Pub Date : 2023-08-02DOI: 10.1080/00029890.2023.2230860
K. Knox
Abstract A quadrilateral contains a closed billiard circuit that bounces consecutively on the interiors of its sides each exactly once per cycle, if and only if it is cyclic and its interior contains the center of its circumscribed circle.
{"title":"Billiard Circuits in Quadrilaterals","authors":"K. Knox","doi":"10.1080/00029890.2023.2230860","DOIUrl":"https://doi.org/10.1080/00029890.2023.2230860","url":null,"abstract":"Abstract A quadrilateral contains a closed billiard circuit that bounces consecutively on the interiors of its sides each exactly once per cycle, if and only if it is cyclic and its interior contains the center of its circumscribed circle.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"755 - 759"},"PeriodicalIF":0.5,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49140242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-27DOI: 10.1080/00029890.2023.2231816
Reed Acton, T. K. Petersen, Blake Shirman, Daniel Toal, Francesco Laudano
Abstract In this paper we study the problem of the Malicious Maitre d’, as described in Peter Winkler’s book Mathematical Puzzles: A Connoisseur’s Collection. This problem, attributed to computer scientist Rob Pike, involves seating diners around a circular table with napkins placed between each pair of adjacent settings. The goal of the maitre d’ is to seat the diners in a way that maximizes the number of diners who arrive at the table to find the napkins on both the left and right of their place already taken by their neighbors. Winkler proposes a solution to the problem that he claims is optimal. We analyze Winkler’s solution using tools from enumerative combinatorics, then present a new strategy that performs better.
{"title":"A More Malicious Maitre d’","authors":"Reed Acton, T. K. Petersen, Blake Shirman, Daniel Toal, Francesco Laudano","doi":"10.1080/00029890.2023.2231816","DOIUrl":"https://doi.org/10.1080/00029890.2023.2231816","url":null,"abstract":"Abstract In this paper we study the problem of the Malicious Maitre d’, as described in Peter Winkler’s book Mathematical Puzzles: A Connoisseur’s Collection. This problem, attributed to computer scientist Rob Pike, involves seating diners around a circular table with napkins placed between each pair of adjacent settings. The goal of the maitre d’ is to seat the diners in a way that maximizes the number of diners who arrive at the table to find the napkins on both the left and right of their place already taken by their neighbors. Winkler proposes a solution to the problem that he claims is optimal. We analyze Winkler’s solution using tools from enumerative combinatorics, then present a new strategy that performs better.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"728 - 745"},"PeriodicalIF":0.5,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43141460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-26DOI: 10.1080/00029890.2023.2231164
B. Chakraborty
{"title":"Sum of a Geometric Series Via the Integral","authors":"B. Chakraborty","doi":"10.1080/00029890.2023.2231164","DOIUrl":"https://doi.org/10.1080/00029890.2023.2231164","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"764 - 764"},"PeriodicalIF":0.5,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48060015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-24DOI: 10.1080/00029890.2023.2230836
K. Williams
Abstract We present a short arithmetic proof of the theorem of Chan, Long, and Yang proved in the Monthly in 2011, which gives explicit formulas for integers x and y such that , where p is a prime satisfying .
{"title":"An Arithmetic Proof of a Theorem of Chan, Long, and Yang","authors":"K. Williams","doi":"10.1080/00029890.2023.2230836","DOIUrl":"https://doi.org/10.1080/00029890.2023.2230836","url":null,"abstract":"Abstract We present a short arithmetic proof of the theorem of Chan, Long, and Yang proved in the Monthly in 2011, which gives explicit formulas for integers x and y such that , where p is a prime satisfying .","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"760 - 763"},"PeriodicalIF":0.5,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46694907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-24DOI: 10.1080/00029890.2023.2231826
Brian Hopkins
The most revelatory experience of my undergraduate mathematics degree was a writing course offered by James Vick. Well on his way to becoming a vice president in the University of Texas administration, Vick still managed to teach the occasional class. This one, “Proofs, Conjectures, and Controversies,” showed us that mathematics was not always as absolute as implied in our other courses. In particular, there is not always consensus about what constitutes a valid proof and what tools are allowed. One example of controversy was the 1976 computer-assisted proof of the four color theorem, establishing that the countries of every map can be colored with at most four colors so that no two neighboring countries are colored the same. The class readings included the recent Dover edition of Saaty and Kainan’s book [18] on the “assaults and conquest” of that graph theory problem and Leonard Gillman’s celebrated guide to mathematical writing [5]. What an honor to learn about recent and contentious research developments while developing skills in mathematical exposition. The year 1976 was an important year for both the United States and graph theory. In that year of the American bicentennial, Kenneth Appel and Wolfgang Haken of the University of Illinois solved Francis Guthrie’s 1852 four color conjecture. This was one of the rare mathematical results to be mentioned (eventually) in The New York Times [19, p. 209]. Another milestone for graph theory that year was the publication of the first book on its history, Graph Theory 1736–1936 by Norman Biggs, Keith Lloyd, and Robin Wilson [2]. That book opened a sort of trilogy on the history of graph theory, all involving the talented and prolific expositor Robin Wilson. The second book was a 2002 solo effort Four Colors Suffice [19]. The third book, based on the 2012 dissertation of David Parks [16] and also featuring John Watkins, focuses on the period 1876–1976 with an emphasis on US and Canadian contributions to the field. This review will focus on the 2023 book while also discussing it in relation to its two predecessors.
