868s^{139}+2918547306910189530506516307375692s^{138}-495729497699611401035280356277680196s^{137}+76089742722234909946196429571333895452s^{136}-10671270893794158877895712177505617523828s^{135}+1375524471618065793880112212598599154653012s^{134}-163680131457973276468171788500327444221444244s^{133}+18041122005484577974071987969780608673747943250s^{132}-1845193374227011301793607038733620442135422735440s^{131}+175195858247481457503205356072668634514707875635464s^{130}-15447131106769150898435521166852265757689347117679048s^{129}+1266589004665859543076277348204652895182867619945255898s^{128}-96872971664861242811412554825190666410176955662558075472s^{127}+6939847565941400486198515625279425362463969118244043805152s^{126}-467586251641955795650527642523133741785627313804139361021412s^{125}+29712916686346155911816329503684280580564307321782343390920945s^{124}-1781869546071794107416303074386560117711940471499978105624204512s^{123}+100692141845407597467752420973765886037444159069768871392617305440s^{122}-5345881769519666404549787030353246578976419001556791199058788111868s^{121}+265745153241256297998761893367366891336827254195720886433343716673314s^{120}-12331900216784265711374361173174853992773142985602439082869045995043620s^{119}+533124642048160878797787516816371559269658695957365696289239653586152536s^{118}-21454346119467121162152752804366671634925654995122131901651080854842653908s^{117}+804137509033612105675830501719617753487476429965727315635094765154608429509s^{116}-28127107800477454224608783588306926887609791011395988349691834833987158467700s^{115}+921246416930684576450726414356413649291858434685374246183966749784918747685898s^{114}-28390357852514647575635304153251006733509079472468682793162434720668520497597508s^{113}+828166533055942389980323072853474773741844516945352000111380951692338661814562750s^{112}-23019859946745245564619050854509564382765776816954463496160143457200568875655263500s^{111}+613584341652374269884917793377224034449208943707706722281248832851205212434542935116s^{110}-15759013359756201601425954086600815916125169227043330927990424168462555090123951648000s^{109}+390943459868126235255883761901077907242607987729337591414340253957672403607173215398650s^{108}-9366575279423493930590320245377238666248026521863698157365356790349162864711782205289356s^{107}+216352869031916941103974168883097076410229240427531516289660856895884573411040138391989008s^{106}-4808372152188842416685049720889157609648589700259906966073914535288536332426660045281665576s^{105}+102730532296869024506560636386673680062422941215075875113492165335062102679114067711595589286s^{104}-2111527924549381695924838025670328572466346746690319171877130768902470515781848199414410720936s^{103}+41831905131678357252471218884771916619975870088217821433418140699364041647672903374550946065106s^{102}-799937696943205886362309315869391648647226626812787165601913537214769018438999347145254423256568s^{101}+14750154008419083794676619125813636103060922241286812273744920967608624399426277671233689970750816s^{100}-260975431069180213161379096976636911786047810516915330900553443161331183641061936532309426166815268s^{99}+4392344885718180821788009252695595386788391929454071852322904840485213772415318679286843092164107326s^{98}-69572449764366603777509810702402733716637668631656846420562636123468612787624389077681592721918843488s^{97}+1027097836001128208475480477047724529415634205788396193786255672967383685469857620307489972997358673526s^{96}-14074624684738338344020983555256300931561811169605846968675883195494170910423934868009815189998913131444s^{95}+180660626905403779319995260410889237792268816586814685932269189175254703565979443977412180365929218465280s^{94}-2245499788361448111606620260028829428953487326596025966685205057335044673291677551140402757171317681417428s^{93}+28671416193365649478884844283419050835810755858475230587266033381784144945601933009018387101662354025540734s^{92}-394217288266871322892333222337975011689661831536401018523536110370589489067761042286000949956400639666042316s^{91}+5703863195259166781025704560694159480317740632678599117279192239656528808349240434327194661001025914590969266s^{90}-80059468446622758198796944179244414275773216304886340967166998450686606958848716898739757721312778409184831024s^{89}+1004138763079804275697043916259570490495614067448884056635127265173895325706519305292407648765439933024547087131s^{88}-10573856961381771762913436897417012389280833263671551808955616987901353562852378162633710797010666619591470102324s^{87}+89592018192513343122608501728306468444080160150570013764942281507872879104525037337900571968575192851768883617072s^{86}-646389054343915846566881857708086454471929561189842086401340461316692430061948367583499858586482550023315255624968s^{85}+6456718039810642099097859415092985378580884294959550540027954767107746733301451637312622694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Improved by David Ellsworth in November 2024, using a similar technique to that of the $s(87)$ improvement found by David W. Cantrell in August 2002. With this, now extends the $s(11)$ found by Walter Trump in 1979.

109.
$s = {}^{4}🔒 = \Nn{10.97240394480333}$ $s^4-24s^3+219s^2-846s+126=0$ 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 converting the "L"-augmented form into a primitive packing.

$s = {19\over 2} + {3\over 2}\sqrt 2 = \Nn{11.62132034355964}$
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.

$s = {}^{8}🔒 = \Nn{11.61393128238062}$ $8s^4-(224+48\sqrt{2})s^3+(2238+1128\sqrt{2})s^2-(9754+8160\sqrt{2})s+17981+16956\sqrt{2}=0$ $64s^8-3584s^7+81376s^6-942112s^5+5554644s^4-11640920s^3-34053200s^2+202670492s-251695511=0$
Base $s(102$) improved by David W. Cantrell in November 2024.

$s = {}^{8}🔒 = \Nn{11.61350257784129}$ $392s^4-(35000+13888\sqrt{2})s^3+(809488+412816\sqrt{2})s^2-(7232424+4071824\sqrt{2})s+22508889+13362320\sqrt{2}=0$ $3136s^8-560000s^7+30079296s^6-804114816s^5+12492967440s^4-118750675952s^3+684185482592s^2-2203093815248s+3051977396929=0$
Base $s(102$) improved by David Ellsworth in November 2024, by adapting the $s(37)$ improvement found by
David W. Cantrell in September 2002.

