{"title":"Repeatedly applying the Combinatorial Nullstellensatz for Zero-sum Grids to Martin Gardner’s minimum no-3-in-a-line problem","authors":"Seunghwan Oh , John R. Schmitt , Xianzhi Wang","doi":"10.1016/j.ejc.2024.104095","DOIUrl":null,"url":null,"abstract":"<div><div>A 1976 question of Martin Gardner asks for the minimum size of a placement of queens on an <span><math><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></math></span> chessboard that is maximal with respect to the property of ‘no-3-in-a-line’. The work of Cooper, Pikhurko, Schmitt and Warrington showed that this number is at least <span><math><mi>n</mi></math></span> in the cases that <span><math><mrow><mi>n</mi><mo>⁄</mo><mo>≡</mo><mn>3</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span>, and at least <span><math><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> in the case that <span><math><mrow><mi>n</mi><mo>≡</mo><mn>3</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span>. When <span><math><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow></math></span> is odd, Gardner conjectured the lower bound to be <span><math><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span>. We prove this conjecture in the case that <span><math><mrow><mi>n</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span>. The proof relies heavily on a recent advancement to the Combinatorial Nullstellensatz for zero-sum grids due to Bogdan Nica.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104095"},"PeriodicalIF":1.0000,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S019566982400180X","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A 1976 question of Martin Gardner asks for the minimum size of a placement of queens on an chessboard that is maximal with respect to the property of ‘no-3-in-a-line’. The work of Cooper, Pikhurko, Schmitt and Warrington showed that this number is at least in the cases that , and at least in the case that . When is odd, Gardner conjectured the lower bound to be . We prove this conjecture in the case that . The proof relies heavily on a recent advancement to the Combinatorial Nullstellensatz for zero-sum grids due to Bogdan Nica.
期刊介绍:
The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.