{"title":"Reviews","authors":"Brian Hopkins","doi":"10.1080/00029890.2023.2231826","DOIUrl":"https://doi.org/10.1080/00029890.2023.2231826","url":null,"abstract":"The most revelatory experience of my undergraduate mathematics degree was a writing course offered by James Vick. Well on his way to becoming a vice president in the University of Texas administration, Vick still managed to teach the occasional class. This one, “Proofs, Conjectures, and Controversies,” showed us that mathematics was not always as absolute as implied in our other courses. In particular, there is not always consensus about what constitutes a valid proof and what tools are allowed. One example of controversy was the 1976 computer-assisted proof of the four color theorem, establishing that the countries of every map can be colored with at most four colors so that no two neighboring countries are colored the same. The class readings included the recent Dover edition of Saaty and Kainan’s book [18] on the “assaults and conquest” of that graph theory problem and Leonard Gillman’s celebrated guide to mathematical writing [5]. What an honor to learn about recent and contentious research developments while developing skills in mathematical exposition. The year 1976 was an important year for both the United States and graph theory. In that year of the American bicentennial, Kenneth Appel and Wolfgang Haken of the University of Illinois solved Francis Guthrie’s 1852 four color conjecture. This was one of the rare mathematical results to be mentioned (eventually) in The New York Times [19, p. 209]. Another milestone for graph theory that year was the publication of the first book on its history, Graph Theory 1736–1936 by Norman Biggs, Keith Lloyd, and Robin Wilson [2]. That book opened a sort of trilogy on the history of graph theory, all involving the talented and prolific expositor Robin Wilson. The second book was a 2002 solo effort Four Colors Suffice [19]. The third book, based on the 2012 dissertation of David Parks [16] and also featuring John Watkins, focuses on the period 1876–1976 with an emphasis on US and Canadian contributions to the field. This review will focus on the 2023 book while also discussing it in relation to its two predecessors.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"779 - 784"},"PeriodicalIF":0.5,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44201560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-17DOI: 10.1080/00029890.2023.2230811
D. Daners, L. Paunescu
The fundamental theorem of algebra states that every polynomial p(z) over C of degree m ≥ 1 has a zero in C. There are many proofs of this theorem, but we have not found the elementary one presented here. Splitting the coefficients of p(z) into their real and imaginary parts,we find polynomials a(z) and b(z) with real coefficients such that p(z) = a(z) + ib(z). Then q(z) := p(z)p(z̄) = a(z)2 + b(z)2 is a polynomial of degree 2m ≥ 2 with real coefficients, and q(x) ≥ 0 for all x ∈ R. Moreover, q(z) = 0 if and only if p(z) = 0 or p(z̄) = 0. If q(z) = 0 for all z ∈ C, then F(z) := ∫ 1 0 z/q(tz) dt defines a primitive of 1/q(z) on C. Hence the integral over the piecewise smooth closed curve given by the interval [−r, r] and the positively oriented semi-cirlce Cr := {reiθ : θ ∈ [0, π ]} vanishes, that is, ∫ r −r 1 q(x) dx + ∫
{"title":"The Fundamental Theorem of Algebra via Real Polynomials","authors":"D. Daners, L. Paunescu","doi":"10.1080/00029890.2023.2230811","DOIUrl":"https://doi.org/10.1080/00029890.2023.2230811","url":null,"abstract":"The fundamental theorem of algebra states that every polynomial p(z) over C of degree m ≥ 1 has a zero in C. There are many proofs of this theorem, but we have not found the elementary one presented here. Splitting the coefficients of p(z) into their real and imaginary parts,we find polynomials a(z) and b(z) with real coefficients such that p(z) = a(z) + ib(z). Then q(z) := p(z)p(z̄) = a(z)2 + b(z)2 is a polynomial of degree 2m ≥ 2 with real coefficients, and q(x) ≥ 0 for all x ∈ R. Moreover, q(z) = 0 if and only if p(z) = 0 or p(z̄) = 0. If q(z) = 0 for all z ∈ C, then F(z) := ∫ 1 0 z/q(tz) dt defines a primitive of 1/q(z) on C. Hence the integral over the piecewise smooth closed curve given by the interval [−r, r] and the positively oriented semi-cirlce Cr := {reiθ : θ ∈ [0, π ]} vanishes, that is, ∫ r −r 1 q(x) dx + ∫","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"727 - 727"},"PeriodicalIF":0.5,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42552116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-17DOI: 10.1080/00029890.2023.2231798
F. Laudano
Henry Perigal was an amateur mathematician and a member of the London Mathematical Society from 1868 to 1897. He is perhaps best known for his proof of the Pythagorean theorem by dissection and transposition [3, 4]. Here we extend the Perigal method to give a new proof for a result that has been called the extended Pythagorean formula [1, 2]. Consider triangle ABC, with BC ≥ AB. Let D be the point on AC such that BD = AB and E the point on the extension of BA where AE = DC. The triangles EAA′ and CDB are congruent, EA′ = CB, and A EA′ = D CB.
