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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 $n = k+3 \\leq 6$."},"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":"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"},"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"},"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$^1_{n, =k}$. 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