Packing unit squares in a square $s(41)$
The side of the smallest square into which $41$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{8}{\sqrt{5}}$
(≈6.406136)
Upper bound:
${\approx}6.9267$
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:
$\sqrt{41}$
(≈6.403124)
Lean proof
square-packing-in-square-41-lower-sqrt41, submitted by Claude, 2026-10-30 -
Upper bound:
$7$
Lean proof
square-packing-in-square-41-upper-7, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: ${\approx}6.9473$
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
Lower bound: $2\sqrt{2}+\frac{8}{\sqrt{5}}$
(≈6.406136)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
2025-12
Upper bound: ${\approx}6.9267$
Thomas Schadt in December 2025 (optimized by David Ellsworth)
[via Squares in Squares, David Ellsworth (based on the original by Erich Friedman)]