$s = {}^{12}🔒 = \Nn{11.60304782603491}$ $16s^6-(808+96\sqrt{2})s^5+(17055+4248\sqrt{2})s^4-(193408+74592\sqrt{2})s^3+(1249979+648684\sqrt{2})s^2-(4396854+2788260\sqrt{2})s+6628736+4726152\sqrt{2}=0$ $256s^{12}-25856s^{11}+1180192s^{10}-32118704s^9+578685345s^8-7241250352s^7+64139810474s^6-401461100180s^5+1735510964089s^4-4911120357412s^3+8091982299332s^2-5580206491008s-732884496512=0$
Improved by David Ellsworth in December 2024, by using the primitive packing as a basis, 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.

$s = {}^{8}🔒 = \Nn{11.60139984236648}$ $2449s^4-(112436+7064\sqrt{2})s^3+(1923195+221470\sqrt{2})s^2-(14552224+2321166\sqrt{2})s+41183566+8147560\sqrt{2}=0$ $2449s^8-224872s^7+8967686s^6-203140728s^5+2862024925s^4-25703580128s^3+143806409196s^2-458545364672s+638350608644=0$
Improved by David Ellsworth in December 2024.

126.
$s = {21\over 2} + {1\over 2}\sqrt 7 = \Nn{11.82287565553229}$ Adds two "L"s to the $s(86)$ found by Erich Friedman in 1997.		$s = {}^{8}🔒 = \Nn{11.78054947822052}$ $137288s^8-10517728s^7+348568536s^6-6519103192s^5+75150255265s^4-545908033276s^3+2435997730348s^2-6093496701376s+6530973936384=0$ Found by David Ellsworth in December 2024, based on the $s(39)$ found by David W. Cantrell in August 2002.		$s = {}^{9}🔒 = \Nn{11.77652079061690}$ $185761s^9-17452914s^8+733790386s^7-18120925928s^6+289457197920s^5-3096648733600s^4+22129577852576s^3-101471852915328s^2+269345322921472s-312506709170176=0$ Improved by David W. Cantrell in December 2024.

127.
$s = {21\over 2} + {1\over 2}\sqrt 7 = \Nn{11.82287565553229}$ Extends the $s(86)$ found by Erich Friedman in 1997.		$s = {21\over 2} + {1\over 2}\sqrt 7 = \Nn{11.82287565553229}$ Alternative packing with a diagonally symmetric background of unrotated squares, with monominoes in the corners.		$s = {21\over 2} + {1\over 2}\sqrt 7 = \Nn{11.82287565553229}$ Alternative packing with a diagonally symmetric background of unrotated squares, with dominoes in the corners.

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.		$s = {}^{3}🔒 = \Nn{11.82547927546988}$ $s^3-16s^2-56s+1246=0$ Improved by David Ellsworth in December 2024.		$s = {}^{32}🔒 = \Nn{11.82538265107608}$ $2401s^{32}-773808s^{31}+116857944s^{30}-10953322436s^{29}+710251339062s^{28}-33509506615256s^{27}+1170211058292596s^{26}-29691425943630216s^{25}+493617200842541351s^{24}-2563093770360430044s^{23}-136153511549406319200s^{22}+5167455238584576722496s^{21}-101317540269324217776128s^{20}+1192703999335242781294276s^{19}-4429951898711733045071610s^{18}-149483973523951297639523200s^{17}+4036268388799162641527101115s^{16}-59541002790231934648142695536s^{15}+630345240522567629425302746066s^{14}-5203751754819678868788726388588s^{13}+36362143340956575547968782237386s^{12}-250162817129143046903025930554608s^{11}+1972546725385414636236445891891470s^{10}-16850524678415933312186738253014836s^9+131393951082365647592910137413049289s^8-845100771668481743674765744921931432s^7+4314039623904025659790955917562980070s^6-17148739311990329953903399127228076344s^5+51998606140673855664408848072800520841s^4-116371241835830474806716427997040997408s^3+181426723023132852968436272803899334368s^2-176109616718430938071768620303187265280s+80171638196482285349976558744374366208=0$ Improved by David Ellsworth in December 2024, after attempting the technique from the $s(69)$ improved by David W. Cantrell in August 2023 and using the technique from the $s(53)$ improved by David W. Cantrell in December 2024.

129.
$s = {21\over 2}+ \sqrt 2 = \Nn{11.91421356237309}$ Provided for comparison. Similar to the $s(70)$ found by Erich Friedman in 1997.		$s = {}^{4}🔒 = \Nn{11.88674602860566}$ $s^4-42s^3+659s^2-4574s+11833=0$ Found by David Ellsworth in December 2024. Adapts and extends the $s(37)$ improvement found by David W. Cantrell in September 2002.		$s = 10 +{4\over 3}\sqrt 2 = \Nn{11.88561808316412}$ Improved by David Ellsworth in December 2024. Now 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.
$s = {21\over 2}+\sqrt 2 = \Nn{11.91421356237309}$ 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 found 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 $s(37)$ improvement 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.		$s = {21\over 2} + {3\over 2}\sqrt 2 = \Nn{12.62132034355964}$ Adds an "L" to the $s(123)$ primitive packing.		$s = {}^{8}🔒 = \Nn{12.61393128238062}$ $8s^4-(256+48\sqrt{2})s^3+(2958+1272\sqrt{2})s^2-(14934+10560\sqrt{2})s+30205+26292\sqrt{2}=0$ $64s^8-4096s^7+108256s^6-1509216s^5+11615764s^4-45037160s^3+44916240s^2+208411140s-470196503=0$ Base $s(102$) improved by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{12.61350257784129}$ $392s^4-(36568+13888\sqrt{2})s^3+(916840+454480\sqrt{2})s^2-(8957968+4939120\sqrt{2})s+30586193+17860848\sqrt{2}=0$ $3136s^8-585088s^7+34087104s^6-996526208s^5+16984550480s^4-177385055408s^3+1123899500496s^2-3981893672992s+6071335095409=0$ Base $s(102$) improved by David Ellsworth in November 2024, by adapting the $s(37)$ improvement found by David W. Cantrell in September 2002.		$s = {}^{12}🔒 = \Nn{12.60304782603491}$ $16s^6-(904+96\sqrt{2})s^5+(21335+4728\sqrt{2})s^4-(270028+92544\sqrt{2})s^3+(1940853+898908\sqrt{2})s^2-(7549392+4326876\sqrt{2})s+12486856+8242032\sqrt{2}=0$ $256s^{12}-28928s^{11}+1481504s^{10}-45399024s^9+925255281s^8-13177364728s^7+133989746070s^6-975129113048s^5+5003168814793s^4-17439265483424s^3+38384597583024s^2-45887339774976s+20059389786688=0$ Base $s(123)$ improved by David Ellsworth in December 2024, by 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.		$s = {}^{8}🔒 = \Nn{12.60139984236648}$ $2449s^4-(122232+7064\sqrt{2})s^3+(2275197+242662\sqrt{2})s^2-(18745718+2785298\sqrt{2})s+57773870+10697260\sqrt{2}=0$ $2449s^8-244464s^7+10610362s^6-261806300s^5+4020285805s^4-39370448492s^3+240260012584s^2-835788126600s+1269480323300=0$ Base $s(123)$ improved by David Ellsworth in December 2024.

