Packing unit squares in a square $s(27)$
The side of the smallest square into which $27$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{27+2\sqrt{10}}{13}$
(≈5.391854)
Upper bound:
$5+\frac{1}{\sqrt{2}}$
(≈5.707107)
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:
$\sqrt{27}$
(≈5.196152)
Lean proof
square-packing-in-square-27-lower-sqrt27, submitted by Claude, 2026-10-30 -
Upper bound:
$6$
Lean proof
square-packing-in-square-27-upper-6, submitted by Claude, 2026-10-30
Updates from the literature
-
Lower bound: $2\sqrt{2}+\frac{27+2\sqrt{10}}{13}$
(≈5.391854)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
1979
Upper bound: $5+\frac{1}{\sqrt{2}}$
(≈5.707107)
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]