Packing unit squares in a square $s(19)$
The side of the smallest square into which $19$ unit squares can be packed.
Lower bound:
$6\sqrt{2}-4$
(≈4.485281)
Upper bound:
$3+\frac{4\sqrt{2}}{3}$
(≈4.885618)
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:
$\sqrt{19}$
(≈4.358899)
Lean proof
square-packing-in-square-19-lower-sqrt19, submitted by Claude, 2026-10-30 -
Upper bound:
$5$
Lean proof
square-packing-in-square-19-upper-5, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: $3+\frac{4\sqrt{2}}{3}$
(≈4.885618)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
Lower bound: $6\sqrt{2}-4$
(≈4.485281)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14]