Squares in Squares
Older and/or Alternative Packings
 SVG, high-precision, and combined list by David Ellsworth
based on original compiled by Erich Friedman

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 a polynomial root is known for $s$ of degree $3$ or higher, 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.

Zoom:              

10.
$s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$
Found by Frits Göbel
in 1979. 		$s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$
Alternative with minimal rotated squares.

11.


$s = {5\over 2}+\sqrt 2 = \Nn{3.91421356237309}$
Side length found by Frits Göbel
in 1979. 		$s = {5\over 2}+\sqrt 2 = \Nn{3.91421356237309}$
Side length found by Frits Göbel in 1979.
Alternative with minimal rotated squares documented by Martin Gardner
in 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. 		$s = 🔒 = \Nn{3.87708359002281}$ $s^8 - 20s^7 + 178s^6 - 842s^5 + 1923s^4 - 496s^3 - 6754s^2 + 12420s -6865 = 0$
Rigid.
Found by Walter Trump
in 1979.

17.

$s = 4 + {1\over 2}\sqrt 2 = \Nn{4.70710678118654}$
Found by Frits Göbel
in 1979. 		$s = {7\over 3} + {5\over 3}\sqrt 2 = \Nn{4.69035593728849}$
Found by Pertti Hämäläinen
in 1980. 		$s = 🔒 = \Nn{4.67553009360455}$ $4775s^{18}-190430s^{17}+3501307s^{16}-39318012s^{15}+300416928s^{14}-1640654808s^{13}+6502333062s^{12}-18310153596s^{11}+32970034584s^{10}-18522084588s^9-93528282146s^8+350268230564s^7-662986732745s^6+808819596154s^5-660388959899s^4+358189195800s^3-126167814419s^2+26662976550s-2631254953=0$
Found by John Bidwell
in 1998.

18.



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

19.



$s = 4 +{2\over 3}\sqrt 2 = \Nn{4.94280904158206}$
Found by Frits Göbel
in 1979. 		$s = {7\over 2}+ \sqrt 2 = \Nn{4.91421356237309}$
Found by Charles Cottingham
in early 1979. 		$s = 🔒 = \Nn{4.88810889245683}$ $s^4-12s^3+51s^2-72s-36=0$
Found by Walter R. Stromquist
in 1984. Didn't set a record. 		$s = 3 +{4\over 3}\sqrt 2 = \Nn{4.88561808316412}$
Found by Robert Wainwright
in 1979. 		$s = 3 +{4\over 3}\sqrt 2 = \Nn{4.88561808316412}$
Alternative packing found by
David W. Cantrell (see also min/max)
in October 2005.

26.


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

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

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

29.

$s = {8\over 3} + {7\over 3}\sqrt 2 = \Nn{5.96649831220388}$
Found by Erich Friedman
in 1997. 		$s = 🔒 = \Nn{5.96480246752266}$ $9s^6-156s^5+653s^4+1084s^3-8100s^2+7344s-39717=0$
Found 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 1979. 		$s = {9\over 2} + {3\over 2}\sqrt 2 = \Nn{6.62132034355964}$
Unpublished, but was likely found before the improvements that followed. 		$s = 🔒 = \Nn{6.62123046462703}$ $9s^5-123s^4+404s^3+146s^2-303s+205=0$
Found by Erich Friedman
in 1997. 		$s = \Nn{6.60323376318593}$
Found by Thierry Gensane and Philippe Ryckelynck in April 2004, but didn't set a record. 		$s = 🔒 = \Nn{6.59861960924436}$ $6(1-\sqrt{2})s^4+16(-5+9\sqrt{2})s^3+(358-1208\sqrt{2})s^2+62(-10+69\sqrt{2})s+159-5661\sqrt{2}=0$ $36s^8-2496s^7+59768s^6-733760s^5+5289248s^4-23462672s^3+63458276s^2-96673872s+64068561=0$
Found by David W. Cantrell
in September 2002.

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

39.

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

41.

$s = 2 + {7\over 2}\sqrt 2 = \Nn{6.94974746830583}$
Found by Charles Cottingham
in 1979. 		$\begin{aligned}s &= 3 + {1\over 4}({7\sqrt 2 + \sqrt{46-8\sqrt 2}}) \\ &= \Nn{6.94725045864072}\end{aligned}$ $16s^4-192s^3+576s^2+112s-1471=0$
Found by David Cantrell
in October 2005. 		$s = 🔒 = \Nn{6.93786550630255}$ $s^4-16s^3+95s^2-218s-34=0$
Found by Joe DeVincentis
in April 2014.

50.


$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$
Found by Frits Göbel
in 1979. 		$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$
Alternative with minimal rotated squares. 		$s = {11\over 2} + {3\over 2}\sqrt 2 = \Nn{7.62132034355964}$
Found by Erich Friedman
in 1997. 		$s = 🔒 = \Nn{7.59861960924436}$ $36s^8-2784s^7+78248s^6-1146800s^5+9944448s^4-53242000s^3+173869324s^2-319180600s+253748689=0$
Found by David W. Cantrell
in September 2002.

