Magic Squares — open problem in number theory

Magic Squares

Does there exist a 3 × 3 matrix such that every entry is a distinct square, and all rows, columns, and diagonals add up to the same value? 0 is excluded, as a Magic Square of Squares with 0 and 8 distinct squares is know is knownn. See Magic Square of Squares

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

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-magic-squares,
  title        = {Magic Squares},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/magic-squares}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

exists_magic_square_squares. Does there exist a $3 \times 3$ matrix such that every entry is a distinct square, and all rows, columns, and diagonals add up to the same value?

0 is excluded, as a Magic Square of Squares with 0 and 8 distinct squares is know is knownn. See Magic Square of Squares

exists_semi_magic_square_cubes. Does there exist a $3 \times 3$ semi-magic square whose entries are all distinct positive integer cubes? A square is semi-magic if all rows and columns sum to the same total.

More precisely, we seek a $3 \times 3$ matrix with entries $a_{ij}$ such that each $a_{ij} = n_{ij}^{3}$ for some positive integer $n_{ij}$ , all nine cubes are distinct, and all row sums and column sums are equal.

Reference: Semi-Magic Square of Cubes

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.MagicSquares (2 statements).

theorem exists_magic_square_squares :
    ∃ m : Fin 3 → Fin 3 → ℕ, ∃ t : ℕ,
       m.Injective2 ∧
       (∀ i j, 0 < (m i j) ∧ IsSquare (m i j)) ∧
       (∀ i, ∑ j, m i j = t) ∧
       (∀ j, ∑ i, m i j = t) ∧
       m 0 0 + m 1 1 + m 2 2 = t ∧
       m 0 2 + m 1 1 + m 2 0 = t

theorem exists_semi_magic_square_cubes :
    ∃ m : Fin 3 → Fin 3 → ℕ, ∃ t : ℕ,
       m.Injective2 ∧
       (∀ i j, ∃ n : ℕ, 0 < n ∧ m i j = n ^ 3) ∧
       (∀ i, ∑ j, m i j = t) ∧
       (∀ j, ∑ i, m i j = t)

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

Magic Square of Squares - Wikipedia
[multimagie.com](http://www.multimagie.com/English/SquaresOfSquaresSearch.htm)
Semi-Magic Square of Cubes
Magic Square of Squares

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.