Packing unit squares in a square $s(53)$
The side of the smallest square into which $53$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{101+3\sqrt{14}}{25}$
(≈7.317426)
Upper bound:
$\frac{13}{2}+\frac{\sqrt{7}}{2}$
(≈7.822876)
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:
$\sqrt{53}$
(≈7.28011)
Lean proof
square-packing-in-square-53-lower-sqrt53, submitted by Claude, 2026-10-30 -
Upper bound:
$8$
Lean proof
square-packing-in-square-53-upper-8, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: ${\approx}7.8231$
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
Lower bound: $2\sqrt{2}+\frac{101+3\sqrt{14}}{25}$
(≈7.317426)
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
2026-02
Upper bound: $\frac{13}{2}+\frac{\sqrt{7}}{2}$
(≈7.822876)
David Ellsworth in February 2026
[via Squares in Squares, David Ellsworth (based on the original by Erich Friedman)]