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A game ends when all pairs have been matched. We determine that, when the game is played optimally, as n tends to infinity: 1) The expected number of moves is (3 - 2 ln 2)n + 7/8 - 2 ln 2 (approximately 1.61 n), 2) The expected number of times two matching cards are unwitt"},"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":"1208.4854","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-08-23T20:04:51Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d4b262c34ea601901cbdf406d991d9cee1cbb4e3d54c641df8b5170c9b4f2c67","abstract_canon_sha256":"a2120d91720b976595cfc3c182e8bf74d1294050b70f822e819c8fc07772cdd2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:47:52.378137Z","signature_b64":"aDynDnhCJxvFKXsc6/3axbPrec3SHAJCp7uQKN/mXBT7hAjTtHzU094EtzasYBrfETgRIBaP6h+wdZQDc+UpAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"412b04617334df878671a80c8008bc031e1688c55bdce5c2536d1c8e4fb42ed0","last_reissued_at":"2026-05-18T03:47:52.377680Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:47:52.377680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Matching expectations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Daniel J. 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