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PapersNumber theory

The Catch-Up game and conjecture

Let TN=∑k=1Nk=N(N+1)2T_N = \sum_{k=1}^{N} k = \frac{N(N+1)}{2}. If TNT_N is even (equivalently N≡0(mod4)N \equiv 0 \pmod 4 or N≡3(mod4)N \equiv 3 \pmod 4), then under optimal play the game Catch-Up($\{1, \ldots, N\}$) ends in a draw.

Number theoryAMS 11AMS 91

Mathematical statement

Let TN=∑k=1Nk=N(N+1)2T_N = \sum_{k=1}^{N} k = \frac{N(N+1)}{2}. If TNT_N is even (equivalently N≡0(mod4)N \equiv 0 \pmod 4 or N≡3(mod4)N \equiv 3 \pmod 4), then under optimal play the game Catch-Up($\{1, \ldots, N\}$) ends in a draw.

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Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

value_of_even_mul_succ_self_div_two

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem value_of_even_mul_succ_self_div_two    (N : ℕ) (h_even : Even (N * (N + 1) / 2)) :    value (.Icc 1 N) = .draw := by  sorry
Statement source
Formal Conjectures
Lean version
v4.27.0
Placeholder
Present; no proof artifact
Source evidence
Pinned source index
Fidelity review
Community formulation

References

Pinned Formal Conjectures sourceformalization