{"id":"https://openalex.org/W2796551842","doi":"https://doi.org/10.1080/00029890.2018.1427392","title":"Maximally Nontransitive Dice","display_name":"Maximally Nontransitive Dice","publication_year":2018,"publication_date":"2018-04-12","ids":{"openalex":"https://openalex.org/W2796551842","doi":"https://doi.org/10.1080/00029890.2018.1427392","mag":"2796551842"},"language":"en","primary_location":{"id":"doi:10.1080/00029890.2018.1427392","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2018.1427392","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020049691","display_name":"Joe Buhler","orcid":"https://orcid.org/0000-0002-2252-2358"},"institutions":[{"id":"https://openalex.org/I4210157118","display_name":"CCI Reprographics (United States)","ror":"https://ror.org/04d7j1v83","country_code":"US","type":"company","lineage":["https://openalex.org/I4210157118"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Joe Buhler","raw_affiliation_strings":["Center for Communications Research, San Diego, CA 92122"],"affiliations":[{"raw_affiliation_string":"Center for Communications Research, San Diego, CA 92122","institution_ids":["https://openalex.org/I4210157118"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071015671","display_name":"Ron Graham","orcid":null},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California, San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ron Graham","raw_affiliation_strings":["University of California, San Diego, CA 92093"],"affiliations":[{"raw_affiliation_string":"University of California, San Diego, CA 92093","institution_ids":["https://openalex.org/I36258959"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5112255690","display_name":"Al Hales","orcid":null},"institutions":[{"id":"https://openalex.org/I4210157118","display_name":"CCI Reprographics (United States)","ror":"https://ror.org/04d7j1v83","country_code":"US","type":"company","lineage":["https://openalex.org/I4210157118"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Al Hales","raw_affiliation_strings":["Center for Communications Research, San Diego, CA 92122"],"affiliations":[{"raw_affiliation_string":"Center for Communications Research, San Diego, CA 92122","institution_ids":["https://openalex.org/I4210157118"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5020049691"],"corresponding_institution_ids":["https://openalex.org/I4210157118"],"apc_list":null,"apc_paid":null,"fwci":2.2767,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.8727342,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"125","issue":"5","first_page":"387","last_page":"399"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12170","display_name":"History and Theory of Mathematics","score":0.9957000017166138,"subfield":{"id":"https://openalex.org/subfields/2614","display_name":"Theoretical Computer Science"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13523","display_name":"Mathematics, Computing, and Information Processing","score":0.9883999824523926,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/dice","display_name":"Dice","score":0.8911433219909668},{"id":"https://openalex.org/keywords/sketch","display_name":"Sketch","score":0.7718514204025269},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6587494611740112},{"id":"https://openalex.org/keywords/pairwise-comparison","display_name":"Pairwise comparison","score":0.6305880546569824},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.559752345085144},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.5345838069915771},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4711509943008423},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.41546663641929626},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3971061110496521},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.24014946818351746},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.19495326280593872},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.17142200469970703},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.13773971796035767},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.1358850598335266}],"concepts":[{"id":"https://openalex.org/C22029948","wikidata":"https://www.wikidata.org/wiki/Q45089","display_name":"Dice","level":2,"score":0.8911433219909668},{"id":"https://openalex.org/C2779231336","wikidata":"https://www.wikidata.org/wiki/Q7534724","display_name":"Sketch","level":2,"score":0.7718514204025269},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6587494611740112},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.6305880546569824},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.559752345085144},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5345838069915771},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4711509943008423},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.41546663641929626},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3971061110496521},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.24014946818351746},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.19495326280593872},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.17142200469970703},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.13773971796035767},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.1358850598335266},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00029890.2018.1427392","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2018.1427392","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/16","score":0.46000000834465027,"display_name":"Peace, Justice and strong institutions"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W193706516","https://openalex.org/W620038288","https://openalex.org/W1974627705","https://openalex.org/W1983337870","https://openalex.org/W2111971073","https://openalex.org/W2148712149","https://openalex.org/W2171037990","https://openalex.org/W2543664787","https://openalex.org/W2585289131","https://openalex.org/W2585490820","https://openalex.org/W2736667771","https://openalex.org/W2765946550","https://openalex.org/W2792711063","https://openalex.org/W2963806048","https://openalex.org/W3037735256","https://openalex.org/W3140802387","https://openalex.org/W4299432846"],"related_works":["https://openalex.org/W3104750253","https://openalex.org/W3021239166","https://openalex.org/W2586273397","https://openalex.org/W4366341510","https://openalex.org/W2390936256","https://openalex.org/W2483429559","https://openalex.org/W2016385589","https://openalex.org/W2009559548","https://openalex.org/W2906397153","https://openalex.org/W2385445039"],"abstract_inverted_index":{"We":[0],"construct":[1],"arbitrarily":[2],"large":[3],"sets":[4],"of":[5,30,34,50,52,55,62,70,82],"dice":[6],"with":[7],"some":[8],"remarkable":[9],"nontransitivity":[10],"properties.":[11],"In":[12],"a":[13,60,63,79,83],"sense":[14],"made":[15],"precise":[16],"later,":[17],"each":[18],"set":[19],"exhibits":[20],"all":[21],"possible":[22],"pairwise":[23],"win/loss":[24],"relationships":[25],"by":[26],"summing":[27],"different":[28],"numbers":[29],"rolls.":[31],"The":[32],"proof":[33,84],"this":[35],"fact":[36],"relies":[37],"on":[38],"an":[39],"asymptotic":[40,68],"formula":[41,58],"for":[42,75],"the":[43,46,71,86],"difference":[44],"between":[45],"median":[47],"and":[48],"mean":[49],"sums":[51],"multiple":[53],"rolls":[54],"dice.":[56],"This":[57],"is":[59],"consequence":[61],"suitable":[64],"Edgeworth":[65],"series":[66],"(an":[67],"refinement":[69],"central":[72],"limit":[73],"theorem),":[74],"which":[76],"we":[77],"give":[78],"detailed":[80],"sketch":[81],"in":[85],"final":[87],"section.":[88]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
