{"title":"Large Independent Set vs. Large Clique","authors":"Melissa Holly","doi":"10.1080/07468342.2023.2237851","DOIUrl":"https://doi.org/10.1080/07468342.2023.2237851","url":null,"abstract":"SummaryLarge cliques with cardinality |ω|>⌈23n⌉ cannot exist with large independent sets where |α|>⌈23n⌉ in the same simple connected graph G of order n≥2. Additional informationNotes on contributorsMelissa Holly Melissa Holly (mholly@vcu.edu) received her bachelor’s degree from University of Illinois at Chicago Circle in 1974. In 2017, she became a graduate student in the Department of Mathematics and Applied Mathematics at Virginia Commonwealth University. Her favorite cycling trail is one with tall trees that reach for the sky where they hold hands.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"150 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jathan Austin, Brian G. Kronenthal, Jonathon Miller, Susanna Molitoris-Miller
{"title":"Caught “Red”-Handed? The Probability of Randomly Constructing a Legal <i>Catan</i> Board","authors":"Jathan Austin, Brian G. Kronenthal, Jonathon Miller, Susanna Molitoris-Miller","doi":"10.1080/07468342.2023.2237850","DOIUrl":"https://doi.org/10.1080/07468342.2023.2237850","url":null,"abstract":"SummaryCatan is a dynamic property-building and trading board game in which players build a new board every time they play by arranging tiles, number tokens, and port markers. In the interest of creating an equitable board, the official Catan rules restrict how the number tokens are placed. In this paper, we use three techniques to count the number of nonequivalent boards satisfying this restriction, as well as determine the probability that a randomly generated board will be legal. AcknowledgmentWe thank the anonymous referees whose time and attention supported the publication of this work.Additional informationNotes on contributorsJathan Austin Jathan Austin (jwaustin@salisbury.edu) is an associate professor of mathematics at Salisbury University in Maryland. He earned a B.S. in mathematics from Salisbury, and both an M.S. in mathematics and a Ph.D. in mathematics education from the University of Delaware. In his spare time he enjoys doing jigsaw puzzles and watching classic sci-fi.Brian G. Kronenthal Brian G. Kronenthal (kronenthal@kutztown.edu) is a professor of mathematics at Kutztown University of Pennsylvania. He earned his B.S. in mathematics from Lafayette College (Easton, Pennsylvania), as well as his M.S. and Ph.D. in mathematics from the University of Delaware. His favorite research problems are combinatorial, often with an algebraic flair. In addition to teaching and research, he enjoys playing ping pong, watching movies, and rooting for Philadelphia sports teams.Jonathon Miller Jonathon Miller (jonathonamiller@gmail.com) is a software developer for Amazon. He earned his degree in mathematics and a minor in computer science from Salisbury University in Maryland. He also earned his MS in mathematics from The University of Delaware. In his free time Jonathon enjoys being a husband and father, playing board games, and serving as an excellent DM for Dungeons and Dragons.Susanna Molitoris-Miller Susanna Molitoris-Miller (susannamolitorismiller@gmail.com) earned her B.S. in mathematics from The University of Scranton, and M.S. in mathematics and Ph.D. in mathematics education from The University of Delaware. Susanna’s research focuses on how students learn mathematical concepts in creative ways. In her free time she enjoys spending time with her family, fiber arts, tea and, of course, game night with friends.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Limiting Behaviour of Pairs with Equal Images for Polynomial Functions","authors":"Peter M. Higgins","doi":"10.1080/07468342.2023.2237854","DOIUrl":"https://doi.org/10.1080/07468342.2023.2237854","url":null,"abstract":"AbstractFor any polynomial p(x) with real coefficients and of degree n≥1, for sufficiently large positive x there is a unique y, distinct from x, such that p(x) and p(y) are equal in absolute value. We show that, in the limit, the mean of x and y is equal to the mean of the roots of p(x).","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An Attractive Attractor","authors":"Hayden Ruff, Kenneth Schilling","doi":"10.1080/07468342.2023.2237849","DOIUrl":"https://doi.org/10.1080/07468342.2023.2237849","url":null,"abstract":"AbstractWe explore a family of closed curves in the plane. Each member of the family is the limit of self-intersecting polygonal curves. Nonetheless, most members of the family are simple closed curves. One, the “attractive attractor,” surrounds an infinite set of disjoint regions. AcknowledgmentsWe are grateful to the referee for a careful reading and many helpful suggestions.Additional informationNotes on contributorsHayden Ruff Hayden