{"id":"https://openalex.org/W1540109178","doi":null,"title":"Generalised Euler's knight","display_name":"Generalised Euler's knight","publication_year":1998,"publication_date":"1998-01-01","ids":{"openalex":"https://openalex.org/W1540109178","mag":"1540109178"},"language":"en","primary_location":{"id":"pmh:oai:HAL:lirmm-01079127v1","doi":null,"is_oa":true,"landing_page_url":"https://hal.science/lirmm-01079127","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4405252976","host_organization_name":"Centre pour la Communication Scientifique Directe","host_organization_lineage":["https://openalex.org/I4405252976"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"1998","raw_type":"info:eu-repo/semantics/preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":[],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://hal.science/lirmm-01079127","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5111458189","display_name":"Nicolas Beldiceanu","orcid":"https://orcid.org/0000-0003-1452-596X","observed_orcids":["https://orcid.org/0000-0003-1452-596X"]},"institutions":[{"id":"https://openalex.org/I4210165258","display_name":"Laboratoire d'informatique de Nantes Atlantique","ror":"https://ror.org/05487pd16","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I205703379","https://openalex.org/I4210127572","https://openalex.org/I4210159245","https://openalex.org/I4210165258","https://openalex.org/I97188460"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Beldiceanu, Nicolas","raw_affiliation_strings":["LINA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LINA","institution_ids":["https://openalex.org/I4210165258"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5083178652","display_name":"Eric Bourreau","orcid":"https://orcid.org/0000-0002-3641-5609","observed_orcids":["https://orcid.org/0000-0002-3641-5609"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bourreau, Eric","raw_affiliation_strings":["Hors \u00c9quipe"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Hors \u00c9quipe","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5039254305","display_name":"Helmut Simonis","orcid":"https://orcid.org/0009-0003-6117-2463","observed_orcids":["https://orcid.org/0009-0003-6117-2463"]},"institutions":[{"id":"https://openalex.org/I27577105","display_name":"University College Cork","ror":"https://ror.org/03265fv13","country_code":"IE","type":"education","lineage":["https://openalex.org/I27577105"]}],"countries":["IE"],"is_corresponding":false,"raw_author_name":"Simonis, Helmut","raw_affiliation_strings":["Cork Constraint Computation Centre"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Cork Constraint Computation Centre","institution_ids":["https://openalex.org/I27577105"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5014471579","display_name":"Abderrahmane Aggoun","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4210093004","display_name":"Cosytec (France)","ror":"https://ror.org/00ek95971","country_code":"FR","type":"company","lineage":["https://openalex.org/I4210093004"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Aggoun, Abderrahmane","raw_affiliation_strings":["Cosytec"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Cosytec","institution_ids":["https://openalex.org/I4210093004"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"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/T11596","display_name":"Constraint Satisfaction and Optimization","score":0.5067999958992004,"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/T11596","display_name":"Constraint Satisfaction and Optimization","score":0.5067999958992004,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.3765000104904175,"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"}},{"id":"https://openalex.org/T12401","display_name":"Scheduling and Timetabling Solutions","score":0.04320000112056732,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/knight-s-tour","display_name":"knight's tour","score":0.9570000171661377},{"id":"https://openalex.org/keywords/combinatorial-enumeration","display_name":"combinatorial enumeration","score":0.6690000295639038}],"concepts":[{"id":"https://openalex.org/C71140485","wikidata":"https://www.wikidata.org/wiki/Q2943226","display_name":"Knight","level":2,"score":0.7990637421607971},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7099239826202393},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.6068955659866333},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5626038908958435},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4617052674293518},{"id":"https://openalex.org/C85079727","wikidata":"https://www.wikidata.org/wiki/Q3560114","display_name":"Quarter (Canadian coin)","level":2,"score":0.4340004622936249},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3393076956272125},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1840325891971588},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.15853798389434814},{"id":"https://openalex.org/C205649164","wikidata":"https://www.wikidata.org/wiki/Q1071","display_name":"Geography","level":0,"score":0.07234743237495422},{"id":"https://openalex.org/C166957645","wikidata":"https://www.wikidata.org/wiki/Q23498","display_name":"Archaeology","level":1,"score":0.0},{"id":"https://openalex.org/C1276947","wikidata":"https://www.wikidata.org/wiki/Q333","display_name":"Astronomy","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"pmh:oai:HAL:lirmm-01079127v1","doi":null,"is_oa":true,"landing_page_url":"https://hal.science/lirmm-01079127","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4405252976","host_organization_name":"Centre pour la Communication Scientifique Directe","host_organization_lineage":["https://openalex.org/I4405252976"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"1998","raw_type":"info:eu-repo/semantics/preprint"}],"best_oa_location":{"id":"pmh:oai:HAL:lirmm-01079127v1","doi":null,"is_oa":true,"landing_page_url":"https://hal.science/lirmm-01079127","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4405252976","host_organization_name":"Centre pour la Communication Scientifique Directe","host_organization_lineage":["https://openalex.org/I4405252976"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"1998","raw_type":"info:eu-repo/semantics/preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W65937437","https://openalex.org/W1533046786","https://openalex.org/W1975129671","https://openalex.org/W1993850412","https://openalex.org/W2059207171","https://openalex.org/W2316727582"],"related_works":["https://openalex.org/W7100771279","https://openalex.org/W3012497277","https://openalex.org/W6922640735","https://openalex.org/W7098504449","https://openalex.org/W3020657308","https://openalex.org/W2354783493","https://openalex.org/W1994484357","https://openalex.org/W195492470","https://openalex.org/W7216672133","https://openalex.org/W2085574526"],"abstract_inverted_index":{"Euler,":[0],"Vandermonde,":[1],"Dudeney,":[2],"Schwesk,":[3],"Berliner,":[4],"Conrad":[5],"and":[6,42,130,159,173,188],"many":[7],"others":[8],"already":[9],"considered":[10],"knight's":[11,17],"tours":[12],"on":[13,24,214],"chessboards.":[14],"The":[15],"classical":[16],"tour":[18],"problem":[19,58,185],"consist":[20],"of":[21,32,71,80,112,167,196],"finding":[22],"out":[23],"a":[25,30,69,78,85,105,215],"N":[26],"\u00d7":[27],"M":[28],"chessboard":[29,76],"sequence":[31],"legal":[33],"knight":[34,221],"moves":[35],"that":[36,87,116,151,190,219],"visit":[37],"every":[38],"cell":[39,89],"exactly":[40,223],"once":[41],"finish":[43],"by":[44,77],"returning":[45],"to":[46,55,59,91,101,182],"the":[47,57,72,99,145,162,165,197],"initial":[48],"cell.":[49],"A":[50],"more":[51,60,192],"challenging":[52],"question":[53],"is":[54,201,206],"generalise":[56],"than":[61,193],"one":[62,92,194],"knight.":[63],"More":[64],"precisely,":[65],"we":[66,96,114,160],"search":[67],"for":[68,164,191,207,211],"partitioning":[70],"m":[73,147],"x":[74,122,124,132,134,148,155,175],"n":[75,149],"set":[79],"C":[81],"cycles":[82,100],"in":[83],"such":[84,150,218],"way":[86],"each":[88,117,220],"belongs":[90],"single":[93],"cycle.":[94],"Moreover":[95],"impose":[97],"all":[98,144],"be":[102],"balanced.":[103],"Since":[104],"cycle":[106,118],"can't":[107],"have":[108],"an":[109],"odd":[110],"number":[111,166],"cells,":[113],"enforce":[115],"visits":[119,222],"between":[120,171],"2":[121,131],"floor(floor((m":[123],"n)":[125,135,156,176],"/":[126,128,136,138,177],"c)":[127,137],"2)":[129,139],"cell(cell((m":[133],"cells.":[140],"We":[141,179],"systematically":[142],"consider":[143],"boards":[146],"1<m<11,":[152],"1<n<11,":[153],"(m":[154],"mod":[157],"2=0":[158],"solve":[161],"problems":[163],"knights":[168,213],"c":[169],"varying":[170],"1":[172],"floor((m":[174],"2).":[178],"show":[180],"how":[181],"express":[183],"this":[184],"with":[186],"CHIP":[187],"prove":[189],"quarter":[195],"820":[198],"instances":[199],"there":[200],"no":[202,209],"balanced":[203],"solution":[204,210],"(there":[205],"instance":[208],"6":[212,224],"6x6":[216],"board":[217],"cells).":[225]},"counts_by_year":[],"updated_date":"2026-10-25T09:29:13.331239","created_date":"2016-06-24T00:00:00"}