147.				148.
$ s = \tfrac{7}{2} + \tfrac{13}{2}\sqrt 2 = \Nn{12.69238815542511} $ Found by David Ellsworth in January 2025. Provided for comparison.		$\begin{aligned}s &= \tfrac{7}{2} + \tfrac{ 1}{2}\sqrt{337} \\ &= \Nn{12.67877987534290}\end{aligned}$ Found by David Ellsworth in January 2025, based on the $s(293)$ improved by David W. Cantrell in January 2025. Didn't set a record.		$ 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		$s = 12 + {1\over 2}\sqrt 2 = \Nn{12.70710678118654}$ 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.						151.
$s = {}^{6}🔒 = \Nn{12.85197533993158}$ $9s^6-444s^5+7490s^4-48910s^3+55012s^2+902432s-6078789=0$ Provided for comparison. Extends the $s(87)$ found by Evert Stenlund in early 1980 and improved by David W. Cantrell in August 2002.		$s = 10 + 2 \sqrt 2 = \Nn{12.82842712474619}$ Provided for comparison. Adds two "L"s to the $s(106)$ alternative found by Károly Hajba in November 2024, which extends the $s(28)$ found by Frits Göbel in early 1979.		$s = {23\over 2} + {1\over 2}\sqrt 7 = \Nn{12.82287565553229}$ Adds an "L" to the $s(127)$ that extends the $s(86)$ found by Erich Friedman in 1997.		$s = {23\over 2} + {1\over 2}\sqrt 7 = \Nn{12.82287565553229}$ Improved by David Cantrell in December 2024.

151.
$s = 10 + 2 \sqrt 2 = \Nn{12.82842712474619}$ Provided for comparison. Adds an "L" to the $s(128)$ that extends the $s(40)$ found by Frits Göbel in early 1979.		$s = {{3154 - 26\sqrt 6}\over 241} = \Nn{12.82287662525990}$ Found by David Ellsworth in November 2024.		$s = {{3154 - 26\sqrt 6}\over 241} = \Nn{12.82287662525990}$ Alternative packing.		$s = {23\over 2} + {1\over 2}\sqrt 7 = \Nn{12.82287565553229}$ Improved by David Cantrell in December 2024.		$s = {23\over 2} + {1\over 2}\sqrt 7 = \Nn{12.82287565553229}$ Alternative with minimal rotated squares.

152.
$s = {}^{4}🔒 = \Nn{12.90310761612765}$ $162s^4-1782s^3-12526s^2+91472s+242873=0$ Found by David Ellsworth in November 2024.		$s = 3 + 7 \sqrt 2 = \Nn{12.89949493661166}$ Found by Károly Hajba in November 2024. Extends the $s(28)$ found by Frits Göbel in early 1979.		$\begin{aligned}s &= 12-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{12.84666719284348}\end{aligned}$ Found by David Ellsworth in December 2024, by extending the $s(107)$ found by Károly Hajba in November 2024.		$\begin{aligned}s &= 12-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{12.84666719284348}\end{aligned}$ Alternative with minimal rotated squares.		$s = {}^{31}🔒 = \Nn{12.83715933768875}$ $s^{31}-243s^{30}+28565s^{29}-2169427s^{28}+119982233s^{27}-5164754427s^{26}+180598466325s^{25}-5285413039907s^{24}+132329890450043s^{23}-2881567052410157s^{22}+55269496948917143s^{21}-942879868360518765s^{20}+14414071643628857463s^{19}-198580879796828825293s^{18}+2475850922437642643625s^{17}-28016581900809820230975s^{16}+288265065806447285536208s^{15}-2698920907734069036001072s^{14}+22988493013427811976878848s^{13}-177913969220085503454472720s^{12}+1248149823403766723781624164s^{11}-7908847126936247649206429640s^{10}+45032826140437864230306945286s^9-228809463292799494481346135746s^8+1027602427084765821498559262080s^7-4026832868122951314576669767688s^6+13523445330012797272995490904144s^5-37931879553269140295461889699156s^4+85472129919234792536246105753425s^3-145207110761483724893237776998139s^2+165279214691288890482883507651107s-94333125210006322277139127916745=0$ Found by David W. Cantrell in January 2025, based on the $s(53)$ he found in September 2002 and improved in December 2024, and the $s(69)$ found by Maurizio Morandi in June 2010.		$s = {}^{84}🔒 = \Nn{12.83100282216725}$ $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/improved by David Ellsworth 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.

