Pub Date : 2022-10-28DOI: 10.1080/0025570X.2022.2126261
D. Jeon, Heonkyu Lee
Summary We determine all natural numbers that can be expressed as the difference of two m-gonal numbers. For each such number, we determine the number of possible expression as the difference of two m-gonal numbers.
{"title":"The Difference of Two Polygonal Numbers","authors":"D. Jeon, Heonkyu Lee","doi":"10.1080/0025570X.2022.2126261","DOIUrl":"https://doi.org/10.1080/0025570X.2022.2126261","url":null,"abstract":"Summary We determine all natural numbers that can be expressed as the difference of two m-gonal numbers. For each such number, we determine the number of possible expression as the difference of two m-gonal numbers.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46626754","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}
Pub Date : 2022-10-26DOI: 10.1080/0025570X.2022.2127300
Julius B. Barbanel
Summary It is known that the three classical geometric construction problems introduced by the ancient Greeks: trisecting an angle, squaring a circle, and doubling a cube, cannot be solved using the Euclidean tools. However, ancient Greek mathematicians solved these three problems using other means. We present solutions to the doubling-the-cube problem using ideas that go beyond the Euclidean tools, and we consider generalizations to higher dimensions.
{"title":"Doubling the Cube and Constructability in Higher Dimensions","authors":"Julius B. Barbanel","doi":"10.1080/0025570X.2022.2127300","DOIUrl":"https://doi.org/10.1080/0025570X.2022.2127300","url":null,"abstract":"Summary It is known that the three classical geometric construction problems introduced by the ancient Greeks: trisecting an angle, squaring a circle, and doubling a cube, cannot be solved using the Euclidean tools. However, ancient Greek mathematicians solved these three problems using other means. We present solutions to the doubling-the-cube problem using ideas that go beyond the Euclidean tools, and we consider generalizations to higher dimensions.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42631716","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}
Pub Date : 2022-10-20DOI: 10.1080/0025570X.2022.2127306
Jacob A. Boswell, Jacob N. Clark, Chip W. Curtis
Summary We present an introduction to the Japanese pencil puzzle Nurikabe and to its basic solution strategies. Further, we establish formulas for the minimum and maximum number of islands in a Nurikabe puzzle made up of one-tile islands.
{"title":"Counting Islands in Nurikabe","authors":"Jacob A. Boswell, Jacob N. Clark, Chip W. Curtis","doi":"10.1080/0025570X.2022.2127306","DOIUrl":"https://doi.org/10.1080/0025570X.2022.2127306","url":null,"abstract":"Summary We present an introduction to the Japanese pencil puzzle Nurikabe and to its basic solution strategies. Further, we establish formulas for the minimum and maximum number of islands in a Nurikabe puzzle made up of one-tile islands.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47327591","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}
Pub Date : 2022-10-20DOI: 10.1080/0025570X.2022.2136462
L. Kauffman, Devika Prasad, Claudia J. Zhu
Summary The topology of knots and links can be studied by examining colorings of their diagrams. We explain how to detect knots and links using the method of Fox tricoloring, and we give a new and elementary proof that an infinite family of Brunnian links are each linked. Our proof is based on the remarkable fact (which we prove) that if a link diagram cannot be tricolored then it must be linked. Our paper introduces readers to the Fox coloring generalization of tricoloring and the further algebraic generalization, called a quandle by David Joyce.
{"title":"Uncolorable Brunnian Links are Linked","authors":"L. Kauffman, Devika Prasad, Claudia J. Zhu","doi":"10.1080/0025570X.2022.2136462","DOIUrl":"https://doi.org/10.1080/0025570X.2022.2136462","url":null,"abstract":"Summary The topology of knots and links can be studied by examining colorings of their diagrams. We explain how to detect knots and links using the method of Fox tricoloring, and we give a new and elementary proof that an infinite family of Brunnian links are each linked. Our proof is based on the remarkable fact (which we prove) that if a link diagram cannot be tricolored then it must be linked. Our paper introduces readers to the Fox coloring generalization of tricoloring and the further algebraic generalization, called a quandle by David Joyce.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44883125","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}
Pub Date : 2022-10-19DOI: 10.1080/0025570X.2022.2126252
F. Gould
Summary There is a simple combinatorial anomaly, making possible some special linear algebra and thereby some special geometry, that occurs only in dimensions 1, 2, 4, and 8. The consequences are wide ranging and in particular lead to the existence of the complex numbers, the quaternions and the octonions. This article explains why the anomaly exists only in these dimensions using elementary linear algebra.
{"title":"Why Quaternions and Octonions Exist","authors":"F. Gould","doi":"10.1080/0025570X.2022.2126252","DOIUrl":"https://doi.org/10.1080/0025570X.2022.2126252","url":null,"abstract":"Summary There is a simple combinatorial anomaly, making possible some special linear algebra and thereby some special geometry, that occurs only in dimensions 1, 2, 4, and 8. The consequences are wide ranging and in particular lead to the existence of the complex numbers, the quaternions and the octonions. This article explains why the anomaly exists only in these dimensions using elementary linear algebra.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44097623","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}
Pub Date : 2022-10-19DOI: 10.1080/0025570X.2022.2125725
D. Daners
Summary We provide a unified approach to three fundamental properties of continuous functions on closed and bounded intervals: the intermediate value theorem, and the uniform continuity theorem. We prove all three using the same building block, only making use of the least upper bound axiom and the definition of continuity.
