{"id":"https://openalex.org/W1708519692","doi":"https://doi.org/10.37236/4579","title":"Sprague-Grundy Values of the $\\mathcal{R}$-Wythoff Game","display_name":"Sprague-Grundy Values of the $\\mathcal{R}$-Wythoff Game","publication_year":2015,"publication_date":"2015-04-21","ids":{"openalex":"https://openalex.org/W1708519692","doi":"https://doi.org/10.37236/4579","mag":"1708519692"},"language":"en","primary_location":{"id":"doi:10.37236/4579","is_oa":true,"landing_page_url":"https://doi.org/10.37236/4579","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v22i2p13/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v22i2p13/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5025386668","display_name":"Albert Gu","orcid":null},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Albert Gu","raw_affiliation_strings":["Carnegie Mellon University"],"affiliations":[{"raw_affiliation_string":"Carnegie Mellon University","institution_ids":["https://openalex.org/I74973139"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5025386668"],"corresponding_institution_ids":["https://openalex.org/I74973139"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.03022976,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"22","issue":"2","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11574","display_name":"Artificial Intelligence in Games","score":0.9904000163078308,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11574","display_name":"Artificial Intelligence in Games","score":0.9904000163078308,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12490","display_name":"Software Engineering and Design Patterns","score":0.9334999918937683,"subfield":{"id":"https://openalex.org/subfields/3303","display_name":"Development"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9146000146865845,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6904557347297668},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.6242622137069702},{"id":"https://openalex.org/keywords/value","display_name":"Value (mathematics)","score":0.585163414478302},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5422990322113037},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.09615421295166016}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6904557347297668},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6242622137069702},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.585163414478302},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5422990322113037},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.09615421295166016}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/4579","is_oa":true,"landing_page_url":"https://doi.org/10.37236/4579","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v22i2p13/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.37236/4579","is_oa":true,"landing_page_url":"https://doi.org/10.37236/4579","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v22i2p13/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1630644263","display_name":null,"funder_award_id":"H98230-11-1-0224","funder_id":"https://openalex.org/F4320311500","funder_display_name":"University of Minnesota Duluth"},{"id":"https://openalex.org/G1967552345","display_name":"REU Site: An Undergraduate Research Program in Discrete Mathematics","funder_award_id":"1062709","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2583634656","display_name":null,"funder_award_id":"NSF/DMS","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8324679094","display_name":null,"funder_award_id":"H98230-11-1-0224","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320309636","display_name":"University of Minnesota","ror":"https://ror.org/03grvy078"},{"id":"https://openalex.org/F4320311500","display_name":"University of Minnesota Duluth","ror":"https://ror.org/01hy4qx27"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1708519692.pdf","grobid_xml":"https://content.openalex.org/works/W1708519692.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W36302007","https://openalex.org/W2159565584","https://openalex.org/W4293258968"],"related_works":["https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694","https://openalex.org/W2042127053","https://openalex.org/W2142036596","https://openalex.org/W4238204885"],"abstract_inverted_index":{"We":[0,66],"examine":[1],"the":[2,6,31,37,48,69,84],"Sprague-Grundy":[3,63,72,87],"values":[4],"of":[5,8,12,28,40,50,55,62,71,83,86],"game":[7,14],"$\\mathcal{R}$-Wythoff,":[9],"a":[10,25,80],"restriction":[11],"Wythoff's":[13,56],"introduced":[15],"by":[16],"Ho,":[17],"where":[18],"each":[19],"move":[20],"is":[21],"either":[22],"to":[23,35],"remove":[24,36],"positive":[26],"number":[27,39],"tokens":[29,41],"from":[30,42],"larger":[32],"pile":[33],"or":[34],"same":[38],"both":[43],"piles.":[44],"Ho":[45],"showed":[46],"that":[47],"$P$-positions":[49],"$\\mathcal{R}$-Wythoff":[51],"agree":[52],"with":[53],"those":[54],"game,":[57],"and":[58,75,77],"found":[59],"all":[60,68],"positions":[61,70,85],"value":[64,73,88],"$1$.":[65],"describe":[67],"$2$":[74],"$3$,":[76],"also":[78],"conjecture":[79],"general":[81],"form":[82],"$g$.":[89]},"counts_by_year":[],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}