Packing unit squares in a square $s(39)$
The side of the smallest square into which $39$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{113+10\sqrt{3}}{37}$
(≈6.350603)
Upper bound:
${\approx}6.8107$
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:
$\sqrt{39}$
(≈6.244998)
Lean proof
square-packing-in-square-39-lower-sqrt39, submitted by Claude, 2026-10-30 -
Upper bound:
$7$
Lean proof
square-packing-in-square-39-upper-7, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: ${\approx}6.8189$
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] -
2026-01
Upper bound: ${\approx}6.8107$
Thomas Schadt in January 2026 (refined and optimized by David Ellsworth)
[via Squares in Squares, David Ellsworth (based on the original by Erich Friedman)]