{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20585"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20585","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On Tait's color-tiling problem","abstract":"In this thesis, we study a problem that generalizes a color tiling problem studied by the Scottish mathematician P. G. Tait in 1883, which has been of interest in China for many decades.","abstract_html":"In this thesis, we study a problem that generalizes a color tiling problem studied by the Scottish mathematician P. G. Tait in 1883, which has been of interest in China for many decades.","abstract_has_math":false,"creators":["Hu, Zhu-Xin"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Weichsel, Paul M."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:43:28Z","date_published":"2011-05-07T12:43:28Z","updated_at":"2026-07-22T22:25:16Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1996 Hu, Zhu-Xin"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["9780591199055","AAI9712315","(UMI)AAI9712315"],"render_values":[{"text":"9780591199055","href":null,"code":true},{"text":"AAI9712315","href":null,"code":true},{"text":"(UMI)AAI9712315","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20585","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Weichsel, Paul M."]},{"key":"dc:creator","label":"Author","values":["Hu, Zhu-Xin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:43:28Z","10000-01-01","1996"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1996 Hu, Zhu-Xin"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["9780591199055","AAI9712315","(UMI)AAI9712315","http://hdl.handle.net/2142/20585"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis, we study a problem that generalizes a color tiling problem studied by the Scottish mathematician P. G. Tait in 1883, which has been of interest in China for many decades.","Let l, m, t be positive integers with $m\\mid l$ and let $n\\sb1,\\ n\\sb2,\\...,\\ n\\sb{t}$ be nonnegative integers. We consider sequences (also called strings) with $n\\sb1+n\\sb2+\\...+n\\sb{t}+l$ positions in a line. Of these positions, $n\\sb1$ are filled with tiles of color 1, $n\\sb2$ are filled with tiles of color 2, $\\...,$ $n\\sb{t}$ are filled with tiles of color t, and the remaining l are left empty, indicated by 0. A segment of a string is a substring consisting of tiles or empty positions in contiguous positions. A solid segment is a segment containing no empty positions. An O-segment is a segment consisting of contiguous empty positions (like 00$\\...$0). We say an O-segment is a maximal O-segment if it is not a part of a longer O-segment. A sequence is called a $(l,m;n\\sb2,\\...,n\\sb{t})$-sequence if (a) the length of its longest solid segment is at least m, and (b) the length of each of its maximal O-segments is a multiple of m. An m-embedding permutation (m-EP) on a $(l,m;n\\sb1,n\\sb2,\\... n\\sb{t})$-sequence is a transformation that moves m contiguous pieces (without alternating their relative positions) into m contiguous empty positions such that the resulting sequence is also a $(l,m;n\\sb1,n\\sb2,\\... n\\sb{t})$-sequence. For any two $(l,m;n\\sb1,n\\sb2,\\... n\\sb{t})$-sequences X and Y, we define $d(X, Y)$ to be the minimum number of m-EPs to transform X into Y if it can be done so otherwise we define $d(X, Y)=\\infty.$","In Chapter 1, we study the general t-color $(l,m;n\\sb1,n\\sb2,\\...,n\\sb{t})$-sequences. We obtained conditions with which the distance between any two $(l,m;n\\sb1,n\\sb2,\\..., n\\sb{t})$-sequences is bounded above by a linear function of $l+n\\sb1+n\\sb2+\\...+n\\sb{t}.$","In Chapter 2, we prove a long-standing conjecture. Let $X\\sb0=(12)\\sp{n}0\\sp3,$ and $B(3, 3; n, n)=\\{1\\sp{n}2\\sp{n}0\\sp3,\\ 2\\sp{n}1\\sp{n}0\\sp3,\\ 0\\sp32\\sp{n}1\\sp{n}\\}.