{"id":"https://openalex.org/W2993673685","doi":"https://doi.org/10.37236/1874","title":"Tiling Parity Results and the Holey Square Solution","display_name":"Tiling Parity Results and the Holey Square Solution","publication_year":2005,"publication_date":"2005-05-16","ids":{"openalex":"https://openalex.org/W2993673685","doi":"https://doi.org/10.37236/1874","mag":"2993673685"},"language":"en","primary_location":{"id":"doi:10.37236/1874","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1874","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i2r17/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":null,"license_id":null,"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/v11i2r17/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5001438565","display_name":"Bridget Eileen Tenner","orcid":"https://orcid.org/0000-0003-0150-9653"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Bridget Eileen Tenner","raw_affiliation_strings":["Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5001438565"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":0.8721,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.73358526,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"11","issue":"2","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11428","display_name":"Advanced Mathematical Identities","score":0.9966999888420105,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11797","display_name":"graph theory and CDMA systems","score":0.9952999949455261,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/domino","display_name":"Domino","score":0.8813972473144531},{"id":"https://openalex.org/keywords/parity","display_name":"Parity (physics)","score":0.8312902450561523},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7826183438301086},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7164048552513123},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.716080367565155},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.7118815779685974},{"id":"https://openalex.org/keywords/substitution-tiling","display_name":"Substitution tiling","score":0.610292911529541},{"id":"https://openalex.org/keywords/square-tiling","display_name":"Square tiling","score":0.5583020448684692},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.5579596161842346},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3868985176086426},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09441739320755005},{"id":"https://openalex.org/keywords/grid","display_name":"Grid","score":0.07559269666671753},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06268495321273804}],"concepts":[{"id":"https://openalex.org/C2776416436","wikidata":"https://www.wikidata.org/wiki/Q3751781","display_name":"Domino","level":3,"score":0.8813972473144531},{"id":"https://openalex.org/C2777151079","wikidata":"https://www.wikidata.org/wiki/Q141160","display_name":"Parity (physics)","level":2,"score":0.8312902450561523},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7826183438301086},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7164048552513123},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.716080367565155},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.7118815779685974},{"id":"https://openalex.org/C21550011","wikidata":"https://www.wikidata.org/wiki/Q7632143","display_name":"Substitution tiling","level":2,"score":0.610292911529541},{"id":"https://openalex.org/C178650486","wikidata":"https://www.wikidata.org/wiki/Q1128619","display_name":"Square tiling","level":3,"score":0.5583020448684692},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.5579596161842346},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3868985176086426},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09441739320755005},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.07559269666671753},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06268495321273804},{"id":"https://openalex.org/C161790260","wikidata":"https://www.wikidata.org/wiki/Q82264","display_name":"Catalysis","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C109214941","wikidata":"https://www.wikidata.org/wiki/Q18334","display_name":"Particle physics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/1874","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1874","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i2r17/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":null,"license_id":null,"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:10.37236/1874","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1874","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i2r17/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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2993673685.pdf","grobid_xml":"https://content.openalex.org/works/W2993673685.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W1653035750","https://openalex.org/W1715153673","https://openalex.org/W2039134431","https://openalex.org/W2095754669","https://openalex.org/W2125416850","https://openalex.org/W2952336157","https://openalex.org/W6806866848"],"related_works":["https://openalex.org/W1567962740","https://openalex.org/W2952996678","https://openalex.org/W2188248099","https://openalex.org/W2046267131","https://openalex.org/W4214482358","https://openalex.org/W2950147731","https://openalex.org/W3102996575","https://openalex.org/W2024646371","https://openalex.org/W3213562055","https://openalex.org/W4320931267"],"abstract_inverted_index":{"We":[0,40],"prove":[1],"combinatorially":[2],"that":[3,52],"the":[4,7,18,21,36,53],"parity":[5,19,49],"of":[6,9,12,20,23,26,45,55,58,62],"number":[8,22,54],"domino":[10,24,56],"tilings":[11,25,57],"a":[13,27,59],"region":[14],"is":[15,64],"equal":[16],"to":[17],"particular":[28,60],"subregion.":[29],"Using":[30],"this":[31],"result":[32],"we":[33],"can":[34],"resolve":[35],"holey":[37],"square":[38],"conjecture.":[39],"furthermore":[41],"give":[42],"combinatorial":[43],"proofs":[44],"several":[46],"other":[47],"tiling":[48],"results,":[50],"including":[51],"family":[61],"rectangles":[63],"always":[65],"odd.":[66]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}