Squares in Squares
SVG, high-precision, and other updates by David Ellsworth
based on original compiled by Erich Friedman

The following pictures show $n$ unit squares packed inside the smallest known square (of side length $s$). For the $n ≤ 100$ not pictured, the trivial packing (with no tilted squares) is the best known packing. 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 $s$.

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1.
$s = 1$
Trivial.
2.
$s = 2$
Proved by Frits Göbel
in 1979.
3.
$s = 2$
Proved by Frits Göbel
in 1979.
4.
$s = 2$
Trivial.
5.
$s = 2 + {1\over 2}\sqrt 2 = \Nn{2.70710678118654}$
Rigid.
Proved by Frits Göbel
in 1979.
6.
$s = 3$
Proved by Michael Kearney
and Peter Shiu in April 2002.
7.
$s = 3$
Proved by Erich Friedman
in 1999.
8.
$s = 3$
Proved by Erich Friedman
in 1999.
9.
$s = 3$
Trivial.
10.
$s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$
Found by Frits Göbel in 1979.
Proved by Walter Stromquist in 2003.
11.
$s = 🔒 = \Nn{3.87708359002281}$ $s^8 - 20s^7 + 178s^6 - 842s^5 + 1923s^4 - 496s^3 - 6754s^2 + 12420s - 6865 = 0$
Rigid.
Found by Walter Trump
in 1979.
13.
$s = 4$
Proved by Wolfram Bentz
in 2009.
14.
$s = 4$
Proved by Erich Friedman
in 1999.
15.
$s = 4$
Proved by Erich Friedman
in 1999.
17.
$s = 🔒 = \Nn{4.67553009360455}$ $4775s^{18}-190430s^{17}+3501307s^{16}-39318012s^{15}+300416928s^{14}-1640654808s^{13}+6502333062s^{12}-18310153596s^{11}+32970034584s^{10}-18522084588s^9-93528282146s^8+350268230564s^7-662986732745s^6+808819596154s^5-660388959899s^4+358189195800s^3-126167814419s^2+26662976550s-2631254953=0$
Found by John Bidwell
in 1998.
18.
$s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$
Found by Pertti Hämäläinen
in 1980.
19.
$s = 3 + {4\over 3} \sqrt 2 = \Nn{4.88561808316412}$
Found by Robert Wainwright
in 1979.
23.
$s = 5$
Proved by Hiroshi Nagamochi
in 2005.
24.
$s = 5$
Proved by Erich Friedman
in 1999.
26.
$s = {7\over 2} + {3\over 2}\sqrt 2 = \Nn{5.62132034355964}$
Found by Erich Friedman
in 1997.
27.
$s = 5 + {1\over 2}\sqrt 2 = \Nn{5.70710678118654}$
Found by Frits Göbel
in 1979.
28.
$s = 3 + 2 \sqrt 2 = \Nn{5.82842712474619}$
Found by Frits Göbel
in 1979.
29.
$s = \Nn{5.93434180499654}$
Found by Thierry Gensane
and Philippe Ryckelynck
in April 2004.
34.
$s = 6$
Proved by Hiroshi Nagamochi
in 2005.
35.
$s = 6$
Proved by Erich Friedman
in 1999.
37.
$s = 🔒 = \Nn{6.59861960924436}$ $36s^8-2496s^7+59768s^6-733760s^5+5289248s^4-23462672s^3+63458276s^2-96673872s+64068561=0$ $6(1-\sqrt{2})s^4+16(-5+9\sqrt{2})s^3+(358-1208\sqrt{2})s^2+62(-10+69\sqrt{2})s+159-5661\sqrt{2}=0$
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.
39.
$s = 🔒 = \Nn{6.81880916998841}$ $5184s^8-197568s^7+3200144s^6-28651016s^5+154197141s^4-506760114s^3+981374688s^2-1005617394s+408278853=0$
Found by David W. Cantrell
in August 2002.
40.
$s = 4 + 2 \sqrt 2 = \Nn{6.82842712474619}$
Rigid.
Found by Frits Göbel
in 1979.
41.
$s = 🔒 = \Nn{6.93786550630255}$ $s^4-16s^3+95s^2-218s-34=0$
Found by Joe DeVincentis
in April 2014.
46.
$s = 7$
Proved by Wolfram Bentz
in 2009.
47.
$s = 7$
Proved by Hiroshi Nagamochi
in 2005.
48.
$s = 7$
Proved by Hiroshi Nagamochi
in 2005.
50.
$s = 🔒 = \Nn{7.59861960924436}$ $36s^8-2784s^7+78248s^6-1146800s^5+9944448s^4-53242000s^3+173869324s^2-319180600s+253748689=0$