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.		$s = {}^{4}🔒 = \Nn{12.93786550630255}$ $s^4-40s^3+599s^2-3950s+9446=0$ Found by David Ellsworth in December 2024, by combining two slightly modified copies of the $s(41)$ found by Joe DeVincentis in April 2014 that fits an $s(n^2\!-\!n\!-\!1)$ pattern.

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 $n=13$.		$s = \Nn{12.99016864798692}$ Improved by David Ellsworth in December 2024.

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 converting the "L"-augmented form into a primitive packing, found by Károly Hajba in November 2024.

171.
$s = {23\over 2} + {3\over 2}\sqrt 2 = \Nn{13.62132034355964}$ Adds three "L"s to the $s(102)$ found by Károly Hajba in September 2024.		$s = {23\over 2} + {3\over 2}\sqrt 2 = \Nn{13.62132034355964}$ Alternative converting the "L"-augmented form into a primitive packing.		$s = {23\over 2} + {3\over 2}\sqrt 2 = \Nn{13.62132034355964}$ Alternative combining the $s(65)$ found by Frits Göbel in early 1979, and the $s(26)$ found by Erich Friedman in 1997.		$s = {}^{8}🔒 = \Nn{13.59898804184451}$ $7112s^4-(452776+82464\sqrt{2})s^3+(10157384+2969160\sqrt{2})s^2-(97196936+35534808\sqrt{2})s+338837443+141507324\sqrt{2}=0$ $56896s^8-7244416s^7+377822400s^6-10799970048s^5+187464298608s^4-2038984215280s^3+13638438367088s^2-51467019311152s+84096926082673=0$ Found by David Ellsworth in December 2024, by adapting the $s(37)$ improvement found by David W. Cantrell in September 2002. Didn't set a record.		$s = {}^{8}🔒 = \Nn{13.59861960924436}$ $6s^4-(376+64\sqrt2)s^3+(8190+2194\sqrt2)s^2-(75556+24966\sqrt2)s+253307+94710\sqrt2=0$ $36s^8-4512s^7+231464s^6-6503888s^5+110915328s^4-1184746768s^3+7780100524s^2-28819607944s+46224468049=0$ Combines two copies of the $s(37)$ found by David W. Cantrell in September 2002.		$s = {}^{8}🔒 = \Nn{13.59569998314336}$ $96s^4-(5968+928\sqrt{2})s^3+(133541+34462\sqrt{2})s^2-(1291883+427186\sqrt{2})s+4598280+1769696\sqrt{2}=0$ $9216s^8-1145856s^7+59534528s^6-1714063968s^5+30175010609s^4-334467918206s^3+2288154693249s^2-8856922100656s+14880531093568=0$ Improved by David Ellsworth in December 2024.

172.
$s = {23\over 2} + {3\over 2}\sqrt 2 = \Nn{13.62132034355964}$ Found by Károly Hajba in November 2024, extending the $s(102)$ he found 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 found 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 $s(37)$ improvement found by David W. Cantrell in September 2002.		$s = {}^{8}🔒 = \Nn{13.61898898660160}$ $54s^4-(3936+864\sqrt{2})s^3+(98658+33466\sqrt{2})s^2-(1047336+434398\sqrt{2})s+4049739+1891234\sqrt{2}=0$ $2916s^8-425088s^7+24654168s^6-774089568s^5+14674175968s^4-173849337008s^3+1265419604436s^2-5196681822080s+9246853882609=0$ Improved by David Ellsworth in December 2024.

174.
$s = 13 + {5\over 7} = \Nn{13.71428571428571}$ Found by David Ellsworth in December 2024. Didn't set a record, but along with $s(104)$ is the first competitively-intended packing found with a rational side length. Its tilt angle is determined by the Pythagorean triple $\{3, 4, 5\}$.		$s = 13 + {1\over 2}\sqrt 2 = \Nn{13.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group		$s = 13 + {1\over 2}\sqrt 2 = \Nn{13.70710678118654}$ simplified

175.
$s = {}^{8}🔒 = \Nn{13.81880916998841}$ $5184s^8-487872s^7+19993424s^6-465928808s^5+6752167261s^4-62298031950s^3+357291488120s^2-1164321164284s+1650202854142=0$ Combines two copies of the $s(39)$ found by David W. Cantrell in August 2002.		$s = {}^{14}🔒 = \Nn{13.80947149866680}$ $1098304s^{14}-211209728s^{13}+15283515712s^{12}-509534777984s^{11}+3694501484994s^{10}+365523875419736s^9-17221364049171985s^8+414143656230594416s^7-6519120689517020588s^6+71523094872408242792s^5-554043158737579047017s^4+2982041007895185591520s^3-10630760279788360964424s^2+22580834037349273632512s-21627361507768752012176=0$ Found by David Ellsworth in December 2024, based on the $s(39)$ found by David W. Cantrell in August 2002.		$s = {}^{10}🔒 = \Nn{13.80768845606169}$ $371522s^{10}-47042788s^9+2700662827s^8-92597452640s^7+2099449835056s^6-32863492145304s^5+359147226918610s^4-2699592486410208s^3+13313176546344054s^2-38708933874939204s+50018146931740075=0$ Improved by David W. Cantrell in December 2024.		$s = {11\over 3} + {43\over 6}\sqrt 2 = \Nn{13.80186386367384}$ Found by David Ellsworth in December 2024, by removing an "L" from the $s(203)$ he found and enlarging the enclosing square to fit.		$s = 6 + {11\over 2}\sqrt 2 = \Nn{13.77817459305202}$ Improved by David Elllsworth in December 2024, by choosing a different slice and removing the need for the enlargement.

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

178.
$s = 4 + 7 \sqrt 2 = \Nn{13.89949493661166}$ Extends the $s(40)$ found by Frits Göbel in early 1979.		$s = {15\over 2} + {9\over 2}\sqrt 2 = \Nn{13.86396103067892}$ Found by David Ellsworth in November 2024. Continues 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 that fits an $s(n^2\!-\!n\!-\!1)$ pattern.		$s = {}^{4}🔒 = \Nn{13.92523715071339}$ $142s^4-7712s^3+154283s^2-1336942s+4184908=0$ Improved by David Ellsworth in December 2024.

