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For a balanced and nontransitive $n$-tuple of dice $(A_1,A_2,\\dots,A_n)$, we define the winning probability $w(A_1,A_2,\\dots,A_n) := P(A_1 < A_2)$. The works of Trybula and Kim et al. together show that for a balanced and nontransitve triple of dice $(A_1,A_2,A_3)$, the least upper bound on the winning probability is $\\frac{-1+\\sqrt{5}}{2}$. Kim et al. th"},"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":"2505.21950","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-28T04:03:03Z","cross_cats_sorted":[],"title_canon_sha256":"32bb7cebc35c441ddc38be17e05b0bfa9b99c7146055aa9080e89e2b20953f63","abstract_canon_sha256":"968107306c0de1d686154a94a72005cef482d6d56b8e4c9685c3fb851f8c14d4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:11:06.673562Z","signature_b64":"XluBui20hI9IML9GfqHcQjEDIuihzvUSu/tgckM4T7fvbnC3d8fBfBZOB3T/7odCR9LEaRTo/LJp9xtu9vr/AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37a8755bb2f99881e4c45d5120976cd13fe5dd2713dfd4bfb216dd362caa4dff","last_reissued_at":"2026-07-05T11:11:06.673148Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:11:06.673148Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Winning Probabilities of Balanced and Nontransitive n-tuples of Dice","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Joshua Rooney","submitted_at":"2025-05-28T04:03:03Z","abstract_excerpt":"For a positive integer $n$, an $n$-tuple of dice $(A_1,A_2,\\dots,A_n)$ is called balanced if $P(A_1<A_2) = P(A_2<A_3) = \\cdots = P(A_n<A_1)$ and nontransitive if $P(A_1<A_2), P(A_2<A_3), \\dots, P(A_n<A_1)$ are each greater than $\\frac{1}{2}$. For a balanced and nontransitive $n$-tuple of dice $(A_1,A_2,\\dots,A_n)$, we define the winning probability $w(A_1,A_2,\\dots,A_n) := P(A_1 < A_2)$. The works of Trybula and Kim et al. together show that for a balanced and nontransitve triple of dice $(A_1,A_2,A_3)$, the least upper bound on the winning probability is $\\frac{-1+\\sqrt{5}}{2}$. 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