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Level A · Machine-checkable Hard Geometry P-square-packing

Packing

What is the smallest square that can contain 11 unit squares? Reference: Wikipedia

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

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Claims
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

least_eleven_square_packing_in_square. What is the smallest square that can contain 11 unit squares?

Reference: Wikipedia

least_seventeen_square_packing_in_square. What is the smallest square that can contain 17 unit squares?

Reference: Wikipedia

least_twenty_one_circle_packing_in_square. What is the smallest square that can contain 21 unit circles?

least_fifteen_circle_packing_in_circle. What is the smallest circle that can contain 15 unit circles?

Reference: Graham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154. Pirl (1969) conjectured this configuration to be optimal.

This file contains a number of open problems related to the minimal size of a square (or circle) that can contain a given number of unit squares (or circles). In each case, we provide a known upper bound, and ask for the least such size.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.SquarePacking (4 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem least_eleven_square_packing_in_square :
    IsLeast {x : ℝ | Nonempty (Packing 11 UnitSquare (Square x))} answer(sorry)
theorem least_seventeen_square_packing_in_square :
    IsLeast {x : ℝ | Nonempty (Packing 17 UnitSquare (Square x))} answer(sorry)
theorem least_twenty_one_circle_packing_in_square :
    IsLeast {x : ℝ | Nonempty (Packing 21 UnitCircle (Square x))} answer(sorry)
theorem least_fifteen_circle_packing_in_circle :
    IsLeast {r : ℝ≥0 | Nonempty (Packing 15 UnitCircle (Circle r))} answer(sorry)

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

Source and licence

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