{"title":"There is no perfect Mondrian partition for squares of side lengths less than 1001","authors":"Natalia García-Colín , Dimitri Leemans , Mia Müßig , Érika Roldán","doi":"10.1016/j.dam.2024.09.021","DOIUrl":null,"url":null,"abstract":"<div><div>In mathematics, a dissection of a square (or rectangle) into non-congruent rectangles is a Mondrian partition. If all the rectangles have the same area, it is called a perfect Mondrian partition. In this paper, we present a computational result by which we can affirm that there is no perfect Mondrian partition of a length <span><math><mi>n</mi></math></span> square for <span><math><mrow><mi>n</mi><mo>≤</mo><mn>1000</mn></mrow></math></span>. Using the same algorithm we have been able to establish that there is no perfect Mondrian partition of a <span><math><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></math></span> rectangle for <span><math><mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>≤</mo><mn>400</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"359 ","pages":"Pages 400-406"},"PeriodicalIF":1.0000,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X24004116","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
In mathematics, a dissection of a square (or rectangle) into non-congruent rectangles is a Mondrian partition. If all the rectangles have the same area, it is called a perfect Mondrian partition. In this paper, we present a computational result by which we can affirm that there is no perfect Mondrian partition of a length square for . Using the same algorithm we have been able to establish that there is no perfect Mondrian partition of a rectangle for .
期刊介绍：
The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal.
Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.
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