{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:optimal-strategies-in-the-all-heads-coin-game",
  "title": "Optimal Strategies in the All-Heads Coin Game",
  "canonical_statement": "In the all-heads coin game a player starts with $n$ coins, each showing heads with\nprobability $p$; each round all remaining coins are flipped, the player must set aside at\nleast one head (losing if none shows), and wins once all coins are set aside. Determine\noptimal strategies and the winning probability $w_{n,p}$. Resolved: for $p=\\tfrac12$ every\nstrategy achieves $w_{n,1/2}=\\tfrac12$; for $p>\\tfrac12$ the single-head strategy One is\noptimal, $n\\mapsto w_{n,p}$ is strictly increasing, and $W(p)=\\lim_n w_{n,p}$ has an explicit\nseries representation. In the regime $p<\\tfrac12$, explicitly left open by van Doorn, a\nfirst-order perturbation in $\\delta=\\tfrac12-p$ gives a closed-form description: the deficit\nsatisfies $\\tfrac12-w_{n,1/2-\\delta}\\approx\\delta c_n$, where $c_n$ obeys a linear recursion\nfor $n\\ge7$ with limit $L\\approx1.7035$, and to first order the optimal-value sequence has a\nstrict local minimum at $n=5$ and no local maximum.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:optimal-strategies-in-the-all-heads-coin-game:event",
      "problem_id": "vibemathed:optimal-strategies-in-the-all-heads-coin-game",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-04-24T00:00:00.000Z",
      "title": "arXiv:2604.22991 [math.PR]",
      "summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:optimal-strategies-in-the-all-heads-coin-game:attempt"
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      "source_assertion_ids": [
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      ]
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  ],
  "attempts": [
    {
      "id": "vibemathed:optimal-strategies-in-the-all-heads-coin-game:attempt",
      "problem_id": "vibemathed:optimal-strategies-in-the-all-heads-coin-game",
      "model": "Claude Opus 4.6 / 4.7 / 4.8",
      "provider": "Anthropic",
      "attempted_at": "2026-04-24T00:00:00.000Z",
      "human_collaborators": [
        "Peter Pfaffelhuber"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Per the paper's authorship disclosure: Claude (Anthropic; versions Opus 4.6, 4.7, 4.8), used\ninteractively, produced the mathematical text, the numerical code, and the complete Lean\n4/Mathlib formalization. The underlying ideas, choice of research question, the structuring\nof the joint induction, and the decision to formally verify are the author's; Claude's role\nwas execution: drafting exposition, proposing and debugging Lean proof tactics, selecting\nMathlib lemmas and producing numerical scripts, with every edit reviewed by the author.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Lean formalization + full transcript:",
          "url": "https://github.com/pfaffelh/coins",
          "kind": "lean-proof",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "arXiv:2604.22991 [math.PR]",
          "url": "https://arxiv.org/abs/2604.22991",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Optimal Strategies in the All-Heads Coin Game",
          "url": "https://vibemathed.com/problem/optimal-strategies-in-the-all-heads-coin-game",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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      "independence": "unknown"
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  "verifications": [
    {
      "id": "vibemathed:optimal-strategies-in-the-all-heads-coin-game:verification",
      "problem_id": "vibemathed:optimal-strategies-in-the-all-heads-coin-game",
      "solution_event_id": "vibemathed:optimal-strategies-in-the-all-heads-coin-game:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Every numbered result, including the perturbation analysis, is formally verified in Lean 4 with Mathlib: no `sorry`, no custom axioms (only propext, Classical.choice, Quot.sound), no `native_decide`/`unsafe`. Trust surface is two files (`CoinsLean/Challenge.lean`, `CoinsLean/CoinsLean/Defs.lean`), independently checkable via the Lean comparator on the public repository; manuscript↔Lean map in Appendix A. arXiv preprint (v2, June 2026), not peer-reviewed.\n\nStatus set to partially resolved (2026-08-02): p = 1/2 and p > 1/2 are fully resolved, but for p < 1/2 the paper gives only a first-order expansion in δ = 1/2 − p near 1/2, leaving the range of validity δ₀(n) open and the numerically observed local maxima outside its reach. Verification tier unchanged: the Lean checks what the paper claims, and the paper does not claim the full small-p regime.",
      "sources": [
        {
          "label": "VibeMathed: Optimal Strategies in the All-Heads Coin Game",
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          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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          "label": "arXiv:2604.22991 [math.PR]",
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          "kind": "primary-mathematical-source",
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        }
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    {
      "label": "arXiv:2604.22991 [math.PR]",
      "url": "https://arxiv.org/abs/2604.22991",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Optimal Strategies in the All-Heads Coin Game",
      "url": "https://vibemathed.com/problem/optimal-strategies-in-the-all-heads-coin-game",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
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    {
      "id": "vibemathed:optimal-strategies-in-the-all-heads-coin-game:signal:recent_partial_result",
      "problem_id": "vibemathed:optimal-strategies-in-the-all-heads-coin-game",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}