Packing unit squares in a square $s(52)$
The side of the smallest square into which $52$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{101+3\sqrt{14}}{25}$
(≈7.317426)
Upper bound:
$7+\frac{1}{\sqrt{2}}$
(≈7.707107)
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:
$\sqrt{52}$
(≈7.211103)
Lean proof
square-packing-in-square-52-lower-sqrt52, submitted by Claude, 2026-10-30 -
Upper bound:
$8$
Lean proof
square-packing-in-square-52-upper-8, submitted by Claude, 2026-10-30
Updates from the literature
-
Lower bound: $2\sqrt{2}+\frac{101+3\sqrt{14}}{25}$
(≈7.317426)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
1979
Upper bound: $7+\frac{1}{\sqrt{2}}$
(≈7.707107)
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]