180.
$s = {}^{4}🔒 = \Nn{13.97970624703929}$ $s^4-32s^3+389s^2-2062s+2036=0$ Adds an "L" to the $s(155)$ that continues the $s(n^2\!-\!n\!-\!1)$ pattern found by Joe DeVincentis in April 2014.		$s = \Nn{13.96119764029312}$ Found by David Ellsworth in December 2024, by removing 2 squares from the $s(182)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.		$s = \Nn{13.95491767825175}$ Improved by David Ellsworth in December 2024.

181.		182.				181.
$s = \Nn{13.99404229036268}$ Adds an "L" to the $s(156)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.		$s = \Nn{13.98318264138415}$ Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019. Shows $s(n^2-n)<n$ for $n=14$.		$s = {}^{66}🔒 = \Nn{13.98264294789927}$ $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$ Improved by David Ellsworth in December 2024.		$s = \Nn{13.97854770217285}$ Found by David Ellsworth in December 2024, by removing 1 square.

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 $n=14$.		$s = {}^{66}🔒 = \Nn{13.98264294789927}$ $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$ Improved by David Ellsworth in December 2024.

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 $s(37)$ improvement 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 the $s(172)$ found by Károly Hajba in November 2024, which extended the $s(102)$ he found in September 2024.		$s = {25\over 2} + {3\over 2}\sqrt 2 = \Nn{14.62132034355964}$ Alternative primitive packing based on the $s(83)$ found by Károly Hajba in September 2024.		$s = {}^{8}🔒 = \Nn{14.61977561767287}$ $4s^4-(132+36\sqrt{2})s^3+(1508+1044\sqrt{2})s^2-(7634+8910\sqrt{2})s+20393+18648\sqrt{2}=0$ $16s^8-1056s^7+26896s^6-308848s^5+989672s^4+11485576s^3-116867004s^2+353254396s-279621359=0$ Base $s(172$) improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement found by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{14.61970688383567}$ $562s^4-(169072+99232\sqrt{2})s^3+(6033538+3876698\sqrt{2})s^2-(75906020+50352330\sqrt{2})s+322778363+217620564\sqrt{2}=0$ $1124s^8-676288s^7+55776008s^6-2088089168s^5+44091546248s^4-562015927840s^3+4311045608500s^2-18401537158840s+33695558254817=0$ Base $s(172$) improved by David Ellsworth in November 2024, by adapting the $s(37)$ improvement found by David W. Cantrell in September 2002.		$s = {}^{8}🔒 = \Nn{14.61898898660160}$ $54s^4-(4152+864\sqrt{2})s^3+(110790+36058\sqrt{2})s^2-(1256676+503922\sqrt{2})s+5199723+2359962\sqrt{2}=0$ $2916s^8-448416s^7+27711432s^6-931104720s^5+18929518528s^4-240795061296s^3+1883132387964s^2-8311787117640s+15898277993841=0$ Base $s(172$) improved by David Ellsworth in December 2024.

201.
$s = 14 + {5\over 7} = \Nn{14.71428571428571}$ Found by David Ellsworth in December 2024. Didn't set a record, but along with $s(104)$ is the first competitively-intended packing found with a rational side length. Its tilt angle is determined by the Pythagorean triple $\{3, 4, 5\}$.		$s = 14 + {1\over 2}\sqrt 2 = \Nn{14.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group		$s = 14 + {1\over 2}\sqrt 2 = \Nn{14.70710678118654}$ Mirror-symmetric alternative with minimal rotated squares arranged by David Ellsworth in December 2024, based on the technique found by David W. Cantrell in 2005.		$s = 14 + {1\over 2}\sqrt 2 = \Nn{14.70710678118654}$ Rotationally symmetric alternative with minimal rotated squares arranged by David W. Cantrell in December 2024.		$s = 14 + {1\over 2}\sqrt 2 = \Nn{14.70710678118654}$ 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. Didn't set a record.		$s = {}^{8}🔒 = \Nn{14.73657855744445}$ $776s^4-(33320+336\sqrt{2})s^3+(528568+7752\sqrt{2})s^2-(3662244+43296\sqrt{2})s+9301325-28968\sqrt{2}=0$ $6208s^8-533120s^7+19900352s^6-421620160s^5+5543230416s^4-46288934864s^3+239607688384s^2-702396496296s+891886272841=0$ Improved by David Ellsworth in November 2024, by adapting the $s(37)$ improvement found by David W. Cantrell in September 2002. Didn't set a record.		$s = {}^{4}🔒 = \Nn{14.73353239707125}$ $56s^4-1048s^3-16152s^2+457156s-2516321=0$ Improved by David Ellsworth in January 2025. Didn't set a record.		$s = 2 + 9 \sqrt 2 = \Nn{14.72792206135785}$ Extends the $s(18)$ found by Frits Göbel in early 1979. Explore group

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.		$s = {}^{4}🔒 = \Nn{14.82486939991991}$ $s^4-52s^3+524s^2+5998s-82960=0$ Found by David Ellsworth in December 2024, extending the $s(128)$ he found/improved in November/December 2024, based on the $s(69)$ found by Maurizio Morandi in June 2010.		$s = {}^{32}🔒 = \Nn{14.82477941231809}$ $2401s^{32}-1004304s^{31}+196236180s^{30}-23714439252s^{29}+1973810974720s^{28}-118799295693656s^{27}+5239475702813528s^{26}-164394146951926744s^{25}+3152460689003455454s^{24}-1884976158937315692s^{23}-2345986459851718325780s^{22}+94358303978726779765624s^{21}-2128646362836938565815244s^{20}+26472288254826011788050888s^{19}+50417445455769159894431170s^{18}-10978063742803684959108505384s^{17}+290122932432608300467064251159s^{16}-4753744212995936392438627897672s^{15}+54598998036410077751874179544836s^{14}-436714300711809377426805814415712s^{13}+2266568996313109881286948921725668s^{12}-9443930478172092067066120085919936s^{11}+153569285924138821668183737305750246s^{10}-3411260012455643282533016078093314852s^9+48716027835563155199316576521575839846s^8-480634574592358844282212600776916840152s^7+3479558215819949081331510430953273130980s^6-18911274556079579556635482161438674133300s^5+77001514472742842264328954587724200333125s^4-229319366384836787105473363766786317325000s^3+473731806994674488737711970411917315943750s^2-608463749385462932355294697864370957750000s+366697674954705698636823123191988152890625=0$ Improved by David Ellsworth in December 2024, using the technique from the $s(53)$ improved by David W. Cantrell in December 2024.

