Packing unit squares in a square $s(37)$
The side of the smallest square into which $37$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{113+10\sqrt{3}}{37}$
(≈6.350603)
Upper bound:
${\approx}6.5986$
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:
$\sqrt{37}$
(≈6.082763)
Lean proof
square-packing-in-square-37-lower-sqrt37, submitted by Claude, 2026-10-30 -
Upper bound:
$7$
Lean proof
square-packing-in-square-37-upper-7, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: ${\approx}6.5986$
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
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]