{"paper":{"title":"Intransitive Dice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Andrew Liu, Brian Conrey, James Gabbard, Katie Grant, Kent Morrison","submitted_at":"2013-11-25T23:05:45Z","abstract_excerpt":"We consider $n$-sided dice whose face values lie between $1$ and $n$ and whose faces sum to $n(n+1)/2$. For two dice $A$ and $B$, define $A \\succ B$ if it is more likely for $A$ to show a higher face than $B$. Suppose $k$ such dice $A_1,\\dots,A_k$ are randomly selected. We conjecture that the probability of ties goes to 0 as $n$ grows. We conjecture and provide some supporting evidence that---contrary to intuition---each of the $2^{k \\choose 2}$ assignments of $\\succ$ or $\\prec$ to each pair is equally likely asymptotically. For a specific example, suppose we randomly select $k$ dice $A_1,\\dot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6511","kind":"arxiv","version":2},"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"}