Squares in Squares

The following pictures show $n$ unit squares packed inside the smallest known square (of side length $s$). If a pictured packing has multiple numbers in its label above, the picture represents the largest; each smaller is represented by removing any square. For the $n ≤ 324$ 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 (and has no concise closed-form expression), a 🔒 icon is shown; click this to see the polynomial root form of $s$. See also triangular table view (recommended) and older records and/or alternative packings. For more information on each packing, view its SVG's source code. In browsers that don't provide an easy way to do this, you can prepend the URL with "view-source:" (without the quotes). In SVG Edit Mode, which can be turned on either with the button or by pressing [E] while viewing an SVG, the squares can be dragged using the mouse or other pointer device. (Note, recursive pushing isn't yet implemented.) Holding Shift constrains motion to an axis parallel to the square's edges. Holding Ctrl enables rotation-only movement. Pressing Delete will delete a square while the left mouse button is held on it. Pressing [S] will download/save the current edited packing. 1$s = 1$Trivial. 2, 3$s = 2$Proved by Frits Göbelin early 1979. 4$s = 2$Trivial. 5$s = 2 + {1\over 2}\sqrt 2 = \Nn{2.70710678118654}$Rigid.Proved by Frits Göbelin early 1979. 6$s = 3$Proved by Michael Kearneyand Peter Shiu in June 2001. 7, 8$s = 3$Proved by Erich Friedmanin 1999. 9$s = 3$Trivial.

10$s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$Found by Frits Göbel in early 1979.Proved by Walter Stromquist in 2003.Explore group 11 $s = {}^{8}🔒 = \Nn{3.87708359002281}$ $s^8 - 20s^7 + 178s^6 - 842s^5 + 1923s^4 - 496s^3 - 6754s^2 + 12420s - 6865 = 0$ Rigid.Found by Walter Trumpin 1979. 13$s = 4$Proved by Wolfram Bentzin August 2009. 14, 15$s = 4$Proved by Erich Friedmanin 1999. 17 $s = {}^{18}🔒 = \Nn{4.67553009360455}$ $4775s^{18}-190430s^{17}+3501307s^{16}-39318012s^{15}+300416928s^{14}-1640654808s^{13}+6502333062s^{12}-18310153596s^{11}+32970034584s^{10}-18522

084588s^9-93528282146s^8+350268230564s^7-662986732745s^6+808819596154s^5-660388959899s^4+358189195800s^3-126167814419s^2+26662976550s-2631254953=0$ Found by John Bidwellin 1998.Based on packing found by Pertti Hämäläinen in 1980. 18$s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$Found by Pertti Hämäläinenin 1980.Pictured alternative with minimal rotated squares found by Mats Gustafsson in 1981. 19$s = 3 + {4\over 3}\sqrt 2 = \Nn{4.88561808316412}$Found first by Robert Wainwrightin late 1979.Based on packing found by Charles F. Cottingham in early 1979. 22$s = 5$Proved by Wolfram Bentzin October 2018. 23$s = 5$Proved by Hiroshi Nagamochiin 2005. 24$s = 5$Proved by Erich Friedmanin 1999. 26$s = {7\over 2} + {3\over 2}\sqrt 2 = \Nn{5.62132034355964}$Found by Erich Friedmanin 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 Göbelin early 1979.Explore group 28 $s = {}^{6}🔒 = \Nn{5.82444461667405}$ $s^6-24s^5+212s^4-812s^3+1025s^2+882s-1615=0$ Rigid.Found by David Ellsworthin December 2025, using his modified version of Thomas Schadt's simulated annealing program, starting from randomness. 29$s = \Nn{5.93383346267692}$Found by Thomas Schadt in December 2025, using a simulated annealing program he wrote, starting from randomness.Similar to the packing found by Thierry Gensane and Philippe Ryckelynck in April 2004, using a computer program they wrote.Optimized by David Ellsworthin December 2025. 33$s = 6$Proved by Wolfram Bentzin October 2018. 34$s = 6$Proved by Hiroshi Nagamochiin 2005. 35$s = 6$Proved by Erich Friedmanin 1999. 37 $s = {}^{8}🔒 = \Nn{6.59861960924436}$ $6s^4-(208+64\sqrt{2})s^3+(2058+850\sqrt{2})s^2-(7936+3658\sqrt{2})s+11163+5502\sqrt{2}=0$

$36s^8-2496s^7+59768s^6-733760s^5+5289248s^4-23462672s^3+63458276s^2-96673872s+64068561=0$ Found by David W. Cantrell in September 2002.Improves upon the $s(37)$ found by Evert Stenlund in early 1980. 38$s = 6 + {1\over 2}\sqrt 2 = \Nn{6.70710678118654}$Found by Frits Göbelin early 1979.Explore group 39 $s = {}^{5}🔒 = \Nn{6.81072208306864}$ $9s^5-171s^4+999s^3-1959s^2+1636s+166=0$ Found by Thomas Schadt in January 2026, using a simulated annealing program he wrote, starting from randomness.Refound and refined by David Ellsworth in January 2026, starting from randomness.Optimized by David Ellsworth in January 2026. 40$s = 4 + 2 \sqrt 2 = \Nn{6.82842712474619}$Rigid.Found by Frits Göbelin early 1979.Explore group 41 $s = {}^{42}🔒 = \Nn{6.92669309446880}$ $144s^{42}-33248s^{41}+3740531s^{40}-273229120s^{39}+14568177368s^{38}-604345329616s^

