{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JFWNFTFHORDPDUMXEDQJTQ4I3Z","short_pith_number":"pith:JFWNFTFH","canonical_record":{"source":{"id":"1508.05777","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-24T12:24:39Z","cross_cats_sorted":[],"title_canon_sha256":"85ee88e2b0803c4bf8f9a65e90cdf112dc85b7d890b35862521c3f112f5663a6","abstract_canon_sha256":"e41d08db7d38181707008b2850b963ded9ac6585791268e65194185beb54866c"},"schema_version":"1.0"},"canonical_sha256":"496cd2cca77446f1d19720e099c388de7766ff6aa0d9b342c27fcb4be82109fc","source":{"kind":"arxiv","id":"1508.05777","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.05777","created_at":"2026-05-18T01:34:51Z"},{"alias_kind":"arxiv_version","alias_value":"1508.05777v1","created_at":"2026-05-18T01:34:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.05777","created_at":"2026-05-18T01:34:51Z"},{"alias_kind":"pith_short_12","alias_value":"JFWNFTFHORDP","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JFWNFTFHORDPDUMX","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JFWNFTFH","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JFWNFTFHORDPDUMXEDQJTQ4I3Z","target":"record","payload":{"canonical_record":{"source":{"id":"1508.05777","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-24T12:24:39Z","cross_cats_sorted":[],"title_canon_sha256":"85ee88e2b0803c4bf8f9a65e90cdf112dc85b7d890b35862521c3f112f5663a6","abstract_canon_sha256":"e41d08db7d38181707008b2850b963ded9ac6585791268e65194185beb54866c"},"schema_version":"1.0"},"canonical_sha256":"496cd2cca77446f1d19720e099c388de7766ff6aa0d9b342c27fcb4be82109fc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:51.673049Z","signature_b64":"h+LyRaJ5b64aHYPF03L2JMtrpMG8N5ow7hVjivOCbEZBiM0EMjP9AcCX/zOiQv+WeFJRWLQ5J5tNTeDjWoM0Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"496cd2cca77446f1d19720e099c388de7766ff6aa0d9b342c27fcb4be82109fc","last_reissued_at":"2026-05-18T01:34:51.672534Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:51.672534Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.05777","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:34:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gPQlygG4d18HarA/2lTgtPonAaJg/64hNChtfD9SCSa5g19ZA3Wf3DS/4Rr2lnP4kPom7P3Igthc8hLppNVpBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:39:59.519086Z"},"content_sha256":"dade28ea88eadf149a718f34d3e866a37dd0cf588081ab026c3ffb9e9f5d06fe","schema_version":"1.0","event_id":"sha256:dade28ea88eadf149a718f34d3e866a37dd0cf588081ab026c3ffb9e9f5d06fe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JFWNFTFHORDPDUMXEDQJTQ4I3Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Slow $k$-Nim","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nhan Bao Ho, Vladimir Gurvich","submitted_at":"2015-08-24T12:24:39Z","abstract_excerpt":"Given $n$ piles of tokens and a positive integer $k \\leq n$, we study the following two impartial combinatorial games Nim$^1_{n, \\leq k}$ and Nim$^1_{n, =k}$. In the first (resp. second) game, a player, by one move, chooses at least $1$ and at most (resp. exactly) $k$ non-empty piles and removes one token from each of these piles. For the normal and mis\\`ere version of each game we compute the Sprague-Grundy function for the cases $n = k = 2$ and $n = k+1 = 3$. For game Nim$^1_{n, \\leq k}$ we also characterize its P-positions for the cases $n \\leq k+2$ and $n = k+3 \\leq 6$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05777","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:34:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mhu3iQ8tHPmqPU5l1troNvjpUZtHpHwEnp99bBHRIdlqv7rbivvNc8NU2ofAQjRSMN2iJ+jgkRszz2IQHVtUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:39:59.519481Z"},"content_sha256":"924ab84d54b67cfdb247af6df7764438aa21ecb9f96059205ba909a4a16563ea","schema_version":"1.0","event_id":"sha256:924ab84d54b67cfdb247af6df7764438aa21ecb9f96059205ba909a4a16563ea"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JFWNFTFHORDPDUMXEDQJTQ4I3Z/bundle.json","state_url":"https://pith.science