Mathoverflow 34145

Can a unit square be covered by rectangles of width 1 / (n + 1) and height 1 / (n + 2)?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-mathoverflow-34145,
  title        = {Mathoverflow 34145},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/mathoverflow-34145}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

rectangles_cover_unit_square. Can a unit square be covered by rectangles of width 1 / (n + 1) and height 1 / (n + 2)?

rectangles_pack_unit_square. Equivalently, can a unit square be packed with rectangles of width 1 / (n + 1) and height 1 / (n + 2)?

Can the unit square be covered by $1/ k$ -by- $1/ (k + 1)$ rectangles (across $1 \leq k$ natural)?

I am deliberately not requiring that the rotations can only be $0^{\circ}, 9 0^{\circ}, 18 0^{\circ}, or 27 0^{\circ}$ .

Because of indexing, since n : ℕ starts at 0, we change the side lengths to $1/ (n + 1)$ and $1/ (n + 2)$ , so that the first rectangle is $1/1$ by $1/2$ , the second is $1/2$ by $1/3$ , etc.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Mathoverflow.«34145» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem rectangles_cover_unit_square :
    answer(sorry) ↔ ∃ c : Configuration, ∀ p ∈ unitSquare, ∃ n, p ∈ (c.rect n).toSet

theorem rectangles_pack_unit_square :
    answer(sorry) ↔ ∃ c : Configuration, (∀ n, (c.rect n).toSet ⊆ unitSquare) ∧ c.IsPacking

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

mathoverflow/34145 asked by user Kaveh

Source and licence

Imported from Formal Conjectures (MathOverflow), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.