{"title":"Summing Squares and Cubes Using Triangular Numbers","authors":"Roger Nelsen","doi":"10.1080/07468342.2023.2264720","DOIUrl":"https://doi.org/10.1080/07468342.2023.2264720","url":null,"abstract":"SummaryFormulas for the sums of the squares and for the cubes of the first n positive integers are usually proved by mathematical induction. We present alternative proofs using triangular numbers. AcknowledgmentThe author wishes to thank an anonymous reviewer and the Editor for helpful suggestions on an earlier draft of this Capsule.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135729792","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":"Extracting Pi from Chaos","authors":"Milton F. Maritz","doi":"10.1080/07468342.2023.2265282","DOIUrl":"https://doi.org/10.1080/07468342.2023.2265282","url":null,"abstract":"SummaryWe present an experimental method to find some digits of π, by extracting it from chaos generated by a quadratic polynomial. In fact, we iterate the logistic map in its chaotic regime, and show how to extract the digits of π out of those chaotic numbers. Additional informationNotes on contributorsMilton F. MaritzMilton F. Maritz (mfmaritz@sun.ac.za) holds a Ph.D. in Applied Mathematics from the University of the Free State (UFS). He has taught applied mathematics at UFS for 11 years, then physics at UFS for 5 years, and then applied mathematics again at Stellenbosch University for 22 years. His research interests include partial differential equations, image processing, and the mechanics of eccentrically loaded rolling hoops. He has also done research for industry, in particular in the modeling of shaped charge jet penetration.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135730610","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}
Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, Michael Voigt
{"title":"The Stable Marriage Problem and Sudoku","authors":"Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, Michael Voigt","doi":"10.1080/07468342.2023.2261183","DOIUrl":"https://doi.org/10.1080/07468342.2023.2261183","url":null,"abstract":"Are you having trouble getting married? These days, there are lots of products on the market for dating, from apps to websites and matchmakers, but we know a simpler way! That’s right—your path to coupled life isn’t through Tinder; it’s through Sudoku! Read our fabulous paper, where we explore the Stable Marriage Problem to help you find happiness and stability in marriage through math. As a bonus, you get two Sudoku puzzles with a new flavor.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135730458","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":"Zig-Zag Paths and Neusis Constructions of a Heptagon and a Nonagon","authors":"Dan Lawson, David Richeson","doi":"10.1080/07468342.2023.2261347","DOIUrl":"https://doi.org/10.1080/07468342.2023.2261347","url":null,"abstract":"SummaryAlthough it is impossible to construct a regular heptagon and a regular nonagon using a compass and unmarked straightedge, it is possible to construct them with a compass and marked straightedge using the neusis technique. We give a geometric proof of Johnson’s neusis construction of the regular heptagon, which he had proven using trigonometry. We do so using so-called central triangles and zig-zag paths in the polygons. We then give efficient neusis constructions of the regular heptagon and the regular nonagon. Additional informationNotes on contributorsDan Lawson Dan Lawson (dlawson@peralta.edu) received his MS in mathematics from San Jose State University in 1994. He teaches at Merritt College in Oakland, CA. He enjoys working on problems in recreational mathematics, including solving and creating sudoku puzzles.David Richeson David Richeson (richesod@dickinson.edu) is a professor of Mathematics and the John J. and Ann Curley Chair in Liberal Arts at Dickinson College. He is the author of Tales of Impossibility (Princeton University Press, 2019) and Euler’s Gem (Princeton University Press, 2008), and he is a past editor of Math Horizons.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135855183","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":"Analysis and Visualization of Fractional Reflections","authors":"Milton F. Maritz, Marèt Cloete","doi":"10.1080/07468342.2023.2253129","DOIUrl":"https://doi.org/10.1080/07468342.2023.2253129","url":null,"abstract":"SummaryOne can reflect once, twice or m times, where m is an integer. Can m be a real number? In this paper we show how fractional (i.e., not necessarily integer) reflections are performed. The result relies on the fact that a reflection matrix has eigenvalues ±1, and since (−1)m= cos (πm)+i sin (πm), a fractional reflection may be interpreted visually if the R3 space is augmented by including an imaginary axis. Additional informationNotes on contributorsMilton F. Maritz Milton F. Maritz (mfmaritz@sun.ac.za) holds a Ph.D. in Applied Mathematics from the University of the Free State (UFS). He has taught applied mathematics at UFS for 11 years, then physics at UFS for 5 years, and then applied mathematics again at Stellenbosch University for 22 years. His research interests include partial differential equations, image processing, and the mechanics of eccentrically loaded rolling hoops. He has also done research for industry, in particular in the modelling of shaped charge jet formation and penetration.Marèt CloeteMarèt Cloete (mcloete@sun.ac.za) received a Ph.D. in applied mathematics from the University of Stellenbosch, South Africa. Her academic interests include fluid dynamics, mechanics and struggling with mathematical problems. Currently she is lecturing classical mechanics and PDEs at her alma mater and in her spare time she enjoys all kinds of sporting activities, camping and hiking in the mountains.