Pith Number

pith:B4DZJ5G2

pith:2026:B4DZJ5G2EVNRBKDEBXOU4JEUX5

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A Subtraction Nim with a Pass

Hikaru Manabe, Ryohei Miyadera, Urban Larsson

Adding a one-time pass to this subtraction Nim leaves its reverse-mex Grundy property unchanged.

arxiv:2605.14321 v1 · 2026-05-14 · math.CO

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\pithnumber{B4DZJ5G2EVNRBKDEBXOU4JEUX5}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We prove that this game still satisfies the reverse-mex property of Grundy numbers when a pass move is available.

C2weakest assumption

The subtraction set must be exactly {2, 4n, 4n+2} for integer n ≥ 3; the reverse-mex property is stated not to hold for n=1 or n=2, and the proofs rely on this specific arithmetic form.

C3one line summary

Subtraction Nim with moves {2,4n,4n+2} (n≥3) and its one-time-pass variant both satisfy the reverse-mex property for Grundy numbers.

References

13 extracted · 13 resolved · 0 Pith anchors

[1] M. H. Albert, R. J. Nowakowski, and D. W olfe, Lessons In Play: An Introduction to Combi- natorial Game Theory , second edition, A K Peters/CRC Press, Boca Raton, FL, 2019 2019

[2] Anjali Bhagat, Urban Larsson, Hikaru Manabe, Takahiro Y amashita, Additive sink subtrac- tion, Preprint arXiv:2601.18715 (2026) 2026

[3] W. H. Chan, R. M. Low, S. C. Locke, and O.L. W ong, A map of the P-positions in ‘Nim With a Pass’ played on heap sizes of at most four, Discrete Applied Mathematics 244 (2018), 44-55 2018

[4] S. W. Golomb, A mathematical investigation of games of “t ake-away”, J. Combinatorial Theory, 1(4) (1966), 443– 458 1966

[5] D. G. Horrocks and R. J. Nowakowski, Regularity in the G–S equences of Octal Games with a Pass, Integers 3 (2003), #G1 2003

Receipt and verification

First computed	2026-05-17T23:39:09.832341Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

0f0794f4da255b10a8640ddd4e2494bf622327aae8afc2ec77e8c53aae6610b2

Aliases

arxiv: 2605.14321 · arxiv_version: 2605.14321v1 · doi: 10.48550/arxiv.2605.14321 · pith_short_12: B4DZJ5G2EVNR · pith_short_16: B4DZJ5G2EVNRBKDE · pith_short_8: B4DZJ5G2

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/B4DZJ5G2EVNRBKDEBXOU4JEUX5 \
  | 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: 0f0794f4da255b10a8640ddd4e2494bf622327aae8afc2ec77e8c53aae6610b2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a2a42acba97399b2d184611aa46873dad439558909003856cb00ae9edd3444fc",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-14T03:34:08Z",
    "title_canon_sha256": "51162b8cfa560ebc9be84d3add6601169d47f85da43e3419ef31d1d51947f659"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14321",
    "kind": "arxiv",
    "version": 1
  }
}