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.
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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.
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.
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.
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).