$ C. Y. Chiang, in a paper in 1936, made the following conjecture: If n is an even integer $\\ge$6, then for any $X\\in B(3, 3; n, n),$ we have $d(X\\sb0,X)=n+1.$","A consequence of the results of Chapter 2 is that Chiang's conjecture is true.","Made available in DSpace on 2011-05-07T12:43:28Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9712315.pdf: 3662160 bytes, checksum: 7041f55e2f26726b8ed881f1a616af0d (MD5) Previous issue date: 1996","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:44:53Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:19:49-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["On Tait's color-tiling problem"]}]}],"canonical_facts":{"dc:contributor":["Weichsel, Paul M."],"dc:creator":["Hu, Zhu-Xin"],"dc:date":["2011-05-07T12:43:28Z","10000-01-01","1996"],"dc:description":["In this thesis, we study a problem that generalizes a color tiling problem studied by the Scottish mathematician P. G. Tait in 1883, which has been of interest in China for many decades.","Let l, m, t be positive integers with $m\\mid l$ and let $n\\sb1,\\ n\\sb2,\\...,\\ n\\sb{t}$ be nonnegative integers. We consider sequences (also called strings) with $n\\sb1+n\\sb2+\\...+n\\sb{t}+l$ positions in a line. Of these positions, $n\\sb1$ are filled with tiles of color 1, $n\\sb2$ are filled with tiles of color 2, $\\...,$ $n\\sb{t}$ are filled with tiles of color t, and the remaining l are left empty, indicated by 0. A segment of a string is a substring consisting of tiles or empty positions in contiguous positions. A solid segment is a segment containing no empty positions. An O-segment is a segment consisting of contiguous empty positions (like 00$\\...$0). We say an O-segment is a maximal O-segment if it is not a part of a longer O-segment. A sequence is called a $(l,m;n\\sb2,\\...,n\\sb{t})$-sequence if (a) the length of its longest solid segment is at least m, and (b) the length of each of its maximal O-segments is a multiple of m. An m-embedding permutation (m-EP) on a $(l,m;n\\sb1,n\\sb2,\\... n\\sb{t})$-sequence is a transformation that moves m contiguous pieces (without alternating their relative positions) into m contiguous empty positions such that the resulting sequence is also a $(l,m;n\\sb1,n\\sb2,\\... n\\sb{t})$-sequence. For any two $(l,m;n\\sb1,n\\sb2,\\... n\\sb{t})$-sequences X and Y, we define $d(X, Y)$ to be the minimum number of m-EPs to transform X into Y if it can be done so otherwise we define $d(X, Y)=\\infty.$","In Chapter 1, we study the general t-color $(l,m;n\\sb1,n\\sb2,\\...,n\\sb{t})$-sequences. We obtained conditions with which the distance between any two $(l,m;n\\sb1,n\\sb2,\\..., n\\sb{t})$-sequences is bounded above by a linear function of $l+n\\sb1+n\\sb2+\\...+n\\sb{t}.$","In Chapter 2, we prove a long-standing conjecture. Let $X\\sb0=(12)\\sp{n}0\\sp3,$ and $B(3, 3; n, n)=\\{1\\sp{n}2\\sp{n}0\\sp3,\\ 2\\sp{n}1\\sp{n}0\\sp3,\\ 0\\sp32\\sp{n}1\\sp{n}\\}.$ C. Y. Chiang, in a paper in 1936, made the following conjecture: If n is an even integer $\\ge$6, then for any $X\\in B(3, 3; n, n),$ we have $d(X\\sb0,X)=n+1.$","A consequence of the results of Chapter 2 is that Chiang's conjecture is true.","Made available in DSpace on 2011-05-07T12:43:28Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9712315.pdf: 3662160 bytes, checksum: 7041f55e2f26726b8ed881f1a616af0d (MD5) Previous issue date: 1996","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:44:53Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:19:49-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["9780591199055","AAI9712315","(UMI)AAI9712315","http://hdl.handle.net/2142/20585"],"dc:language":["eng"],"dc:rights":["Copyright 1996 Hu, Zhu-Xin"],"dc:subject":["Mathematics"],"dc:title":["On Tait's color-tiling problem"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:16Z"}