Found by David W. Cantrell
in September 2002.
51.
$s = 🔒 = \Nn{7.70435372947124}$ $36864s^{28}-6340608s^{27}+502050816s^{26}-24636665856s^{25}+847410746368s^{24}-21857473382400s^{23}+441080063406080s^{22}-7168010813250560s^{21}+95780257115813376s^{20}-1068807979173627904s^{19}+10079527432131681024s^{18}-81076261200222141184s^{17}+560144016315152943424s^{16}-3340596154679285521280s^{15}+17248154822575215485952s^{14}-77154000981112955287360s^{13}+298459379274993606556192s^{12}-993878287428748511469056s^{11}+2827560064086331516654992s^{10}-6798876862709272559608016s^9+13620557443692132080422196s^8-22318847802748398169997192s^7+29208515137302727559556744s^6-29572821499263810227200404s^5+22139938904533199326391397s^4-11407627350518593079154528s^3+3525180882798952592954446s^2-436679755165931930913236s-28766318325274882531199=0$
Found by Károly Hajba
in July 2009.
52.
$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$
Found by Frits Göbel
in 1979.
53.
$s = 🔒 = \Nn{7.82303789616673}$ $s^3-17s^2+36s+280=0$
Found by David W. Cantrell
in September 2002.
54.
\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.
$s = \Nn{7.95424222760119}$
Found by Joe DeVincentis
in April 2014.
62.
$s = 8$
Proved by Hiroshi Nagamochi
in 2005.
63.
$s = 8$
Proved by Hiroshi Nagamochi
in 2005.
65.
$s = 5 + {5\over 2}\sqrt 2 = \Nn{8.53553390593273}$
Found by Frits Göbel
in 1979.
66.
$s = 3 + 4 \sqrt 2 = \Nn{8.65685424949238}$
Found by Evert Stenlund
in 1980.
67.
$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$
Found by Frits Göbel
in 1979.
68.
$s={13\over 3}+2\sqrt 5=\Nn{8.80546928833291}$
Found by Sigvart Brendberg
in June 2023.
69.
$s = \Nn{8.82721205592900}$
Found by David W. Cantrell
in August 2023.
70.
$s = 🔒 = \Nn{8.88166675700900}$ $23s^4-742s^3+8848s^2-45876s+86229=0$
Found by Joe DeVincentis
in April 2014.
71.
$s = 🔒 = \Nn{8.96028765944389}$ $s^4-20s^3+151s^2-468s+12=0$
Found by Joe DeVincentis
in April 2014.
79.
$s = 9$
Proved by Hiroshi Nagamochi
in 2005.
80.
$s = 9$
Proved by Hiroshi Nagamochi
in 2005.
82.
$s = 6 + {5\over 2}\sqrt 2 = \Nn{9.53553390593273}$
Found by Frits Göbel
in 1979.
83.
$s = 4 + 4 \sqrt 2 = \Nn{9.65685424949238}$
Found by Evert Stenlund
in 1980.
84.
$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$
Found by Frits Göbel
in 1979.
85.
$s = {11\over 2} + 3 \sqrt 2 = \Nn{9.74264068711928}$
Found by Erich Friedman
in 1997.
86.
$s = {17\over 2} + {1\over 2}\sqrt 7 = \Nn{9.82287565553229}$
Found by Erich Friedman
in 1997.
87.
$s = 🔒 = \Nn{9.85197533993158}$ $9s^6-282s^5+2045s^4+5870s^3-89663s^2+554156s-3691596=0$
Found by David W. Cantrell
in August 2002.
88.
$s = 🔒 = \Nn{9.90177651254408}$ $4s^6+(-320-56\sqrt{2})s^5+(9952+2828\sqrt{2})s^4+(-55956\sqrt{2}-156632)s^3+(543586\sqrt{2}+1329945)s^2+(-5815938-2595866\sqrt{2})s+10276983+4876156\sqrt{2}=0$ $16s^{12}-2560s^{11}+175744s^{10}-6988864s^9+181397032s^8-3260558096s^7+41816558200s^6-386837301552s^5+2568083107241s^4-11953457726884s^3+37081298010138s^2-68909201625724s+58062584909617=0$
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.
272.
$s = 🔒 = \Nn{16.9915164682460}$ $s^5-49s^4+872s^3-6894s^2+24437s-34521=0$
Found by Lars Cleemann
between 1991 and 1998.

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).