OP-00011 Open

A 3 × 3 magic square of distinct squares

Is there a $3 \times 3$ magic square whose nine entries are distinct positive perfect squares? In a magic square, every row, every column and both main diagonals have the same sum.

Source: M. LaBar, Problem 270, College Math. J. 15 (1984), 69; popularised by M. Gardner, Quantum 6 (1996).

Classification

Primary: NT — Number Theory
Secondary: CO
Keywords: Diophantine equations, magic squares

References

@article{labar1984,
  author = {LaBar, Martin},
  title = {Problem 270},
  journal = {College Mathematics Journal},
  volume = {15},
  number = {1},
  pages = {69},
  year = {1984}
}

@article{gardner1996,
  author = {Gardner, Martin},
  title = {The magic of 3 x 3},
  journal = {Quantum},
  volume = {6},
  number = {3},
  pages = {24--26},
  year = {1996},
  url = {https://static.nsta.org/pdfs/QuantumV6N3.pdf}
}

@misc{boyer,
  author = {Boyer, Christian},
  title = {Magic squares of squares: open problems and searches},
  howpublished = {multimagie.com},
  url = {http://www.multimagie.com/English/SquaresOfSquaresSearch.htm}
}

@misc{wikipedia,
  title = {Magic square of squares},
  howpublished = {Wikipedia},
  url = {https://en.wikipedia.org/wiki/Magic_square_of_squares}
}

@misc{formalconjectures,
  author = {{The Formal Conjectures Authors}},
  title = {Formal Conjectures},
  url = {https://github.com/google-deepmind/formal-conjectures/blob/0771383387505c96d1b2f6a3d35088ad00892c5c/FormalConjectures/Wikipedia/MagicSquares.lean#L35},
  note = {Lean statement under Apache-2.0; problem text under CC BY 4.0, or CC BY-SA 4.0 where it is based on Wikipedia}
}

Canonical Lean statement

A submission resolves this problem by proving the constant Problem_OP_00011.

/-- OP-00011: A 3 × 3 magic square of distinct squares -/
def Problem_OP_00011 : Prop :=
  ∃ m t,
      Function.Injective2 m ∧
        (∀ (i j : Fin 3), 0 < m i j ∧ IsSquare (m i j)) ∧
          (∀ (i : Fin 3), ∑ j, m i j = t) ∧
            (∀ (j : Fin 3), ∑ 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

Formalised by The Formal Conjectures Authors (Apache-2.0); reviewed by clemens-admin.

Available to match against in: Lean v4.33.1 with Mathlib v4.33.1.

Resolving this problem in Lean

Add the problems package as a dependency, import this problem's module, and prove the constant. Any published revision works.

In lakefile.toml:

[[require]]
name = "RegistryProblems"
git = "https://github.com/cthalhammer/registry-problems"
rev = "<published revision>"

In your proof:

import RegistryProblems.OP_00011

theorem resolution : RegistryProblems.Problem_OP_00011 := by
  sorry

In preprint.toml, mapping the claim to the declaration that establishes it:

[[statements]]
label = "Theorem 1"
lean = "resolution"
latex = "..."
problem = "OP-00011"
claim = "full"

To disprove it instead, prove the negation and claim "disproof", or "counterexample" if the proof exhibits one:

theorem disproof : ¬ RegistryProblems.Problem_OP_00011 := by
  sorry

You can also write the statement out in full; the verifier checks it matches up to definitional equality. Partial results and reductions are not matched against this statement.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00011, "A 3 × 3 magic square of distinct squares", clopen.solutions.