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

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

53.

$s = 5 + 2 \sqrt 2 = \Nn{7.82842712474619}$
Found by Evert Stenlund
in 1980. 		$s = 5 + 2 \sqrt 2 = \Nn{7.82842712474619}$
Alternative removing the "L" to yield a primitive packing. 		$s = 🔒 = \Nn{7.82303789616673}$ $s^3-17s^2+36s+280=0$
Found by David W. Cantrell
in September 2002.

54.



$s = {13\over 2} + \sqrt 2 = \Nn{7.91421356237309}$
Provided for comparison. 		$s = 6 + {4\over 3}\sqrt 2 = \Nn{7.88561808316412}$
by Erich Friedman in 1997, extending $s(19)$ found by Robert Wainwright
in 1979. 		$s = 🔒 = \Nn{7.85109314217532}$ $18432s^{24}-3446784s^{23}+304664576s^{22}-16968519680s^{21}+669449619968s^{20}-19932214017536s^{19}+465775491192832s^{18}-8768302695054848s^{17}+135403072780311424s^{16}-1737169753235173760s^{15}+18680996284830988288s^{14}-169371496869909229568s^{13}+1299023794031261063520s^{12}-8437530238970805727264s^{11}+46364613247151797971136s^{10}-214838591130149411496032s^9+834615127459846686846440s^8-2694590714870358438730024s^7+7138799170846114834605888s^6-15241751239015139566326992s^5+25549157414710670461692094s^4-32328821957836025603803370s^3+28978412289382495954074792s^2-16361602577473933571257634s+4362099385839709619937827=0$
Apparently found by Maurizio Morandi
in June 2010 or slightly earlier, but didn't set a record. 		$s = 🔒 = \Nn{7.84878975975240}$ $4096s^{26}-696320s^{25}+56739840s^{24}-2949122048s^{23}+109766839552s^{22}-3113698237440s^{21}+69948465260800s^{20}-1276594350771968s^{19}+19265599452587584s^{18}-243443276207946880s^{17}+2598512723829467904s^{16}-23570528241163997888s^{15}+182367913759609388096s^{14}-1205666564490332194624s^{13}+6810634124400001375824s^{12}-32812890065132800032688s^{11}+134342994605566431921940s^{10}-464769276437493261704728s^9+1347820021349550459367488s^8-3241044285502742122448756s^7+6369772559544867940758561s^6-10037363738549635149843920s^5+12358928187941177640594208s^4-11472837613066933113730048s^3+7610215413527273079398656s^2-3277082686415784635207680s+712999405005519711846400=0$
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.
$s = 🔒 = \Nn{7.98701198965972}$ $481s^6-15166s^5+186713s^4-1132594s^3+3533776s^2-5430600s+3263162=0$
Found by David W. Cantrell
in August 2002. 		$s = \Nn{7.95424222760119}$
Found by Joe DeVincentis
in April 2014.

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

67.

$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$
by Erich Friedman in 1997, extending $s(52)$ found by Frits Göbel
in 1979. 		$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$
by Erich Friedman in 1997, extending $s(52)$ found by Frits Göbel
in 1979. 		$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$
Alternative with minimal rotated squares found by David W. Cantrell
in 2023.

68.



$s = 5 + 2 \sqrt 2 = \Nn{8.82842712474619}$
Found by Evert Stenlund
in 1980. 		$s = 5 + 2 \sqrt 2 = \Nn{8.82842712474619}$
Alternative removing the "L" to yield a primitive packing. 		$s = {15\over 2} + {1\over 2}\sqrt 7 = \Nn{8.82287565553229}$
Found by David W. Cantrell
in September 2002. 		$s = {13\over 3} + 2 \sqrt 5 = \Nn{8.80546928833291}$
Found by Sigvart Brendberg
in June 2023. 		$s = {13\over 3} + 2 \sqrt 5 = \Nn{8.80546928833291}$
Alternative 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. 		$s = 🔒 = \Nn{8.85617974640243}$ $16s^6-504s^5+6117s^4-34854s^3+86322s^2-53182s+19237=0$
Found by David W. Cantrell
in September 2002. 		$s = 🔒 = \Nn{8.85109314217532}$ $18432s^{24}-3889152s^{23}+389027840s^{22}-24580483072s^{21}+1102466163200s^{20}-37385083702784s^{19}+996598199576064s^{18}-21433776649410560s^{17}+378654550444888448s^{16}-5564791746074648960s^{15}+68634423179436341888s^{14}-714589619551925523840s^{13}+6301668522983200715232s^{12}-47124317021386371701792s^{11}+298546848767309267273056s^{10}-1597315404347371966113984s^9+7177015917694030181782984s^8-26850335202074919621125800s^7+82608242230494967829610872s^6-205341156374029643891989208s^5+401949708478623153033594262s^4-596112355282004302680524122s^3+629100458661410266649761474s^2-420590216248553598525473360s+133760979570165275895770709=0$
Apparently found by Maurizio Morandi
in June 2010 or slightly earlier. 		$s = 🔒 = \Nn{8.84878975975240}$ $4096s^{26}-802816s^{25}+75479040s^{24}-4530423808s^{23}+194919613696s^{22}-6398615904768s^{21}+166556019374080s^{20}-3527133377275648s^{19}+61862476687325248s^{18}-910107143467106304s^{17}+11332922227072242880s^{16}-120197511537209237184s^{15}+1090196653398278259840s^{14}-8474103290188848197504s^{13}+56471863481873946657168s^{12}-322223865161764417071024s^{11}+1569458277137839730778564s^{10}-6493082769370743372961904s^9+22652899039672064985204476s^8-65980365270454114217274836s^7+158260034464024031096444789s^6-306765404716748327800087354s^5+467930910665959192575853315s^4-540089919969690029016072512s^3+443048213754547127055903719s^2-230214712619268014090364786s+57068214173713936231485937=0$
Found by Maurizio Morandi (by adding an "L" to the $s(54)$ found by David W. Cantrell)
in June 2010 or slightly earlier. 		$s = 🔒 = \Nn{8.82868705685148}$ $s^6-32s^5+423s^4-3296s^3+18806s^2-75584s+142565=0$
Found by Maurizio Morandi
in June 2010. 		$s = \Nn{8.82721205592900}$
Found by David W. Cantrell
in August 2023.