Ruff (hr442@drexel.edu) is a pre-candidacy mathematics Ph.D. student enrolled at Drexel University in Philadelphia, PA. He completed a B.S. in mathematics and physics at the University of Michigan – Flint, where he collaborated with and received instruction from Dr. Schilling. His research interests include signal processing, psychoacoustics, and applications of mathematics to the study of musicality.Kenneth Schilling Kenneth Schilling (ksch@umich.edu) received his Ph.D. from the University of California, Berkeley in 1981. He is professor emeritus of mathematics from the University of Michigan-Flint. During his years on the faculty, he loved working with students and interactiing with colleagues and pretty much everything about university life except grading exams.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Problems and Solutions","authors":"Greg Oman, Charles N. Curtis","doi":"10.1080/07468342.2023.2229224","DOIUrl":"https://doi.org/10.1080/07468342.2023.2229224","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Not Solving Differential Equations","authors":"James S. Wolper","doi":"10.1080/07468342.2023.2240204","DOIUrl":"https://doi.org/10.1080/07468342.2023.2240204","url":null,"abstract":"AbstractOne can learn a lot about a differential equation without solving it. Using the relationship between the sign of the derivative of a function and its behavior, given by the Mean Value Theorem, makes many important properties of solutions clear, and allows one to consider variations that are difficult or ugly to solve exactly. Additional informationNotes on contributorsJames S. WolperJames Wolper (wolpjame@isu.edu) received his Ph.D. in Mathematics from Brown University. He is an Emeritus Professor at Idaho State University. Most of his research has been in Algebraic Geometry, most recently into the statistical properties of periods of Riemann Surfaces.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"<i>Proof Without Words: Tangent of π/8.</i>","authors":"Armengol Gasull","doi":"10.1080/07468342.2023.2238581","DOIUrl":"https://doi.org/10.1080/07468342.2023.2238581","url":null,"abstract":"\"Proof Without Words: Tangent of π/8..\" The College Mathematics Journal, ahead-of-print(ahead-of-print), p. 1","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Up the Hill and Down Again","authors":"Don Chakerian, Stephen Erfle","doi":"10.1080/07468342.2023.2223509","DOIUrl":"https://doi.org/10.1080/07468342.2023.2223509","url":null,"abstract":"Click to increase image sizeClick to decrease image size AcknowledgmentsWe are thankful for the LaTeX wizardry of our colleagues Xiaozhou Ding and David Richeson.Additional informationNotes on contributorsDon Chakerian Don Chakerian (dmandgd@aol.com) received his Ph.D. from UC Berkeley and is Emeritus Professor of Mathematics at the University of California, Davis. His area of research is the theory of convex sets and geometric inequalities. He earned the George Pólya Award from the MAA in 1981 for his paper Circles and Spheres, College Mathematics Journal, vol.11, pp.26–41. In it, he mentions an elegant proof of a theorem of N.A. Court, found by one of his favorite undergraduate students, by name of Steve Erfle.Stephen Erfle Stephen Erfle (erfle@dickinson.edu) took as many classes as he could from Don Chakerian while attending UC Davis as an undergraduate. He received his Ph.D. in Economics from Harvard and is Professor of International Business and Management at Dickinson College. He is currently working on a recreational mathematics book entitled Playing with Polygons. Some of the Excel files that form the backbone of this book allow users to produce electronic string art on a polygonal vertex frame. Some images were sufficiently interesting that he enticed his mentor and now coauthor into jointly working on this paper.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"147 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two Quasigroup Elements Can Commute With Any Positive Rational Probability","authors":"Ron Lycan","doi":"10.1080/07468342.2023.2237853","DOIUrl":"https://doi.org/10.1080/07468342.2023.2237853","url":null,"abstract":"SummaryA quasigroup is a set with a binary operation in which both left and right division are unique. Equivalently, every row and column in a quasigroup table is a permutation of its elements. The commuting probability of a quasigroup is the probability that two of its elements, chosen at random, will commute. In this paper, we show that a quasigroup may have any rational number in (0,1] as a commuting probability. AcknowledgmentsThe author would like to thank their advisor, Vadim Ponomarenko, for helping and supporting them throughout the process of writing this article.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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