Packing unit squares in a square $s(51)$
The side of the smallest square into which $51$ unit squares can be packed.
Lower bound:
$2\sqrt{2}+\frac{101+3\sqrt{14}}{25}$
(≈7.317426)
Upper bound:
${\approx}7.7008$
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:
$\sqrt{51}$
(≈7.141428)
Lean proof
square-packing-in-square-51-lower-sqrt51, submitted by Claude, 2026-10-30 -
Upper bound:
$8$
Lean proof
square-packing-in-square-51-upper-8, submitted by Claude, 2026-10-30
Updates from the literature
-
Upper bound: ${\approx}7.7044$
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] -
2026-01
Upper bound: ${\approx}7.7008$
Thomas Schadt in January 2026 (refined and optimized by David Ellsworth)
[via Squares in Squares, David Ellsworth (based on the original by Erich Friedman)]