{"title":"Unified Proofs of Three Fundamental Properties of Continuous Functions","authors":"D. Daners","doi":"10.1080/0025570X.2022.2125725","DOIUrl":"https://doi.org/10.1080/0025570X.2022.2125725","url":null,"abstract":"Summary We provide a unified approach to three fundamental properties of continuous functions on closed and bounded intervals: the intermediate value theorem, and the uniform continuity theorem. We prove all three using the same building block, only making use of the least upper bound axiom and the definition of continuity.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49446639","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}
Pub Date : 2022-10-18DOI: 10.1080/0025570X.2022.2125254
Ezra Brown, A. Rice
Summary We give a simple proof, intelligible to undergraduates, that a particular multiplicative formula for sums of n squares can only occur when or 8, a result originally proved by Hurwitz in 1898. We begin with a brief survey of the history of sums of squares, leading to a discussion of the related topic of normed division algebras over the real numbers. This story culminates with a crucial paper by Dickson in 1919 that not only contained an exposition of Hurwitz’s 1898 proof, but which also outlined a new process for producing division algebras over the reals. That process, now called the Cayley-Dickson construction, is intimately connected with the product formula for sums of squares and the dimensions necessary for its existence. For this reason, we present an introduction to the Cayley-Dickson construction for beginners, together with a proof of Hurwitz’s theorem accessible to anyone with a basic knowledge of undergraduate algebra.
{"title":"An Accessible Proof of Hurwitz’s Sums of Squares Theorem","authors":"Ezra Brown, A. Rice","doi":"10.1080/0025570X.2022.2125254","DOIUrl":"https://doi.org/10.1080/0025570X.2022.2125254","url":null,"abstract":"Summary We give a simple proof, intelligible to undergraduates, that a particular multiplicative formula for sums of n squares can only occur when or 8, a result originally proved by Hurwitz in 1898. We begin with a brief survey of the history of sums of squares, leading to a discussion of the related topic of normed division algebras over the real numbers. This story culminates with a crucial paper by Dickson in 1919 that not only contained an exposition of Hurwitz’s 1898 proof, but which also outlined a new process for producing division algebras over the reals. That process, now called the Cayley-Dickson construction, is intimately connected with the product formula for sums of squares and the dimensions necessary for its existence. For this reason, we present an introduction to the Cayley-Dickson construction for beginners, together with a proof of Hurwitz’s theorem accessible to anyone with a basic knowledge of undergraduate algebra.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42692833","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}
Pub Date : 2022-10-18DOI: 10.1080/0025570X.2022.2126159
Günhan Caglayan
Summary We give a visual proof for an identity relating triangular and star numbers.
摘要我们给出了一个与三角形数和星形数有关的恒等式的直观证明。
{"title":"A Relationship of Triangular and Star Numbers","authors":"Günhan Caglayan","doi":"10.1080/0025570X.2022.2126159","DOIUrl":"https://doi.org/10.1080/0025570X.2022.2126159","url":null,"abstract":"Summary We give a visual proof for an identity relating triangular and star numbers.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41970929","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}
Pub Date : 2022-10-17DOI: 10.1080/0025570X.2022.2127302
E. Berkove, Michael A. Brilleslyper
Summary We describe a general method that finds closed forms for partial sums of power series whose coefficients arise from linear recurrence relations. These closed forms allow one to derive a vast collection of identities involving the Fibonacci numbers and other related sequences. Although motivated by a polynomial long division problem, the method fits naturally into a standard generating function framework. We also describe an explicit way to calculate the generating function of the Hadamard product of two generating functions, a construction on power series which resembles the dot product. This allows one to use the method for many examples where the recurrence relation for the coefficients is not initially known.
{"title":"Summation Formulas, Generating Functions, and Polynomial Division","authors":"E. Berkove, Michael A. Brilleslyper","doi":"10.1080/0025570X.2022.2127302","DOIUrl":"https://doi.org/10.1080/0025570X.2022.2127302","url":null,"abstract":"Summary We describe a general method that finds closed forms for partial sums of power series whose coefficients arise from linear recurrence relations. These closed forms allow one to derive a vast collection of identities involving the Fibonacci numbers and other related sequences. Although motivated by a polynomial long division problem, the method fits naturally into a standard generating function framework. We also describe an explicit way to calculate the generating function of the Hadamard product of two generating functions, a construction on power series which resembles the dot product. This allows one to use the method for many examples where the recurrence relation for the coefficients is not initially known.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48407109","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}
Pub Date : 2022-10-13DOI: 10.1080/0025570X.2022.2127301
A. Berele, S. Catoiu
Summary Zindler’s theorem of 1920 says that each planar convex set admits two perpendicular lines that divide it into four parts of equal area. Call the intersection of the two lines a Zindler point. We show that each triangle admits either one, two or three Zindler points, and we classify all triangles according to these three numbers.
{"title":"Zindler Points of Triangles","authors":"A. Berele, S. Catoiu","doi":"10.1080/0025570X.2022.2127301","DOIUrl":"https://doi.org/10.1080/0025570X.2022.2127301","url":null,"abstract":"Summary Zindler’s theorem of 1920 says that each planar convex set admits two perpendicular lines that divide it into four parts of equal area. Call the intersection of the two lines a Zindler point. We show that each triangle admits either one, two or three Zindler points, and we classify all triangles according to these three numbers.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45957415","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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