206.
$s = {27\over 2} + \sqrt 2 = \Nn{14.91421356237309}$ Provided for comparison. Similar to the $s(70)$ found by Erich Friedman in 1997.		$s = {}^{4}🔒 = \Nn{14.88674602860566}$ $s^4-54s^3+1091s^2-9770s+32701=0$ Found by David Ellsworth in December 2024. Adapts and extends the $s(37)$ improvement found by David W. Cantrell in September 2002.		$s = 13 +{4\over 3}\sqrt 2 = \Nn{14.88561808316412}$ Improved by David Ellsworth in December 2024. Now 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.
$s = {27\over 2} + \sqrt 2 = \Nn{14.91421356237309}$ 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 found 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 $s(37)$ improvement found by David W. Cantrell in September 2002, and the $s(130)$ improvement found by David W. Cantrell in November 2024.

208.
$s = \Nn{14.95969377477006}$ Found by David Ellsworth in December 2024, by removing 2 squares from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024.		$s = 10 + {7\over 2}\sqrt 2 = \Nn{14.94974746830583}$ Found by David Ellsworth in December 2024. Based on the $s(238)$ that extends the $s(109)$ which continues a pattern found by Frits Göbel in early 1979.

209.		210.				209.
$s = {}^{4}🔒 = \Nn{14.98444916984191}$ $s^4-32s^3+391s^2-2082s+654=0$ 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 $n=15$.		$s = {}^{66}🔒 = \Nn{14.98264294789927}$ $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$ Improved by David Ellsworth in December 2024.		$s = \Nn{14.96442179744201}$ Found by David Ellsworth in December 2024, by removing 1 square from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024.

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 $n=15$.		$s = {}^{66}🔒 = \Nn{14.98264294789927}$ $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$ Improved by David Ellsworth in December 2024.

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 $s(37)$ improvement 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 found 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 $s(37)$ improvement found by David W. Cantrell in September 2002.		$s = {}^{12}🔒 = \Nn{15.60902282132495}$ $24s^6-(3008+632\sqrt{2})s^5+(143990+46836\sqrt{2})s^4-(3498934+1393560\sqrt{2})s^3+(46342192+20798868\sqrt{2})s^2-(320302448+155625644\sqrt{2})s+907787225+466763736\sqrt{2}=0$ $576s^{12}-144384s^{11}+15160736s^{10}-915791264s^9+36096990788s^8-988137514952s^7+19384693200660s^6-275725098032832s^5+2830331680270636s^4-20490408263563724s^3+99460046477154240s^2-290971312817869664s+388340875383845233=0$ Improved by David Ellsworth in December 2024.

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.		$s = {{7212 - 34\sqrt{10}}\over 449} = \Nn{15.82290102350618}$ Found by David Ellsworth in December 2024. Extends the $s(151)$ found by David Ellsworth in November 2024.		$s = {}^{32}🔒 = \Nn{15.82288272542124}$ $2401s^{32}-1081136s^{31}+227575796s^{30}-29658121472s^{29}+2666248732772s^{28}-173774204141352s^{27}+8338731572952864s^{26}-287804287422674920s^{25}+6315119157104953822s^{24}-28041637023421982020s^{23}-4200365752197925106408s^{22}+198645587687666998327868s^{21}-5061129096506078245699908s^{20}+73726267308823082442945172s^{19}-87947049283563327476026870s^{18}-27216842619744961591124737032s^{17}+835654086887645374368674033753s^{16}-15205890184479011864998900236148s^{15}+191651587009563650893651694385492s^{14}-1660301063115615092298384133605124s^{13}+8686223135028209936867163410143500s^{12}-19641905504914869202249570317433868s^{11}+287644910082291844950022660238680758s^{10}-11246330960198056495231711623386810364s^9+202355897018377179156570542109032137704s^8-2302635316094370535736927517505458541780s^7+18637198014925689744736021134333289992832s^6-111686247212255633894905670328457247786180s^5+497848050391431476660300176121042489717121s^4-1616634353567457905781858976959940153654232s^3+3632890484832687237588233953926466020404880s^2-5068769120809153666731317435277329464903680s+3315870865953295517111766154888558886723584=0$ Improved by David Ellsworth in December 2024, 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.

