{"id":"https://openalex.org/W2141025769","doi":"https://doi.org/10.37236/1701","title":"Packing 10 or 11 Unit Squares in a Square","display_name":"Packing 10 or 11 Unit Squares in a Square","publication_year":2003,"publication_date":"2003-03-18","ids":{"openalex":"https://openalex.org/W2141025769","doi":"https://doi.org/10.37236/1701","mag":"2141025769"},"language":"en","primary_location":{"id":"doi:10.37236/1701","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1701","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v10i1r8/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":false,"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":"bronze","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v10i1r8/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088710002","display_name":"Walter Stromquist","orcid":"https://orcid.org/0000-0002-7361-4965"},"institutions":[{"id":"https://openalex.org/I102373834","display_name":"Bryn Mawr College","ror":"https://ror.org/05sjwtp51","country_code":"US","type":"education","lineage":["https://openalex.org/I102373834"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Walter Stromquist","raw_affiliation_strings":["Department of Mathematics Bryn Mawr College, Bryn Mawr, Pennsylvania, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics Bryn Mawr College, Bryn Mawr, Pennsylvania, USA","institution_ids":["https://openalex.org/I102373834"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5088710002"],"corresponding_institution_ids":["https://openalex.org/I102373834"],"apc_list":null,"apc_paid":null,"fwci":0.354,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.65901588,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"10","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9929999709129333,"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"}},"topics":[{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9929999709129333,"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"}},{"id":"https://openalex.org/T12176","display_name":"Optimization and Packing Problems","score":0.9896000027656555,"subfield":{"id":"https://openalex.org/subfields/2209","display_name":"Industrial and Manufacturing Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11729","display_name":"Product Development and Customization","score":0.9674000144004822,"subfield":{"id":"https://openalex.org/subfields/1405","display_name":"Management of Technology and Innovation"},"field":{"id":"https://openalex.org/fields/14","display_name":"Business, Management and Accounting"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8791012763977051},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.845721960067749},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8385387063026428},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.7518503665924072},{"id":"https://openalex.org/keywords/unit","display_name":"Unit (ring theory)","score":0.7161089181900024},{"id":"https://openalex.org/keywords/unit-square","display_name":"Unit square","score":0.5623006820678711},{"id":"https://openalex.org/keywords/circle-packing","display_name":"Circle packing","score":0.5431167483329773},{"id":"https://openalex.org/keywords/sphere-packing","display_name":"Sphere packing","score":0.4458268880844116},{"id":"https://openalex.org/keywords/packing-problems","display_name":"Packing problems","score":0.43609803915023804},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1245567798614502}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8791012763977051},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.845721960067749},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8385387063026428},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.7518503665924072},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.7161089181900024},{"id":"https://openalex.org/C7356347","wikidata":"https://www.wikidata.org/wiki/Q1307838","display_name":"Unit square","level":2,"score":0.5623006820678711},{"id":"https://openalex.org/C123115066","wikidata":"https://www.wikidata.org/wiki/Q5121501","display_name":"Circle packing","level":2,"score":0.5431167483329773},{"id":"https://openalex.org/C183893376","wikidata":"https://www.wikidata.org/wiki/Q900117","display_name":"Sphere packing","level":2,"score":0.4458268880844116},{"id":"https://openalex.org/C130253271","wikidata":"https://www.wikidata.org/wiki/Q3851477","display_name":"Packing problems","level":2,"score":0.43609803915023804},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1245567798614502},{"id":"https://openalex.org/C145420912","wikidata":"https://www.wikidata.org/wiki/Q853077","display_name":"Mathematics education","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.37236/1701","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1701","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v10i1r8/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":false,"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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.9.4417","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.9.4417","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/Volume_10/PDF/v10i1r8.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1701","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1701","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v10i1r8/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":false,"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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2141025769.pdf","grobid_xml":"https://content.openalex.org/works/W2141025769.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W2128421821","https://openalex.org/W2139004098"],"related_works":["https://openalex.org/W1993644547","https://openalex.org/W2731736014","https://openalex.org/W3100495456","https://openalex.org/W2568571446","https://openalex.org/W2135620139","https://openalex.org/W4251859436","https://openalex.org/W1977424097","https://openalex.org/W2057875651","https://openalex.org/W4301653932","https://openalex.org/W2131547375"],"abstract_inverted_index":{"Let":[0],"$s(n)$":[1],"be":[2],"the":[3,6,60],"side":[4,48],"of":[5,34],"smallest":[7],"square":[8],"into":[9],"which":[10,64],"it":[11],"is":[12,59],"possible":[13],"pack":[14],"$n$":[15],"unit":[16,36],"squares.":[17],"We":[18,27],"show":[19,29],"that":[20,24,30,57],"$s(10)=3+\\sqrt{1\\over":[21],"2}\\approx3.707$":[22],"and":[23],"$s(11)\\geq2+2\\sqrt{4\\over":[25],"5}\\approx3.789$.":[26],"also":[28],"an":[31,65],"optimal":[32,66],"packing":[33],"$11$":[35],"squares":[37],"with":[38],"orientations":[39],"limited":[40],"to":[41],"$0$":[42],"degrees":[43,46],"or":[44],"$45$":[45],"has":[47],"$2+2\\sqrt{8\\over":[49],"9}\\approx3.886$.":[50],"These":[51],"results":[52],"prove":[53],"Martin":[54],"Gardner's":[55],"conjecture":[56],"$n=11$":[58],"first":[61],"case":[62],"in":[63],"result":[67],"requires":[68],"a":[69],"non-$45$":[70],"degree":[71],"packing.":[72]},"counts_by_year":[{"year":2020,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
