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OP-10030Status unassessedNumber theory

Magic Squares

Magic Squares

MSC 11 · Number theory

Inferred from the community field; finer classification has not been reviewed.

Specialist source · status unassessedAs of 2026-09-10Source & evidence ↓
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Problem statement

exists_magic_square_squares

Does there exist a 3×33 \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 3×33 \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 3×33 \times 3 matrix with entries aija_{ij} such that each aij=nij3a_{ij} = n_{ij}^3 for some positive integer nijn_{ij}, all nine cubes are distinct, and all row sums and column sums are equal.

Reference: Semi-Magic Square of Cubes

Lean 4

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Formal statement

Lean 4
/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

/-!
# Magic Squares

*References:*

* [Magic Square of Squares - Wikipedia](https://en.wikipedia.org/wiki/Magic_square_of_squares)
* [multimagie.com](http://www.multimagie.com/English/SquaresOfSquaresSearch.htm)
* [Semi-Magic Square of Cubes](https://unsolvedproblems.org/index_files/SquareofCubes.htm)
* [Magic Square of Squares](https://static.nsta.org/pdfs/QuantumV6N3.pdf)
-/

namespace MagicSquares

/--
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](https://static.nsta.org/pdfs/QuantumV6N3.pdf)
-/
@[category research open, AMS 11]
theorem exists_magic_square_squares :
    answer(sorry) ↔ ∃ 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 := by
  sorry

/--
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](https://unsolvedproblems.org/index_files/SquareofCubes.htm)
-/
@[category research open, AMS 5 11]
theorem exists_semi_magic_square_cubes :
    answer(sorry) ↔ ∃ 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) := by
  sorry

end MagicSquares

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Background

Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.

Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 2 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.

Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.

References
  1. [1]Formal Conjectures — Magic Squares link
  2. [2]Original reference cited in the formalization link
  3. [3]Original reference cited in the formalization link
  4. [4]Original reference cited in the formalization link
  5. [5]Original reference cited in the formalization link

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