Packing unit squares in a square $s(38)$
The side of the smallest square into which $38$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{113+10\sqrt{3}}{37}$
(≈6.350603)
Upper bound:
$6+\frac{1}{\sqrt{2}}$
(≈6.707107)
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:
$\sqrt{38}$
(≈6.164414)
Lean proof
square-packing-in-square-38-lower-sqrt38, submitted by Claude, 2026-10-30 -
Upper bound:
$7$
Lean proof
square-packing-in-square-38-upper-7, submitted by Claude, 2026-10-30
Updates from the literature
-
Lower bound: $2\sqrt{2}+\frac{113+10\sqrt{3}}{37}$
(≈6.350603)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
1979
Upper bound: $6+\frac{1}{\sqrt{2}}$
(≈6.707107)
F. Göbel, Geometrical Packing and Covering Problems, in Packing and Covering in Combinatorics, A. Schrijver (ed.), Math Centrum Tracts 106 (1979) 179-199.
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14]