/pith/JFWNFTFHORDPDUMXEDQJTQ4I3Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JFWNFTFHORDPDUMXEDQJTQ4I3Z/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T02:39:59Z","links":{"resolver":"https://pith.science/pith/JFWNFTFHORDPDUMXEDQJTQ4I3Z","bundle":"https://pith.science/pith/JFWNFTFHORDPDUMXEDQJTQ4I3Z/bundle.json","state":"https://pith.science/pith/JFWNFTFHORDPDUMXEDQJTQ4I3Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JFWNFTFHORDPDUMXEDQJTQ4I3Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JFWNFTFHORDPDUMXEDQJTQ4I3Z","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e41d08db7d38181707008b2850b963ded9ac6585791268e65194185beb54866c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-24T12:24:39Z","title_canon_sha256":"85ee88e2b0803c4bf8f9a65e90cdf112dc85b7d890b35862521c3f112f5663a6"},"schema_version":"1.0","source":{"id":"1508.05777","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.05777","created_at":"2026-05-18T01:34:51Z"},{"alias_kind":"arxiv_version","alias_value":"1508.05777v1","created_at":"2026-05-18T01:34:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.05777","created_at":"2026-05-18T01:34:51Z"},{"alias_kind":"pith_short_12","alias_value":"JFWNFTFHORDP","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JFWNFTFHORDPDUMX","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JFWNFTFH","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:924ab84d54b67cfdb247af6df7764438aa21ecb9f96059205ba909a4a16563ea","target":"graph","created_at":"2026-05-18T01:34:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Given $n$ piles of tokens and a positive integer $k \\leq n$, we study the following two impartial combinatorial games Nim$^1_{n, \\leq k}$ and Nim$^1_{n, =k}$. In the first (resp. second) game, a player, by one move, chooses at least $1$ and at most (resp. exactly) $k$ non-empty piles and removes one token from each of these piles. For the normal and mis\\`ere version of each game we compute the Sprague-Grundy function for the cases $n = k = 2$ and $n = k+1 = 3$. For game Nim$^1_{n, \\leq k}$ we also characterize its P-positions for the cases $n \\leq k+2$ and $n = k+3 \\leq 6$.","authors_text":"Nhan Bao Ho, Vladimir Gurvich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-24T12:24:39Z","title":"Slow $k$-Nim"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05777","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:dade28ea88eadf149a718f34d3e866a37dd0cf588081ab026c3ffb9e9f5d06fe","target":"record","created_at":"2026-05-18T01:34:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e41d08db7d38181707008b2850b963ded9ac6585791268e65194185beb54866c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-24T12:24:39Z","title_canon_sha256":"85ee88e2b0803c4bf8f9a65e90cdf112dc85b7d890b35862521c3f112f5663a6"},"schema_version":"1.0","source":{"id":"1508.05777","kind":"arxiv","version":1}},"canonical_sha256":"496cd2cca77446f1d19720e099c388de7766ff6aa0d9b342c27fcb4be82109fc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"496cd2cca77446f1d19720e099c388de7766ff6aa0d9b342c27fcb4be82109fc","first_computed_at":"2026-05-18T01:34:51.672534Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:51.672534Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h+LyRaJ5b64aHYPF03L2JMtrpMG8N5ow7hVjivOCbEZBiM0EMjP9AcCX/zOiQv+WeFJRWLQ5J5tNTeDjWoM0Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:51.673049Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.05777","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dade28ea88eadf149a718f34d3e866a37dd0cf588081ab026c3ffb9e9f5d06fe","sha256:924ab84d54b67cfdb247af6df7764438aa21ecb9f96059205ba909a4a16563ea"],"state_sha256":"e860911672cf6501a81e049f0460b713f6eb5c3aa9410c779017b1d9cb9029e3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oMzqemJBeE87p2DNAPcCAnvEOVB9pAXFsV15BVWVfAFxUxPMIJtWlDouuXkOurM4p5RR+fJhbD8CAnAxIiLZAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T02:39:59.521847Z","bundle_sha256":"c3c9c3a89cf837b70f1400e01cdfd23dec8c1796f58b9df29e874c598e703c4e"}}