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For $k=2$ the game was solved for $n \\leq 6$. For $n \\leq 4$ the Sprague-Grundy function was efficiently computed (for both the normal and mis\\`ere versions). For $n = 5,6$ a polynomial algorithm computing P-positions was obtained. Here we consider the case $2 \\leq k = n-1$ and compute Smith's remoteness function, w"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2304.06498","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-04-13T13:13:26Z","cross_cats_sorted":[],"title_canon_sha256":"fca2e741e0d3b1eb51065bca30427115f828ef206ce294805ebcbe00838bd502","abstract_canon_sha256":"cac4ed7b6b3860128312ee108bd538eb73454104b60f4a1ff2e709b63ae71b7a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:00:44.410154Z","signature_b64":"8PpfERwz9AB1AR9BFhCeknqQPFIAa7LzSgtsbjxglAi+54bnPsnXvSpSUMfJh/nXEujGDGythpRErpOy+J17AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"acb3344a7c84b7338763db1e23a3dd1121d66bc49068eef768ea973ba1dc4e85","last_reissued_at":"2026-07-05T06:00:44.409649Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:00:44.409649Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Remoteness Functions of Exact Slow $k$-NIM with $k+1$ Piles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D. Martynov, M. Vyalyi, V. Gurvich, V. Maximchuk","submitted_at":"2023-04-13T13:13:26Z","abstract_excerpt":"Given integer $n$ and $k$ such that $0 < k \\leq n$ and $n$ piles of stones, two player alternate turns. By one move it is allowed to choose any $k$ piles and remove exactly one stone from each. The player who has to move but cannot is the loser. Cases $k=1$ and $k = n$ are trivial. For $k=2$ the game was solved for $n \\leq 6$. For $n \\leq 4$ the Sprague-Grundy function was efficiently computed (for both the normal and mis\\`ere versions). For $n = 5,6$ a polynomial algorithm computing P-positions was obtained. Here we consider the case $2 \\leq k = n-1$ and compute Smith's remoteness function, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.06498","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2304.06498/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2304.06498","created_at":"2026-07-05T06:00:44.409713+00:00"},{"alias_kind":"arxiv_version","alias_value":"2304.06498v1","created_at":"2026-07-05T06:00:44.409713+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2304.06498","created_at":"2026-07-05T06:00:44.409713+00:00"},{"alias_kind":"pith_short_12","alias_value":"VSZTIST4QS3T","created_at":"2026-07-05T06:00:44.409713+00:00"},{"alias_kind":"pith_short_16","alias_value":"VSZTIST4QS3THB3D","created_at":"2026-07-05T06:00:44.409713+00:00"},{"alias_kind":"pith_short_8","alias_value":"VSZTIST4","created_at":"2026-07-05T06:00:44.409713+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2404.06608","citing_title":"On the $\\mathcal{P}$-positions of some infinite families of Slow $A$-Nim","ref_index":11,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VSZTIST4QS3THB3D3MPCHI65CE","json":"https://pith.science/pith/VSZTIST4QS3THB3D3MPCHI65CE.json","graph_json":"https://pith.science/api/pith-number/VSZTIST4QS3THB3D3MPCHI65CE/graph.json","events_json":"https://pith.science/api/pith-number/VSZTIST4QS3THB3D3MPCHI65CE/events.json","paper":"https://pith.science/paper/VSZTIST4"},"agent_actions":{"view_html":"https://pith.science/pith/VSZTIST4QS3THB3D3MPCHI65CE","download_json":"https://pith.science/pith/VSZTIST4QS3THB3D3MPCHI65CE.json","view_paper":"https://pith.science/paper/VSZTIST4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2304.06498&json=true","fetch_graph":"https://pith.science/api/pith-number/VSZTIST4QS3THB3D3MPCHI65CE/graph.json","fetch_events":"https://pith.science/api/pith-number/VSZTIST4QS3THB3D3MPCHI65CE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VSZTIST4QS3THB3D3MPCHI65CE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VSZTIST4QS3THB3D3MPCHI65CE/action/storage_attestation","attest_author":"https://pith.science/pith/VSZTIST4QS3THB3D3MPCHI65CE/action/author_attestation","sign_citation":"https://pith.science/pith/VSZTIST4QS3THB3D3MPCHI65CE/action/citation_signature","submit_replication":"https://pith.science/pith/VSZTIST4QS3THB3D3MPCHI65CE/action/replication_record"}},"created_at":"2026-07-05T06:00:44.409713+00:00","updated_at":"2026-07-05T06:00:44.409713+00:00"}