Packing unit squares in a square $s(65)$
The side of the smallest square into which $65$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{71}{13}$
(≈8.289966)
Upper bound:
$5+\frac{5}{\sqrt{2}}$
(≈8.535534)
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:
$\sqrt{65}$
(≈8.062258)
Lean proof
square-packing-in-square-65-lower-sqrt65, submitted by Claude, 2026-10-30 -
Upper bound:
$9$
Lean proof
square-packing-in-square-65-upper-9, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: $5+\frac{5}{\sqrt{2}}$
(≈8.535534)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
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]