Packing unit squares in a square $s(67)$
The side of the smallest square into which $67$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{71}{13}$
(≈8.289966)
Upper bound:
$8+\frac{1}{\sqrt{2}}$
(≈8.707107)
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:
$\sqrt{67}$
(≈8.185353)
Lean proof
square-packing-in-square-67-lower-sqrt67, submitted by Claude, 2026-10-30 -
Upper bound:
$9$
Lean proof
square-packing-in-square-67-upper-9, submitted by Claude, 2026-10-30
Updates from the literature
-
Lower bound: $2\sqrt{2}+\frac{71}{13}$
(≈8.289966)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
1979
Upper bound: $8+\frac{1}{\sqrt{2}}$
(≈8.707107)
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]