235.
$\begin{aligned}s &= 15-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{15.84666719284348}\end{aligned}$ Combines two copies of the $s(54)$ found by Joe DeVincentis in April 2014.		$s = {}^{31}🔒 = \Nn{15.82714315573223}$ $s^{31}-300s^{30}+43628s^{29}-4109912s^{28}+282829346s^{27}-15202732736s^{26}+666362946480s^{25}-24541552616092s^{24}+776214320977993s^{23}-21431425419755588s^{22}+523001856772598488s^{21}-11388322649038624180s^{20}+222877433745524086052s^{19}-3941799128134579888096s^{18}+63253414917377651428798s^{17}-923497619134878250370552s^{16}+12287827518414885631755236s^{15}-149102803935155789292396760s^{14}+1649382178383300745118205622s^{13}-16610866997050850232948382312s^{12}+151924948652570323876118489848s^{11}-1257223240992861378924264620612s^{10}+9364018065577642515615580510104s^9-62325471574294519742065067109632s^8+367120316751912153269115040730624s^7-1888648828522551311026426466607272s^6+8331583444645583296360903393584416s^5-30698235854618682436313114803951912s^4+90800509932103681915982638707767665s^3-202137562956221374464690253300307748s^2+300599154296824607166874955716670128s-223231333612259231542708349001297600=0$ Found by David W. Cantrell in January 2025, based on the $s(53)$ he found in September 2002 and improved in December 2024, and the $s(69)$ found by Maurizio Morandi in June 2010.		$s = {}^{83}🔒 = \Nn{15.82660563342856}$ $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/improved by David Ellsworth 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.
$s = {29\over 2}+\sqrt 2 = \Nn{15.91421356237309}$ 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 found by David W. Cantrell in November 2024.		$s = {}^{12}🔒 = \Nn{15.87607676541001}$ $6596s^6-(619856+19000\sqrt{2})s^5+(24226944+1444540\sqrt{2})s^4-(504036792+43818500\sqrt{2})s^3+(5886496273+662906450\sqrt{2})s^2-(36585732598+5001593450\sqrt{2})s+94529982167+15055629320\sqrt{2}=0$ $26384s^{12}-4958848s^{11}+426380416s^{10}-22179443008s^9+777414394344s^8-19344554351312s^7+350411708641272s^6-4655972006018640s^5+45039272938487593s^4-309348405383979564s^3+1432057511182322314s^2-4011939813305628468s+5144071303851334561=0$ Improved by David Ellsworth in December 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.		$s = {}^{66}🔒 = \Nn{15.98264294789927}$ $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$ Converted by David Ellsworth in December 2024, from the $s(210)$ improved by David Ellsworth in December 2024.

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.

257.
$s = 13 + {5\over 2}\sqrt 2 = \Nn{16.53553390593273}$ Adds eight "L"s to the $s(65)$ found by Frits Göbel in early 1979.		$s = 13 + {5\over 2}\sqrt 2 = \Nn{16.53553390593273}$ Alternative converting the "L"-augmented form into a primitive packing, found by David Ellsworth in December 2024 by combining two copies of the $s(65)$ found by Frits Göbel in early 1979.

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 $s(37)$ improvement 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 the $s(228)$ that extends the $s(102)$ found by Károly Hajba in September 2024.		$s = {29\over 2} + {3\over 2}\sqrt 2 = \Nn{16.62132034355964}$ Alternative converting the "L"-augmented form into a primitive packing.		$s = {}^{8}🔒 = \Nn{16.61235446152284}$ $8s^4-(304+80\sqrt{2})s^3+(3970+2680\sqrt{2})s^2-(22466+26440\sqrt{2})s+65381+64020\sqrt{2}=0$ $64s^8-4864s^7+143136s^6-1915616s^5+7640724s^4+85791512s^3-1060595304s^2+3833056108s-3922445639=0$ Base $s(228$) improved by David Ellsworth in November 2024, by adapting the $s(102)$ improvement found by David W. Cantrell in November 2024.		$s = {}^{8}🔒 = \Nn{16.61205342220945}$ $1568s^4-(158944+46784\sqrt{2})s^3+(5044684+2111296\sqrt{2})s^2-(65137908+31700320\sqrt{2})s+299182329+158437976\sqrt{2}=0$ $50176s^8-10172416s^7+749098752s^6-28833010176s^5+658086103248s^4-9284536988320s^3+79870392630248s^2-385430283384616s+802140438936961=0$ Base $s(228$) improved by David Ellsworth in November 2024, by adapting the $s(37)$ improvement 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 $s(37)$ improvement found by David W. Cantrell in September 2002.		$s = {}^{12}🔒 = \Nn{16.60255251726339}$ $64s^6-(4352+288\sqrt{2})s^5+(112386+18636\sqrt{2})s^4-(1295184+466536\sqrt{2})s^3+(4902306+5553822\sqrt{2})s^2+(19879740-30390606\sqrt{2})s-143635023+56063664\sqrt{2}=0$ $4096s^{12}-557056s^{11}+33159424s^{10}-1122522624s^9+23299339236s^8-290070700224s^7+1703658414312s^6+5714085847824s^5-182330704044288s^4+1346739639595488s^3-4105729915848180s^2+1104381069893496s+14344750990000737=0$ Improved by David Ellsworth in December 2024.

260.
$s = 11 + 4 \sqrt 2 = \Nn{16.65685424949238}$ Adds four "L"s to the $s(148)$ that continues a pattern found by Frits Göbel in early 1979.		$s = 11 + 4 \sqrt 2 = \Nn{16.65685424949238}$ Alternative converting the "L"-augmented form into a primitive packing, found by David Ellsworth in December 2024. Extends the $s(124)$ that continues a pattern found by Frits Göbel in early 1979.

266.
$s = {}^{4}🔒 = \Nn{16.82313609425533}$ $s^4-62s^3+819s^2+6346s-123453=0$ Found by David Ellsworth in November 2024, based on the $s(53)$ found by David W. Cantrell in September 2002.		$s = {}^{32}🔒 = \Nn{16.82306208283780}$ $2401s^{32}-1162084s^{31}+263199482s^{30}-36971795756s^{29}+3593304360857s^{28}-254523535144036s^{27}+13408548240776810s^{26}-519901936408593644s^{25}+13805430523268790240s^{24}-171760035794017558004s^{23}-4426818799267672804198s^{22}+326745541238344524670064s^{21}-10052823560617182774582207s^{20}+181074087488548519346667052s^{19}-1180165534713863709787052484s^{18}-39814767996810201813233201880s^{17}+1608559573416436468095371051805s^{16}-32123383071505854643466513964696s^{15}+412614513240021198536387912093202s^{14}-3166929683248511957762311522115552s^{13}+4864413227687588362660593637481982s^{12}+178773674416065677195226856423404204s^{11}-1078437021027255344046160578887057944s^{10}-30113420634289372798330615891075770272s^9+791230647693671970529285145442783714082s^8-10342818482011428376427887130138935213892s^7+91890490062472897914498395018088233984940s^6-595883639785360944056309451122760067196636s^5+2858083210066295032341927117330041150532457s^4-9961121615142990990156112009477337084333564s^3+23996879793794528190240708983850819688279982s^2-35874639749283350605126520242841070953510704s+25142156270060598069723106950936800398659193=0$ Improved by David Ellsworth in December 2024, based on the $s(53)$ improved by David W. Cantrell in December 2024.

