Pith Number

pith:U6JXXOJD

pith:2026:U6JXXOJDY3NFKVTJIBXF7NURGQ

not attested not anchored not stored refs pending

Optimal strategies in the all-heads coin game

Peter Pfaffelhuber

When the coin is biased toward heads, always setting aside exactly one head is optimal and the winning probability increases with the number of coins.

arxiv:2604.22991 v2 · 2026-04-24 · math.PR

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4 Citations open

5 Replications open

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Claims

C1strongest claim

For p > 1/2 the strategy One is optimal and n ↦ w_{n,p} is strictly increasing; the limit W(p) admits an explicit series representation and satisfies p ≤ W(p) < 1. For p < 1/2 near 1/2 the deficit 1/2 - w_{n,1/2-δ} ≈ δ c_n to first order, where c_n satisfies a linear recursion for n ≥ 7 with limit L ≈ 1.7035; consequently the optimal value sequence has a strict local minimum at n=5 and no local maximum.

C2weakest assumption

That the nonlinear suffix-maximum Bellman operator for this MDP admits an optimal policy that is one of the two natural strategies (One or All) for all n and p, allowing the value function to be obtained by direct comparison rather than full policy iteration.

C3one line summary

In the all-heads coin game the optimal winning probability equals 1/2 for any strategy when p=1/2, is strictly increasing in n for p>1/2 with explicit limit series W(p) satisfying p ≤ W(p) < 1, and to first order in bias δ=1/2-p exhibits a strict local minimum at n=5 with coefficient limit ≈1.7035.

Receipt and verification

First computed	2026-06-02T03:04:41.481417Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

a7937bb923c6da555669406e5fb691340237f22d0f148ac26c466f02a68fc558

Aliases

arxiv: 2604.22991 · arxiv_version: 2604.22991v2 · doi: 10.48550/arxiv.2604.22991 · pith_short_12: U6JXXOJDY3NF · pith_short_16: U6JXXOJDY3NFKVTJ · pith_short_8: U6JXXOJD

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/U6JXXOJDY3NFKVTJIBXF7NURGQ \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: a7937bb923c6da555669406e5fb691340237f22d0f148ac26c466f02a68fc558

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "85a59425a28e2c60ff815606b9b957a3547a54b2be113893cd49a1f977ef53a6",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-04-24T20:16:05Z",
    "title_canon_sha256": "45c75a61893447e9c4b4c7b8fe3ec08111161cf3bf2d84ad067fc997facba86b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.22991",
    "kind": "arxiv",
    "version": 2
  }
}