{"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 $\\frac{-1+\\sqrt{5}}{2}$. Kim et al. th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.21950","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.21950/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"}