Packing unit squares in a square $s(40)$
The side of the smallest square into which $40$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{8}{\sqrt{5}}$
(≈6.406136)
Upper bound:
$4+2\sqrt{2}$
(≈6.828427)
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:
$\sqrt{40}$
(≈6.324555)
Lean proof
square-packing-in-square-40-lower-sqrt40, submitted by Claude, 2026-10-30 -
Upper bound:
$7$
Lean proof
square-packing-in-square-40-upper-7, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: $4+2\sqrt{2}$
(≈6.828427)
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]