MagicSquares.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 2 contain sorry, 0 declared as axioms

Magic Squares

References:

Imports (1)

FormalConjecturesUtil

theorem MagicSquares.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? sorry line 43 source
Source documentation
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

L4DEL explanation
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
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Exists, Fin, OfNat.ofNat, Nat, And, Function.Injective2, LT.lt, IsSquare, Eq, Finset.sum, Finset.univ, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
theorem MagicSquares.exists_semi_magic_square_cubes Does there exist a $3 \times 3$ semi-magic square whose entries are all distinct positive integer cubes? sorry line 65 source
Source documentation
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

L4DEL explanation
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
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Exists, Fin, OfNat.ofNat, Nat, And, Function.Injective2, LT.lt, Eq, HPow.hPow, Finset.sum, Finset.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```