Packing unit squares in a square $s(68)$
The side of the smallest square into which $68$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{71}{13}$
(≈8.289966)
Upper bound:
${\approx}8.7988$
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:
$\sqrt{68}$
(≈8.246211)
Lean proof
square-packing-in-square-68-lower-sqrt68, submitted by Claude, 2026-10-30 -
Upper bound:
$9$
Lean proof
square-packing-in-square-68-upper-9, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: $\frac{15}{2}+\frac{\sqrt{7}}{2}$
(≈8.822876)
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] -
2026-09
Upper bound: ${\approx}8.7988$
Jake Loyd in September 2026
[via Squares in Squares, David Ellsworth (based on the original by Erich Friedman)]