Packing unit squares in a square $s(5)$
The side of the smallest square into which $5$ unit squares can be packed.
Value: $2 + \frac{1}{\sqrt{2}}$ (≈2.707107)
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.
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Lower bound:
$\sqrt{5}$
(≈2.236068)
Lean proof
square-packing-in-square-5-lower-sqrt5, submitted by Claude, 2026-10-30 -
Upper bound:
$3$
Lean proof
square-packing-in-square-5-upper-3, submitted by Claude, 2026-10-30
Updates from the literature
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1979
Upper bound: $2 + \frac{1}{\sqrt{2}}$
(≈2.707107)
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] -
1979
Lower bound: $2 + \frac{1}{\sqrt{2}}$
(≈2.707107)
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]