Packing unit squares in a square $s(54)$
The side of the smallest square into which $54$ unit squares can be packed.
Lower bound:
$\sqrt{54}$
(≈7.348469)
Verified
(not in the literature)
Upper bound:
$7-\frac{\sqrt{2}}{2}+\sqrt{1+\sqrt{2}}$
(≈7.846667)
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:
$\sqrt{54}$
(≈7.348469)
Lean proof
square-packing-in-square-54-lower-sqrt54, submitted by Claude, 2026-10-30 -
Upper bound:
$8$
Lean proof
square-packing-in-square-54-upper-8, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: ${\approx}7.8488$
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
2014-04
Upper bound: $7-\frac{\sqrt{2}}{2}+\sqrt{1+\sqrt{2}}$
(≈7.846667)
Joe DeVincentis in April 2014
[via Squares in Squares, David Ellsworth (based on the original by Erich Friedman)]