{"title":"a2=b2+cd, an Extended Pythagorean Formula","authors":"F. Laudano","doi":"10.1080/00029890.2023.2231798","DOIUrl":"https://doi.org/10.1080/00029890.2023.2231798","url":null,"abstract":"Henry Perigal was an amateur mathematician and a member of the London Mathematical Society from 1868 to 1897. He is perhaps best known for his proof of the Pythagorean theorem by dissection and transposition [3, 4]. Here we extend the Perigal method to give a new proof for a result that has been called the extended Pythagorean formula [1, 2]. Consider triangle ABC, with BC ≥ AB. Let D be the point on AC such that BD = AB and E the point on the extension of BA where AE = DC. The triangles EAA′ and CDB are congruent, EA′ = CB, and A EA′ = D CB.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"746 - 746"},"PeriodicalIF":0.5,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41666839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-03DOI: 10.1080/00029890.2023.2188815
D. Ullman, Daniel J. Velleman, S. Wagon, D. West
Proposed problems, solutions, and classics should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed problems must not be under consideration concurrently at any other journal, nor should they be posted to the internet before the deadline date for solutions. Proposed solutions to the problems below must be submitted by October 31, 2023. Proposed classics should include the problem statement, solution, and references. More detailed instructions are available online. An asterisk (*) after the number of a problem or a part of a problem indicates that no solution is currently available.
{"title":"Problems and Solutions","authors":"D. Ullman, Daniel J. Velleman, S. Wagon, D. West","doi":"10.1080/00029890.2023.2188815","DOIUrl":"https://doi.org/10.1080/00029890.2023.2188815","url":null,"abstract":"Proposed problems, solutions, and classics should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed problems must not be under consideration concurrently at any other journal, nor should they be posted to the internet before the deadline date for solutions. Proposed solutions to the problems below must be submitted by October 31, 2023. Proposed classics should include the problem statement, solution, and references. More detailed instructions are available online. An asterisk (*) after the number of a problem or a part of a problem indicates that no solution is currently available.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"587 - 596"},"PeriodicalIF":0.5,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46913055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-23DOI: 10.1080/00029890.2023.2219174
Soumya Bhattacharya
The pigeonhole principle states that if n pigeons are put into m < n pigeonholes, then at least two pigeons must be in the same hole. What happens if there are more pigeonholes than pigeons and the pigeons are placed in the pigeonholes randomly? For example, if each among 50 shades of grey are chosen at random from 256 possibilities, can one assert that there are at least two identical choices? Yes, almost surely one can! See Corollary 1. Theorem (Probabilistic Pigeonhole Principle). Given a positive integer m and p ∈ [0, 1), let n be an integer that is larger than or equal to
{"title":"The Probabilistic Pigeonhole Principle","authors":"Soumya Bhattacharya","doi":"10.1080/00029890.2023.2219174","DOIUrl":"https://doi.org/10.1080/00029890.2023.2219174","url":null,"abstract":"The pigeonhole principle states that if n pigeons are put into m < n pigeonholes, then at least two pigeons must be in the same hole. What happens if there are more pigeonholes than pigeons and the pigeons are placed in the pigeonholes randomly? For example, if each among 50 shades of grey are chosen at random from 256 possibilities, can one assert that there are at least two identical choices? Yes, almost surely one can! See Corollary 1. Theorem (Probabilistic Pigeonhole Principle). Given a positive integer m and p ∈ [0, 1), let n be an integer that is larger than or equal to","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":" 10","pages":"678 - 678"},"PeriodicalIF":0.5,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41254397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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