{"title":"Problems and Solutions","authors":"Greg Oman, Charles N. Curtis","doi":"10.1080/07468342.2023.2237385","DOIUrl":"https://doi.org/10.1080/07468342.2023.2237385","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"33 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":"135840447","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":"A New Take on Classic ‘Pen Problems’","authors":"David A. Nash","doi":"10.1080/07468342.2023.2237843","DOIUrl":"https://doi.org/10.1080/07468342.2023.2237843","url":null,"abstract":"AbstractIn this article we generalize the classic “farm pen” optimization problem from a first course in calculus in a handful of different ways. We describe the solution to an n-dimensional rectangular variant, and then study the situation when the pens are either regular polygons or platonic solids. Additional informationNotes on contributorsDavid A. Nash David A. Nash (prof.nash@gmail.com) is a data scientist at the consulting firm Kin + Carta. After an undergraduate career at Santa Clara University, he earned his Ph.D. from University of Oregon in 2010 with an emphasis in representation theory. From 2010 to 2021, he served as an associate professor of mathematics at Le Moyne College. He enjoys sharing his passion for mathematics and problem solving with students, colleagues, and his children.","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":"135746223","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":"Integration by Substitution: A Case Study in the Relationship Between Calculus and Analysis","authors":"Daniel J. Velleman","doi":"10.1080/07468342.2023.2228673","DOIUrl":"https://doi.org/10.1080/07468342.2023.2228673","url":null,"abstract":"SummaryDo the theorems we prove in our analysis classes justify the calculations we teach in calculus classes? In this article I use some examples of integration by substitution to show that the answer is more complicated than one might think. Notes1 Courant [2, pp. 211–212] gives a different justification for this kind of substitution in definite integrals, using limits of Riemann sums. Surprisingly, he assumes that g has a continuous, nonvanishing derivative, even though his proof uses only (g−1)′ and not g′.Additional informationNotes on contributorsDaniel J. Velleman Daniel J. Velleman (djvelleman@amherst.edu) received his B.A. from Dartmouth College in 1976 and his Ph.D. from the University of Wisconsin–Madison in 1980. He taught at the University of Texas before joining the faculty of Amherst College in 1983. Since 2011, he has also been an adjunct professor at the University of Vermont. He was the editor of the American Mathematical Monthly from 2007 to 2011. In his spare time he enjoys singing, bicycling, and playing volleyball.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"38 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":"135840445","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":"Media Highlights","authors":"Lowell Beineke, Philip Straffin","doi":"10.1080/07468342.2023.2237382","DOIUrl":"https://doi.org/10.1080/07468342.2023.2237382","url":null,"abstract":"In the 1960s Robert Berger constructed a set of 20,426 tiles that tile the plane aperiodically but cannot tile it periodically. The race to find smaller sets of such tiles culminated in the 1970s with Roger Penrose’s discovery of a set of just two tiles with this property—his famous kite and dart tiles. Since then, mathematicians have wondered whether there might be one single tile that tiles the plane aperiodically but not periodically, a so-called “einstein,” playing on the German “ein stein” (one stone). Fifty years of searching could not produce an einstein. Does one really exist? In November 2022, David Smith, a retired print technician in northern England, was experimenting with tiles formed by assembling pieces of hexagons in different ways, and found that a 13-sided tile made of four pieces of hexagons, which he called the “hat,” seemed to be able to cover larger and larger areas without repeating a pattern. “It’s a tricky little tile.” Smith enlisted the help of Craig Kaplan, a computer scientist at the University of Waterloo, Chaim Goodman-Strauss at the National Museum of Mathematics and the University of Arkansas, and Joseph Samuel Myers, a software engineer in Cambridge, England, and on March 20, 2023 they announced that the hat did tile the plane, but only aperiodically. It was the long-sought einstein, and so simple, “hiding in plain sight!” Marjorie Senechal called the discovery “Just mind-boggling!” Doris Schattschneider described herself as “flabbergasted.” Meanwhile, Smith announced that he had found another, even simpler, tile he called the “turtle,” which also turned out to be an einstein, and in fact the team discovered that they could morph the hat into the turtle through a continuum of tiles that tile the plane only aperiodically. The Quanta article has a moving graphic showing a hat tiling morphing into a turtle tiling. You should look at the pictures, but the turtle is easy to describe. Think of a hexagon as made up of six equilateral triangles, then add one more equilateral triangle on one edge of the hexagon, and another equilateral triangle on a side of that one. It’s a hexagon with a snout.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"1 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":"135840439","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":"The Largest Quadrilateral is Cyclic: A New Geometric Proof","authors":"Jyotirmoy Sarkar","doi":"10.1080/07468342.2023.2239227","DOIUrl":"https://doi.org/10.1080/07468342.2023.2239227","url":null,"abstract":"AbstractAlthough treated as “obvious” since antiquity, the first complete proof that “a quadrilateral with given sides achieves the maximum area when it is cyclic” is attributed to Bretschneider (1842), who proved it using trigonometry. Peter (2003) proved it using calculus. It also follows from the isoperimetric inequality, proved geometrically in [Citation5] and [Citation11]. Here we give a new Euclidean geometric proof, starting from a different maximization problem: Find the tallest vertical line segment sandwiched between two semi-circles in a plane lying on opposite sides of a horizontal line with (partially) overlapping diameters.MSC: 51-01 AcknowledgmentI thank my colleague Professor Patrick Morton for permitting me to attend his Geometry for High School Teachers class and for reading an earlier draft of this paper. I am grateful to two referees for giving me many helpful suggestions.Additional informationNotes on contributorsJyotirmoy Sarkar Jyotirmoy Sarkar(jsarkar@iupui.edu) received his Ph.D. in Statistics from the University of Michigan, Ann Arbor. He is a professor and statistics consultant at Indiana University-Purdue University Indianapolis. His research interests are enumeration, probability, stochastic processes, and reliability theory. When he has time, he reads novels, cares for plants, and invents mathematical puzzles. ORCiD: 0000-0001-5002-5845","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"59 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":"135840434","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":"The Product of Two Natural Numbers Equals the Product of their GCD and LCM","authors":"G. Lawlor","doi":"10.1080/07468342.2023.2231311","DOIUrl":"https://doi.org/10.1080/07468342.2023.2231311","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46475721","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":"A Simple Introduction to the Exponential Function","authors":"V. Komornik, R. Schäfke","doi":"10.1080/07468342.2023.2234256","DOIUrl":"https://doi.org/10.1080/07468342.2023.2234256","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45363118","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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