70.

$s = {15\over 2}+\sqrt 2 = \Nn{8.91421356237309}$
Found by Erich Friedman
in 1997 		$s = 🔒 = \Nn{8.91209113581151}$ $16s^7-960s^6+24610s^5-346304s^4+2850209s^3-13406809s^2+31814883s-25624307=0$
Found by David W. Cantrell
in August 2002. 		$s = 🔒 = \Nn{8.88166675700900}$ $23s^4-742s^3+8848s^2-45876s+86229=0$
Found by Joe DeVincentis
in April 2014.

71.
$s = 🔒 = \Nn{8.96326750850139}$ $s^8-36s^7+438s^6-1064s^5-21677s^4+211944s^3-784922s^2+1228316s-563063=0$
Found by David Cantrell
in October 2005. 		$s = 🔒 = \Nn{8.96028765944389}$ $s^4-20s^3+151s^2-468s+12=0$
Found by Joe DeVincentis
in April 2014.

82.
$s = 6 + {5\over 2}\sqrt 2 = \Nn{9.53553390593273}$
by Erich Friedman in 1997, extending $s(65)$ found by Frits Göbel
in 1979. 		$s = 6 + {5\over 2}\sqrt 2 = \Nn{9.53553390593273}$
Alternative with minimal rotated squares found by David Ellsworth
in 2023.

84.


$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$
by Erich Friedman in 1997, extending $s(52)$ found by Frits Göbel
in 1979. 		$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$
by Erich Friedman in 1997, extending $s(52)$ found by Frits Göbel
in 1979. 		$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$
by Erich Friedman in 1997, extending $s(52)$ found by Frits Göbel
in 1979. 		$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$
Alternative with minimal rotated squares found by David W. Cantrell
in 2005.

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

87.
$s = {14\over 3}+{11\over 3}\sqrt 2 = \Nn{9.85211639536801}$
Found by Evert Stenlund
in 1980. 		$s = 🔒 = \Nn{9.85197533993158}$ $9s^6-282s^5+2045s^4+5870s^3-89663s^2+554156s-3691596=0$
Found by David W. Cantrell
in August 2002.

88.
$s = {17\over 2}+\sqrt 2 = \Nn{9.91421356237309}$
Found by Erich Friedman
in 1997. 		$s = 🔒 = \Nn{9.90177651254408}$ $4s^6+(-320-56\sqrt{2})s^5+(9952+2828\sqrt{2})s^4+(-55956\sqrt{2}-156632)s^3+(543586\sqrt{2}+1329945)s^2+(-5815938-2595866\sqrt{2})s+10276983+4876156\sqrt{2}=0$ $16s^{12}-2560s^{11}+175744s^{10}-6988864s^9+181397032s^8-3260558096s^7+41816558200s^6-386837301552s^5+2568083107241s^4-11953457726884s^3+37081298010138s^2-68909201625724s+58062584909617=0$
Found by David W. Cantrell
in August 2002.

89.
$s = 5 + {7\over 2}\sqrt 2 = \Nn{9.94974746830583}$
Found by Evert Stenlund
in 1980. 		$s = 5 + {7\over 2}\sqrt 2 = \Nn{9.94974746830583}$
Alternative with minimal rotated squares found by David Ellsworth
in 2023.

272.
$s ≤ 17$
Found by Lars Cleemann
between 1991 and 1998. 		$s = 🔒 = \Nn{16.9915164682460}$ $s^5-49s^4+872s^3-6894s^2+24437s-34521=0$
Found by Lars Cleemann between 1991 and 1998. Optimized by David Ellsworth in May 2023.

For more details, see Erich Friedman's paper on the subject: Packing Unit Squares in Squares: A Survey and New Results (or David Ellsworth's edit).