Packing unit squares in a square $s(18)$
The side of the smallest square into which $18$ unit squares can be packed.
Lower bound:
$\frac{40\sqrt{2}+19}{17}$
(≈4.445208)
Upper bound:
$\frac{7+\sqrt{7}}{2}$
(≈4.822876)
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:
$\sqrt{18}$
(≈4.242641)
Lean proof
square-packing-in-square-18-lower-sqrt18, submitted by Claude, 2026-10-30 -
Upper bound:
$5$
Lean proof
square-packing-in-square-18-upper-5, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: $\frac{7+\sqrt{7}}{2}$
(≈4.822876)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
Lower bound: $\frac{40\sqrt{2}+19}{17}$
(≈4.445208)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14]