{"id":"https://openalex.org/W7225035525","doi":"https://doi.org/10.5281/zenodo.23852983","title":"Six unit squares in a strip of height 1.87 \u2264 H < 2: the minimum width is F6(H)","display_name":"Six unit squares in a strip of height 1.87 \u2264 H < 2: the minimum width is F6(H)","publication_year":2026,"publication_date":"2026-11-01","ids":{"openalex":"https://openalex.org/W7225035525","doi":"https://doi.org/10.5281/zenodo.23852983"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23852983","doi":"https://doi.org/10.5281/zenodo.23852983","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23852983","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23852983","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5159027868","display_name":"Kevin Fang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Fang, Kevin","raw_affiliation_strings":["Independent researcher"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Independent researcher","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5159027868"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12176","display_name":"Optimization and Packing Problems","score":0.9926999807357788,"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"}},"topics":[{"id":"https://openalex.org/T12176","display_name":"Optimization and Packing Problems","score":0.9926999807357788,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.003700000001117587,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.0005000000237487257,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/strip-packing","display_name":"strip packing","score":0.7269999980926514},{"id":"https://openalex.org/keywords/branch-and-bound","display_name":"branch and bound","score":0.8740000128746033},{"id":"https://openalex.org/keywords/computer-assisted-proof","display_name":"computer-assisted proof","score":0.7950000166893005},{"id":"https://openalex.org/keywords/directed-rounding","display_name":"directed rounding","score":0.6679999828338623},{"id":"https://openalex.org/keywords/ieee-754","display_name":"IEEE 754","score":0.7620000243186951}],"concepts":[{"id":"https://openalex.org/C2781302577","wikidata":"https://www.wikidata.org/wiki/Q209","display_name":"Rectangle","level":2,"score":0.849399983882904},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8270999789237976},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6722999811172485},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6067000031471252},{"id":"https://openalex.org/C136625980","wikidata":"https://www.wikidata.org/wiki/Q663208","display_name":"Rounding","level":2,"score":0.5511000156402588},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.48809999227523804},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.4636000096797943},{"id":"https://openalex.org/C128394911","wikidata":"https://www.wikidata.org/wiki/Q1987578","display_name":"Unit interval","level":2,"score":0.39959999918937683},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.3880999982357025},{"id":"https://openalex.org/C199185054","wikidata":"https://www.wikidata.org/wiki/Q552299","display_name":"Chain (unit)","level":2,"score":0.3779999911785126},{"id":"https://openalex.org/C2777026412","wikidata":"https://www.wikidata.org/wiki/Q2684591","display_name":"Statement (logic)","level":2,"score":0.33399999141693115},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.3310999870300293},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.326200008392334},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3147999942302704},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.30820000171661377},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.3066999912261963},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.2937999963760376},{"id":"https://openalex.org/C173801870","wikidata":"https://www.wikidata.org/wiki/Q201413","display_name":"Heuristic","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.5281/zenodo.23852983","doi":"https://doi.org/10.5281/zenodo.23852983","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23852983","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.23852984","doi":"https://doi.org/10.5281/zenodo.23852984","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23852984","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.23865706","doi":"https://doi.org/10.5281/zenodo.23865706","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23865706","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.23852983","doi":"https://doi.org/10.5281/zenodo.23852983","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23852983","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":1,"referenced_works":["https://openalex.org/W7222948279"],"related_works":["https://openalex.org/W6922406383","https://openalex.org/W7223461664","https://openalex.org/W2141025769","https://openalex.org/W1543927914","https://openalex.org/W2139004098","https://openalex.org/W3212402724","https://openalex.org/W7101229957","https://openalex.org/W2128421821","https://openalex.org/W2917904214","https://openalex.org/W3133809200"],"abstract_inverted_index":{"For":[0],"every":[1,9],"H":[2,6,30],"with":[3,15,108,120,126],"1.87":[4],"\u2264":[5],"<":[7],"2,":[8],"packing":[10],"of":[11,25,39],"six":[12],"unit":[13],"squares":[14],"arbitrary":[16],"rotations":[17],"and":[18,28,100,116,129,138],"pairwise":[19],"disjoint":[20],"interiors":[21],"in":[22,144],"a":[23,90,105,113,117],"rectangle":[24],"width":[26],"L":[27,32],"height":[29],"has":[31],"\u2265":[33],"F6(H)":[34,70],"=":[35],"min":[36],"over":[37],"\u03b8":[38,49],"2":[40],"+":[41,50],"(4":[42,51],"\u2212":[43,45,52],"(3H":[44],"2)":[46],"cos":[47],"\u03b8)/sin":[48],"2H":[53],"sin":[54],"\u03b8)/cos":[55],"\u03b8;":[56],"the":[57,72,96,102,145,153],"bound":[58],"is":[59,71,78,157],"attained":[60],"by":[61],"Morandi's":[62],"configuration":[63],"(Friedman,":[64],"Math":[65],"Magic,":[66],"June":[67],"2010),":[68],"so":[69],"exact":[73],"minimum":[74],"width.":[75],"The":[76,148],"proof":[77],"computer-assisted:":[79],"an":[80,109,130],"elementary":[81],"reduction":[82,111],"from":[83],"packings":[84],"to":[85,112],"64":[86],"floor/ceiling":[87],"chain":[88],"functions,":[89],"certified":[91,118],"min-plus":[92],"dynamic":[93],"program":[94],"for":[95,101],"62":[97],"non-alternating":[98],"patterns,":[99],"alternating":[103],"pattern":[104],"support-function":[106],"dual":[107],"analytic":[110],"compact":[114],"box":[115],"branch-and-bound":[119],"outward":[121],"rounding.":[122],"Premises:":[123],"IEEE-754":[124],"binary64":[125],"directed":[127],"rounding":[128],"8-ulp":[131],"libm":[132],"padding.":[133],"Source,":[134],"certificates,":[135],"replay":[136],"commands":[137],"three":[139],"blind":[140],"referee":[141],"reports":[142],"are":[143],"linked":[146],"repositories.":[147],"interval":[149],"[1.8611,":[150],"1.87),":[151],"where":[152],"statement":[154],"remains":[155],"meaningful,":[156],"not":[158],"covered.":[159]},"counts_by_year":[],"updated_date":"2026-11-03T10:50:19.383937","created_date":"2026-11-02T00:00:00"}