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135768575","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":"Disjoint Placement Probability of Line Segments via Induction","authors":"Christopher Ennis, Inge Helland","doi":"10.1080/07468342.2023.2255087","DOIUrl":"https://doi.org/10.1080/07468342.2023.2255087","url":null,"abstract":"AbstractWe give a different proof of the following recent result: Let a finite number n of line segments, the sum Ln of whose lengths is less than one, be placed onto the real line in such a way that their centers fall randomly within the unit interval [0,1]. Then the probability of obtaining a mutually disjoint placement of these segments, entirely within [0,1], is given by (1−Ln)n. The proof presented here uses induction on the number of line segments and provides insight, at each level of the induction, into the relationship between two seemingly different methods of placement: sequential random placement versus simultaneous random placement. From a purely mathematical perspective, these methods can be seen as equivalent. However, physical constraints in performing a simultaneous random placement of actual segments (e.g., toothpicks) might a priori lead to very different outcomes. AcknowledgmentThe authors express their sincere appreciation to the referees and the Editor for their many helpful suggestions. These have greatly improved the clarity and readability of the article. The proof concept here is due to IH and the exposition is due to CE.Additional informationNotes on contributorsChristopher Ennis Christopher Ennis (cjennis10@gmail.com) attended UCLA and UC Berkeley. After holding several academic appointments, he settled into a rewarding 24 year teaching career at Normandale Community College, including several years as department chairperson. Retiring in 2016, he has again found time to do mathematical research.Inge Helland Inge Helland (ingeh@math.uio.no) is a retired professor of statistics from the University of Oslo. After retirement he has spent most of his time trying to understand the foundation of quantum mechanics from his point of view as a statistician. This has resulted in two books and several articles. But he also has some interest in recreational mathematics.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136136105","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":"Exact Expressions for Trigonometric Functions","authors":"S. H. S. Indika, L. Leemis","doi":"10.1080/07468342.2023.2241316","DOIUrl":"https://doi.org/10.1080/07468342.2023.2241316","url":null,"abstract":"Herman Robinson (7 Apr 1912 to 10 Oct 1986) was a scientist at the UC Berkeley Radiation Lab in Berkeley, CA. He worked there from 1945 until he retired in 1973. Herman was a chemical engineer by training, but his interests and skills ranged over a variety of science, math, and technology topics. He enjoyed \"tinkering\" with mathematical calculations after he retired with the Wang 720c calculator he purchased for himself, having used one at the office. In 1990, I discovered that he was my biological father.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49657706","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":"Approximation of Pi Using Infinite Series of Arcsine","authors":"Kyumin Nam","doi":"10.1080/07468342.2023.2242213","DOIUrl":"https://doi.org/10.1080/07468342.2023.2242213","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46031687","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":"On the Center of Surface Area of the Boundary of a Star-Shaped Region","authors":"Thunwa Theerakarn","doi":"10.1080/07468342.2023.2240203","DOIUrl":"https://doi.org/10.1080/07468342.2023.2240203","url":null,"abstract":"AbstractEvery line passing through the center of a circular disk bisects its area. Not every planar region has a point with this property. It turns out that if a compact connected region is star-shaped at its center of area, then it must be centrally symmetric. However, there exists a non-centrally symmetric star-shaped region whose boundary has a center of length, which is a point where every line passing through this point cut its length in half. In this article, we characterize such regions and provide a method to create them. Additionally, we provide new examples and a method to generate analogous objects in higher dimensions. AcknowledgmentThe author is extremely grateful to Professor Thomas F. Banchoff for introducing the research problem and for his invaluable support and guidance during the initial investigation. The author thanks the referees for their insightful comments and valuable suggestions. The author is partially supported by Faculty of Science, Srinakharinwirot University through Grant 170/2563.0090`Additional informationNotes on contributorsThunwa Theerakarn Thunwa Theerakarn (thunwa@g.swu.ac.th) has been teaching at Srinakharinwirot University in Bangkok, Thailand since 2019. He received Ph.D. in mathematics from the University of California, Berkeley. He received M.Sc. in applied mathematics and B.Sc. in mathematics from Brown University.","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":"135840435","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":"Hyperbolic Lunes","authors":"Tevian Dray","doi":"10.1080/07468342.2023.2231314","DOIUrl":"https://doi.org/10.1080/07468342.2023.2231314","url":null,"abstract":"Summary The formula for the area of a hyperbolic triangle in terms of its angle defect is derived using hyperbolic lunes, in analogy with the argument using (elliptic) lunes to express the area of an elliptic triangle in terms of its angle excess. Several pedagogical features of this construction are then discussed.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"23 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":"135840438","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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