Packing unit squares in a square $s(82)$
The side of the smallest square into which $82$ unit squares can be packed.
Lower bound:
${\approx}9.2667$
Upper bound:
$6+\frac{5}{\sqrt{2}}$
(≈9.535534)
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:
$\sqrt{82}$
(≈9.055385)
Lean proof
square-packing-in-square-82-lower-sqrt82, submitted by Claude, 2026-10-30 -
Upper bound:
$10$
Lean proof
square-packing-in-square-82-upper-10, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: $6+\frac{5}{\sqrt{2}}$
(≈9.535534)
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]