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

270.
$s = \Nn{16.98318264138415}$ Adds two "L"s to the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.		$s = {}^{4}🔒 = \Nn{16.96028765944389}$ $s^4-52s^3+1015s^2-8772s+27756=0$ Found by David Ellsworth in December 2024, by combining two slightly modified copies of the $s(41)$ found by Joe DeVincentis in April 2014 that fits an $s(n^2\!-\!n\!-\!1)$ pattern. Didn't set a record.		$s = \Nn{16.95499994790412}$ Found by David Ellsworth in December 2024, by removing 2 squares from the $s(272)$ based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.

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.		$s = {}^{66}🔒 = \Nn{16.98264294789927}$ $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$ Improved by David Ellsworth in December 2024.

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.		$s = {}^{66}🔒 = \Nn{16.98264294789927}$ $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$ Improved by David Ellsworth in December 2024.

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.		$s = {}^{5}🔒 = \Nn{16.99295005377111}$ $10s^5-412s^4+7204s^3-73814s^2+416937s-935217=0$ Improved by David Ellsworth in December 2024.

293.
$s = 12 + 4 \sqrt 2 = \Nn{17.65685424949238}$ Adds five "L"s to the $s(148)$ that continues a pattern found by Frits Göbel in early 1979.		$s = {7\over 2} + 10 \sqrt 2 = \Nn{17.64213562373095}$ Found by David Ellsworth in December 2024. This is the first 3-width strip of its kind, clamped lengthwise but with interlocking teeth (unlike the $s(66)$ found by Evert Stenlund in early 1980, which is clamped lengthwise with opposing teeth).		$s = 17 + {26\over 41} = \Nn{17.63414634146341}$ 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 due to a Pythagorean triple.

294.
$s = 12 + 4 \sqrt 2 = \Nn{17.65685424949238}$ Combines the $s(148)$ that continues a pattern found by by Frits Göbel in 1979, and the $s(26)$ found by Erich Friedman in 1997. Is likely an irreducible primitive packing.		$s = 12 + 4 \sqrt 2 = \Nn{17.65685424949238}$ Alternative found by David Ellsworth in December 2024. Extends the $s(148)$ that continues a pattern found by Frits Göbel in early 1979. Is likely an irreducible primitive packing.

296.
$s = 17 + {1\over 2}\sqrt 2 = \Nn{17.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group		$s = 17 + {1\over 2}\sqrt 2 = \Nn{17.70710678118654}$ Rigid alternative with minimal rotated squares, based on the rigid $s(52)$ found by David W. Cantrell in 2005.

300.
$s = 3 + {21\over 2}\sqrt 2 = \Nn{17.84924240491749}$ Found by David Ellsworth in December 2024. Extends the $s(28)$ found by Frits Göbel in 1979, and unextends the $s(861)$ that continues that pattern, clamped lengthwise.		$s = {}^{32}🔒 = \Nn{17.82442450194827}$ $2401s^{32}-1234800s^{31}+296028208s^{30}-43787349172s^{29}+4447593803222s^{28}-325302474537864s^{27}+17311567981201236s^{26}-645182372198475544s^{25}+13837196473272880383s^{24}+80900249230246410660s^{23}-20568679509060130505064s^{22}+920036760088872328695408s^{21}-23584191790096989323550520s^{20}+301225363617716483395354964s^{19}+3306782666397187070274017342s^{18}-284390314409319888849379255928s^{17}+8149211786420656055342585494971s^{16}-148913864740995395212738327005616s^{15}+1814416877369227440283632143889698s^{14}-12175229320483112740559170564072244s^{13}-24399018072141993476725426289195974s^{12}+1429143592810572657904567529536292992s^{11}-7892142464877656555451102305171862930s^{10}-209223038143589793338954466697147117932s^9+5720426881261352471833294168046744250065s^8-78159708093687389330697852518534676168792s^7+729681673955892025056174878918410390998726s^6-4990349834530796079653914985571847293526368s^5+25309074235966113070386092161305251964840825s^4-93449971845877531351148978361989669182305000s^3+238863061480171470191587079806750175711250000s^2-379348085850264572736715719653525693400000000s+282723151965698669809699808373173551875000000=0$ 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.

302.
$s = {33\over 2} + \sqrt 2 = \Nn{17.91421356237309}$ Provided for comparison. Extends the $s(88)$ found by Erich Friedman in 1997, by generalizing the $s(41)$ technique found by Charles F. Cottingham in 1979.		$s = {}^{4}🔒 = \Nn{17.88674602860566}$ $s^4-66s^3+1631s^2-17882s+73369=0$ Found by David Ellsworth in December 2024 (including re-adapting the techniques from the $s(130)$ and $s(129)$/$s(206)$ he improved) by adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.

305.
$s = {19\over 2} + 6 \sqrt 2 = \Nn{17.98528137423857}$ Found by David Ellsworth in December 2024. Continues the $s(26)$, $s(85)$ series found by Erich Friedman in 1997, which continues on to $s(178)$ before this.		$s = {}^{66}🔒 = \Nn{17.98264294789927}$ $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.

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.		$s = {}^{5}🔒 = \Nn{17.99295005377111}$ $10s^5-462s^4+8952s^3-97998s^2+587875s-1433594=0$ Improved by David Ellsworth and Károly Hajba in December 2024. (see also asymmetric version)

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.