Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 106

claims/: The 2 claim pages of Problem 106, one per claimant's result; the problem's standing derives from them.

Statement. 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 $f(k^2+1)=k$ ?

Status. 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.

Source. erdosproblems.com/106, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #106, https://www.erdosproblems.com/106.

References.

[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).

Formalization. No formal statement is recorded. A Lean proof of $f(17)>4$ by a different 17-square packing, credited to Raj Singh and kept in Boris Alexeev's repository of formalized Erdős problems, was built here at a pinned commit, its axioms checked and its statement audited; it is an accepted disproof, recorded on its own claim page.

Progress

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Linked library material

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Linked from (13)

A 17-square packing with total side length above 4 Another 17-square packing with a Lean proof in Alexeev's repository problems/discrete_geometry/E0106/claims Discrete and Convex Geometry Problem 1026 Alexeev's Lean proof of the exact constant for every n discrete_geometry/baek_2024_note_erdos_conjecture_about_square_packing Theorem 1.1: g(k^2+2c+1) = k + c/k for all integers -k < c < k Theorem 2.1: g(k^2+1) = k for axis-parallel square packings discrete_geometry/singh_2026_square_packing_conjecture_erdos Sections 4 and 5 (p. 3): Erdős's square conjecture holds iff the excesses have a convergent sum Theorem 1 (p. 1): rigidity of the excess f(k^2+1) - k for any f obeying (*)Theorem 2 (p. 2): for triangles in a triangle, f(k^2+1) = k for all k iff the excesses have a convergent sum

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