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Whats the Best Seat in the Game Left, Center, Right
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| Pengarang | : | Benjamin Richeson & David Richeson |
| Nama Majalah/Jurnal | : | Mathematics Magazine |
| Volume / Edisi | : | 99 (No. 3) |
| Halaman | : | 252-262 |
| Abstrak | : | Left, Center, Right is a popular dice game. We analyze the game using Markov chain and Monte Carlo methods. We compute the expected game length for two to eight players and determine the probability of winning for each player in the game. We discuss the surprising conclusions of which players have the highest and lowest chance of winning, and we propose a small rule change that makes the game a little more fair. |
Apollonius Theorem via Ptolemy Theorem
| Pengarang | : | Roger B. Nelsen |
| Nama Majalah/Jurnal | : | Mathematics Magazine |
| Volume / Edisi | : | 99 (No. 3) |
| Halaman | : | 250-251 |
| Abstrak | : | We use Ptolemy’s theorem to prove Apollonius’s theorem for the length of a triangle median. |
Arrow-Chasing in Pascal’s Triangle – Visual Proofs for Summation Formulas Involving Binomial Coefficients
| Pengarang | : | Regula? Krapf |
| Nama Majalah/Jurnal | : | Mathematics Magazine |
| Volume / Edisi | : | 99 (No. 3) |
| Halaman | : | 237-249 |
| Abstrak | : | This article demonstrates, using numerous examples of varying complexity, how one can visually prove summation formulas involving binomial coefficients by exclusively using the recurrence relation for binomial coefficients and its illustration through arrows in Pascal’s triangle. The method developed for this purpose, which we call “arrow-chasing,” is elementary and it is accessible to a very broad audience. |
Doanes Rhyming Sequences
| Pengarang | : | Steven C. Althoen & Kenneth Schilling |
| Nama Majalah/Jurnal | : | Mathematics Magazine |
| Volume / Edisi | : | 99 (No. 3) |
| Halaman | : | 233-236 |
| Abstrak | : | Infinite sequences of automorphic numbers in different bases are introduced via a simple construction. |
Composite Numbers in an Arithmetic Progression
| Pengarang | : | Hùng Vi?t Chu, Steven J. Miller & Joshua M. Siktar |
| Nama Majalah/Jurnal | : | Mathematics Magazine |
| Volume / Edisi | : | 99 (No. 3) |
| Halaman | : | 224-232 |
| Abstrak | : | One challenge (or opportunity!) that many instructors face is how varied the backgrounds, abilities, and interests of students are. In order to simultaneously instill confidence in those with weaker preparations and still challenge those able to go faster, an instructor must be prepared to give problems of different difficulty levels. Using Dirichlet’s theorem as a case study, we create and discuss a family of problems in number theory that highlight the relative strengths and weaknesses of different ways to approach a question and show how to invite students to extend the problems and explore research-level mathematics. |
A Synthetic Geometry Proof of Heron’s Formula
| Pengarang | : | Colin Beveridge |
| Nama Majalah/Jurnal | : | Mathematics Magazine |
| Volume / Edisi | : | 99 (No. 3) |
| Halaman | : | 220-223 |
| Abstrak | : | Heron's formula, which dates back to at least the first century CE, links the area of a triangle to the lengths of its sides. This paper offers a proof of the formula using the triangle's incircle and the properties of similar triangles. |
Optimization and Learning via Stochastic Gradient Search
| Pengarang | : | Marissa Masden |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 604-608 |
| Abstrak | : | What is stochastic gradient search, and who is interested in it? Optimization and Learning via Stochastic Gradient Search by Felisa Vásquez-Abad and Bernd Heidergott is contemporary and timely in answering these questions. This book (henceforth referred to as OLSGS) details some of the mathematics that underpins modern algorithms in fields such as computer science, economics, and control. It is pitched both as a textbook for graduate learners and as a reference for practitioners, aiming for an intuitive, algorithm-first pedagogy that is then supported by theory. The book advertises itself as one which sits at the boundary between theory and application. This balance is extremely difficult to strike well; my interest in finding books that succeed at such a task is the primary lens through that I am writing this review. |
An Intriguing Limit
| Pengarang | : | Lorenzo Fornari, Enrico Laeng & Vittorino Pata |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 589-592 |
| Abstrak | : | The authors thank the anonymous referees for their valuable comments. |
Classification of d for Which the Period Length of the Continued Fraction Expansion of ?d is Odd
| Pengarang | : | Jun Ho Lee |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 584-588 |
| Abstrak | : | Let d be a non-square positive integer. In this paper, we show that if the period length of the continued fraction expansion of √???? is odd, then ????≡1 or 2 (mod?4). This provides a simple classification criterion for d based solely on congruence conditions. Our approach relies only on basic properties of continued fraction expansions, offering an elementary perspective on a classical problem. |
Minkowskis Theorem on Archimedean Lattices
| Pengarang | : | Dana Paquin & Julian Schennach |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 578-583 |
| Abstrak | : | Minkowski’s celebrated theorem, bounding the area of centrosymmetric convex sets intersecting only one point of a lattice, has broad implications from number theory to geometry. But what about discrete sets generated by symmetries other than translations? Here we explore extensions for all lattices constructed from Archimedean tilings of the Euclidean plane and observe how symmetry constraints and optimality interact in intricate ways. |