{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:DUJLT7NTQVSAK7DIITKOFHVONL","short_pith_number":"pith:DUJLT7NT","schema_version":"1.0","canonical_sha256":"1d12b9fdb38564057c6844d4e29eae6adb96d726620b8ae2d70b6d8809030e89","source":{"kind":"arxiv","id":"2009.03708","version":2},"attestation_state":"computed","paper":{"title":"Winning Strategy for the Multiplayer and Multialliance Zeckendorf Games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aidan Dunkelberg, Anna Cusenza, Clayton Mizgerd, Daniel Kleber, Dianhui Ke, Jingkai Ye, Kate Huffman, Steven J. Miller, Vashisth Tiwari, Xiaoyan Zheng","submitted_at":"2020-09-08T12:52:45Z","abstract_excerpt":"Edouard Zeckendorf proved that every positive integer $n$ can be uniquely written \\cite{Ze} as the sum of non-adjacent Fibonacci numbers, known as the Zeckendorf decomposition. Based on Zeckendorf's decomposition, we have the Zeckendorf game for multiple players. We show that when the Zeckendorf game has at least $3$ players, none of the players have a winning strategy for $n\\geq 5$. Then we extend the multi-player game to the multi-alliance game, finding some interesting situations in which no alliance has a winning strategy. This includes the two-alliance game, and some cases in which one al"},"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":"2009.03708","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-09-08T12:52:45Z","cross_cats_sorted":[],"title_canon_sha256":"73bd27b5d3850f02e98bfe0e33e950724aab9f56b2bc5216bfbc734abd406470","abstract_canon_sha256":"c9dbbbd1404ac2cc742e6981209ce5ef12763cd06b11ee6d950df3230da737be"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:44:50.573074Z","signature_b64":"eSOcz6NFTs+sJ1u20H+LFFdnDs+Cingz1ih/P0fAfwQriIBzRr8wTSYa8CervrnbY2168OMLzjyZBBBhHvvrAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d12b9fdb38564057c6844d4e29eae6adb96d726620b8ae2d70b6d8809030e89","last_reissued_at":"2026-07-05T01:44:50.572673Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:44:50.572673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Winning Strategy for the Multiplayer and Multialliance Zeckendorf Games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aidan Dunkelberg, Anna Cusenza, Clayton Mizgerd, Daniel Kleber, Dianhui Ke, Jingkai Ye, Kate Huffman, Steven J. Miller, Vashisth Tiwari, Xiaoyan Zheng","submitted_at":"2020-09-08T12:52:45Z","abstract_excerpt":"Edouard Zeckendorf proved that every positive integer $n$ can be uniquely written \\cite{Ze} as the sum of non-adjacent Fibonacci numbers, known as the Zeckendorf decomposition. Based on Zeckendorf's decomposition, we have the Zeckendorf game for multiple players. We show that when the Zeckendorf game has at least $3$ players, none of the players have a winning strategy for $n\\geq 5$. Then we extend the multi-player game to the multi-alliance game, finding some interesting situations in which no alliance has a winning strategy. This includes the two-alliance game, and some cases in which one al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.03708","kind":"arxiv","version":2},"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/2009.03708/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":"2009.03708","created_at":"2026-07-05T01:44:50.572729+00:00"},{"alias_kind":"arxiv_version","alias_value":"2009.03708v2","created_at":"2026-07-05T01:44:50.572729+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.03708","created_at":"2026-07-05T01:44:50.572729+00:00"},{"alias_kind":"pith_short_12","alias_value":"DUJLT7NTQVSA","created_at":"2026-07-05T01:44:50.572729+00:00"},{"alias_kind":"pith_short_16","alias_value":"DUJLT7NTQVSAK7DI","created_at":"2026-07-05T01:44:50.572729+00:00"},{"alias_kind":"pith_short_8","alias_value":"DUJLT7NT","created_at":"2026-07-05T01:44:50.572729+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DUJLT7NTQVSAK7DIITKOFHVONL","json":"https://pith.science/pith/DUJLT7NTQVSAK7DIITKOFHVONL.json","graph_json":"https://pith.science/api/pith-number/DUJLT7NTQVSAK7DIITKOFHVONL/graph.json","events_json":"https://pith.science/api/pith-number/DUJLT7NTQVSAK7DIITKOFHVONL/events.json","paper":"https://pith.science/paper/DUJLT7NT"},"agent_actions":{"view_html":"https://pith.science/pith/DUJLT7NTQVSAK7DIITKOFHVONL","download_json":"https://pith.science/pith/DUJLT7NTQVSAK7DIITKOFHVONL.json","view_paper":"https://pith.science/paper/DUJLT7NT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2009.03708&json=true","fetch_graph":"https://pith.science/api/pith-number/DUJLT7NTQVSAK7DIITKOFHVONL/graph.json","fetch_events":"https://pith.science/api/pith-number/DUJLT7NTQVSAK7DIITKOFHVONL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DUJLT7NTQVSAK7DIITKOFHVONL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DUJLT7NTQVSAK7DIITKOFHVONL/action/storage_attestation","attest_author":"https://pith.science/pith/DUJLT7NTQVSAK7DIITKOFHVONL/action/author_attestation","sign_citation":"https://pith.science/pith/DUJLT7NTQVSAK7DIITKOFHVONL/action/citation_signature","submit_replication":"https://pith.science/pith/DUJLT7NTQVSAK7DIITKOFHVONL/action/replication_record"}},"created_at":"2026-07-05T01:44:50.572729+00:00","updated_at":"2026-07-05T01:44:50.572729+00:00"}