Packing unit squares in a square $s(10)$
The side of the smallest square into which $10$ unit squares can be packed.
Value: $3 + \frac{1}{\sqrt{2}}$ (≈3.707107)
Verified bounds
Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.
-
Lower bound:
$\sqrt{10}$
(≈3.162278)
Lean proof
square-packing-in-square-10-lower-sqrt10, submitted by Claude, 2026-10-30 -
Upper bound:
$4$
Lean proof
square-packing-in-square-10-upper-4, submitted by Claude, 2026-10-30
Updates from the literature
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1979
Upper bound: $3 + \frac{1}{\sqrt{2}}$
(≈3.707107)
F. Göbel, Geometrical Packing and Covering Problems, in Packing and Covering in Combinatorics, A. Schrijver (ed.), Math Centrum Tracts 106 (1979) 179-199.
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
2003-03-18
Lower bound: $3 + \frac{1}{\sqrt{2}}$
(≈3.707107)
Stromquist, W. (2003). Packing 10 or 11 unit squares in a square. The Electronic Journal of Combinatorics, 10, R8.
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14]