Problem 106

Draw $n$ squares inside the unit square with no common interior point. Let $f(n)$ be the maximum possible sum of the side-lengths of the squares. Is $f(k^2+1)=k$ ?

Source: erdosproblems.com/106

Geometry

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

DISPROVED (LEAN): the site credits a packing of seventeen squares with total side length above $4$ , so $f(17)>4$ , found by Claude Opus 5 prompted by Conner Silverstein (2026); see the claim page (Silverstein, 2026).

BKU24: Baek, J. and Koizumi, J. and Ueoro, T., A note on the Erdős conjecture about square packing. arXiv:2411.07274 (2024).
CaSt05: Campbell, Connie and Staton, William, A square-packing problem of Erdős. Amer. Math. Monthly (2005), 165-167.
Er94b: Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.
ErSo95: Erdős, Paul and Soifer, Alexander, Squares in a square. Geombinatorics (1995), 110-114.
Ha84: Halász, Sylvia, Packing a convex domain with similar convex domains. J. Combin. Theory Ser. A (1984), 85-90.
Pr08: Praton, I., Packing squares in a square. Math. Mag. (2008), 358-361.
Ra26: A. Raj Singh, On a square packing conjecture of Erdős. arXiv:2601.22163 (2026).

Problem 106

Status

Proof claims

Discussion