Packing unit squares in a square $s(85)$
The side of the smallest square into which $85$ unit squares can be packed.
Lower bound:
${\approx}9.2667$
Upper bound:
$\frac{11}{2}+3\sqrt{2}$
(≈9.742641)
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:
$\sqrt{85}$
(≈9.219544)
Lean proof
square-packing-in-square-85-lower-sqrt85, submitted by Claude, 2026-10-30 -
Upper bound:
$10$
Lean proof
square-packing-in-square-85-upper-10, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: $\frac{11}{2}+3\sqrt{2}$
(≈9.742641)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
Lower bound: ${\approx}9.2667$
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14]