{37}+20303247278518s^{36}-567715628580628s^{35}+13476130642163772s^{34}-275622556171657148s^{33}+4913118839607229315s^{32}-77021965442580593792s^{31}+1069597207525632250760s^{30}-13234280063158548374864s^{29}+146588403144234109714492s^{28}-1459056537531761947694412s^{27}+13090219490685085049164304s^{26}-106115059640167069135194108s^{25}+778709778173545540562913112s^{24}-5180160724110239826615572336s^{23}+31267085211757278545052994144s^{22}-171331300125735569491450805184s^{21}+852412555299622931388971786184s^{20}-3849

639590878015114188275848896s^{19}+15771592794879254264477226832440s^{18}-58556476540137831140983424890112s^{17}+196742347065286547712609208667628s^{16}-597075318553361988026330293830592s^{15}+1632807555219691500831155576662224s^{14}-4011703707846363075271797958908992s^{13}+8823126218415607049547609565313808s^{12}-17292499393880618294971765830702496s^{11}+30033784585675389426408059928238624s^{10}-45904080196423917967770013765165584

s^9+61200148145342575539111090507985440s^8-70370773645277985938951580858638528s^7+68756165329665893470887878785349152s^6-55949600498940958320310297578555360s^5+36867848125763978702951438802849616s^4-18873332426700882570902855047275200s^3+7023595398126017089078028623797120s^2-1682258751137636203725622554061120s+192930128676231207430057837613968=0$ Found by Thomas Schadt in December 2025, using a simulated annealing program he wrote, starting from randomness.Refound and refined by David Ellsworth in December-January 2025-2026, using his modified version of Thomas Schadt's simulated annealing program, starting from randomness.Optimized by David Ellsworth in January 2026. 46$s = 7$Proved by Wolfram Bentzin August 2009. 47, 48$s = 7$Proved by Hiroshi Nagamochiin 2005.

50$s = 7 + {4\over 7} = \Nn{7.57142857142857}$ Found by Thomas Schadt in December 2025, using a simulated annealing program he wrote, starting from randomness. Optimized by David Ellsworth. This is the third record-setting packing found with a rational side length, thanks to the Pythagorean triple $\{3, 4, 5\}$ determining its tilt angle. It is the first record-setting packing to use the same Pythagorean triple as the $s(104)$ found by David Ellsworth in December 2024 (the first competitively-intended packing found with a rational side length, though not record-setting). It's the second record-setting packing found that is doubly semi-primitive, i.e. has only non-rotated squares on its leftmost and rightmost sides, but has rotated square(s) poking into its top and bottom sides. It's the first to have both of these properties. 51 $s = {}^{12}🔒 = \Nn{7.70079923541701}$ $s^{12}-52s^{11}+1168s^{10}-14808s^9+116250s^8-584196s^7+1885642s^6-3878332s^5+5185145s^4-4669592s^3-1070690s^2+14600744s-1119939=0$ Found by Thomas Schadt in January 2026, using a simulated annealing program he wrote, starting from randomness.Refound by David Ellsworth in January 2026, using his modified version #2 of Thomas Schadt's simulated annealing program, starting from randomness(statistics gathered).Optimized by David Ellsworth in February 2026.

52$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$Found by Frits Göbelin early 1979.Explore group 53$s = {13\over 2} + {1\over 2}\sqrt 7 = \Nn{7.82287565553229}$Found by David W. Cantrellin September 2002.Improved by David W. Cantrellin December 2024.Improved by David Ellsworth in February 2026, using his modified version #3 of Thomas Schadt's simulated annealing program, starting from randomness.Optimized by David Ellsworthin February 2026. 54$\begin{aligned}s &= 7-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{7.84666719284348}\end{aligned}$Found by David W. Cantrellin October 2005.Improved by Joe DeVincentisin April 2014. 55$s = \Nn{7.94577100750391}$Found by Thomas Schadt in January 2026, using a simulated annealing program he wrote, starting from a cherry-picked state found by his earlier program (started from randomness) which reimplements the Gensane & Ryckelynck algorithm.Refound by David Ellsworth in February 2026, using his modified version #3 of Thomas Schadt's simulated annealing program, starting from randomness (statistics gathered).Improved by David Ellsworth in February 2026, using his modified version of Thomas Schadt's simulated annealing program.Optimized by David Ellsworthin February 2026. 62, 63$s = 8$Proved by Hiroshi Nagamochiin 2005.

65$s = 5 + {5\over 2}\sqrt 2 = \Nn{8.53553390593273}$Found by Frits Göbelin early 1979.Explore group 66$s = 3 + 4 \sqrt 2 = \Nn{8.65685424949238}$Found by Evert Stenlund in early 1980. 67$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$Found by Evert Stenlundin early 1980, extending the $s(52)$found by Frits Göbel in early 1979.Explore group 68$s = \Nn{8.80345993651653}$Found by Sigvart Brendberg in June 2023, using a computer program he wrote followed by manual optimization.Improved by Thomas Schadt in December 2025, using a simulated annealing program he wrote, starting from randomness.Optimized by David Ellsworth in December 2025. 69 $s = {}^{82}🔒 = \Nn{8.82721205592900}$ $52389094428262881s^{82}-28863139436366651460s^{81}+7840436786580754561842s^{80}-1399864630898909951672184s^{79}+184777024966383679131379203s^{78}-19229480097533386652981194668s^{77}+164317800