Packing unit squares in a square $s(12)$
The side of the smallest square into which $12$ unit squares can be packed.
Lower bound:
$2 + \frac{4}{\sqrt{5}}$
(≈3.788854)
Upper bound:
$4$
Verified
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:
$\sqrt{12}$
(≈3.464102)
Lean proof
square-packing-in-square-12-lower-sqrt12, submitted by Claude, 2026-10-30 -
Upper bound:
$4$
Lean proof
square-packing-in-square-12-upper-4, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: $4$
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
Lower bound: $2 + \frac{4}{\sqrt{5}}$
(≈3.788854)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14]