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 $s$). 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.

Zoom: 0.25×  1/3×   0.5×   2/3×   3/4×   1×   1.5×   2× &emsp; SVG Edit Mode: OFF

$s = 1$ Trivial.

$s = 2$ Proved by Frits Göbel in early 1979.

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2, 3

$s = 2$ Proved by Frits Göbel in early 1979.

$s = 2$ Trivial.

$s = 2 + {1\over 2}\sqrt 2 = \Nn{2.70710678118654}$ Rigid. Proved by Frits Göbel in early 1979.

$s = 3$ Proved by Michael Kearney and Peter Shiu in June 2001.

$s = 3$ Proved by Erich Friedman in 1999.

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7, 8

$s = 3$ Proved by Erich Friedman in 1999.

$s = 3$ Trivial.

$s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$ Found by Frits Göbel in early 1979. Proved by Walter Stromquist in 2003. Explore group

$s = {}^{8}🔒 = \Nn{3.87708359002281}$ $s^8 - 20s^7 + 178s^6 - 842s^5 + 1923s^4 - 496s^3 - 6754s^2 + 12420s - 6865 = 0$

Rigid. Found by Walter Trump in 1979.

$s = 4$ Proved by Wolfram Bentz in August 2009.

$s = 4$ Proved by Erich Friedman in 1999.

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14, 15

$s = 4$ Proved by Erich Friedman in 1999.

$s = {}^{18}🔒 = \Nn{4.67553009360455}$ $4775s^{18}-190430s^{17}+3501307s^{16}-39318012s^{15}+300416928s^{14}-1640654808s^{13}+6502333062s^{12}-18310153596s^{11}+32970034584s^{10}-18522084588s^9-93528282146s^8+350268230564s^7-662986732745s^6+808819596154s^5-660388959899s^4+358189195800s^3-126167814419s^2+26662976550s-2631254953=0$

Found by John Bidwell in 1998. Based on packing found by Pertti Hämäläinen in 1980.

$s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$ Found by Pertti Hämäläinen in 1980. Pictured alternative with minimal rotated squares found by Mats Gustafsson in 1981.

$s = 3 + {4\over 3}\sqrt 2 = \Nn{4.88561808316412}$ Found first by Robert Wainwright in late 1979. Based on packing found by Charles F. Cottingham in early 1979.

$s = 5$ Proved by Wolfram Bentz in October 2018.

$s = 5$ Proved by Hiroshi Nagamochi in 2005.

$s = 5$ Proved by Erich Friedman in 1999.

$s = {7\over 2} + {3\over 2}\sqrt 2 = \Nn{5.62132034355964}$ Found by Erich Friedman in 1997. Unextends the $s(37)$ found by Evert Stenlund in early 1980.

$s = 5 + {1\over 2}\sqrt 2 = \Nn{5.70710678118654}$ Found by Frits Göbel in early 1979. Explore group

$s = {}^{6}🔒 = \Nn{5.82444461667405}$ $s^6-24s^5+212s^4-812s^3+1025s^2+882s-1615=0$

Rigid. Found by David Ellsworth in December 2025, using his modified version of Thomas Schadt's simulated annealing program, starting from randomness.

$s = \Nn{5.93383346267692}$ 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 Ellsworth in December 2025.

$s = 6$ Proved by Wolfram Bentz in October 2018.

$s = 6$ Proved by Hiroshi Nagamochi in 2005.

$s = 6$ Proved by Erich Friedman in 1999.

$s = {}^{8}🔒 = \Nn{6.59861960924436}$ $6s^4-(208+64\sqrt{2})s^3+(2058+850\sqrt{2})s^2-(7936+3658\sqrt{2})s+11163+5502\sqrt{2}=0$ $36s^8-2496s^7+59768s^6-733760s^5+5289248s^4-23462672s^3+63458276s^2-96673872s+64068561=0$

Found by David W. Cantrell in September 2002. Improves upon the $s(37)$ found by Evert Stenlund in early 1980.

$s = 6 + {1\over 2}\sqrt 2 = \Nn{6.70710678118654}$ Found by Frits Göbel in early 1979. Explore group

$s = {}^{5}🔒 = \Nn{6.81072208306864}$ $9s^5-171s^4+999s^3-1959s^2+1636s+166=0$

Found by Thomas Schadt in January 2026, using a simulated annealing program he wrote, starting from...

Squares in Squares

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