# Mathematics Magazine - June 2017

For many, it is the end of the school year. For summer reading, check out Seth Zimmerman’s optimal group testing algorithm and its connection to the Fibonacci numbers. Yoga Gerchak and Marc Kilgour get you prepared for tennis by considering whether you should serve differentially on your first or second serves. Leanhardt and Parker consider Fontaine’s method, the precursor to the modern integrating factor method to solve ordinary differential equations. Other topics covered include the Law of Sines, Law of Cosines, Pythagorean theorem, Ptolemy’s theorem, irrational roots, the Maclaurin inequality, hypergeometric functions, and lattice walks.

Enjoy the weather and the reading.

Michael A. Jones, Editor

##### JOURNAL SUBSCRIBERS AND MAA MEMBERS:

Vol. 90, No. 3, pp. 165 – 240

## Articles

### Detecting Deficiencies: An Optimal Group Testing Algorithm

p. 167.

Seth Zimmerman

The use of group testing to locate all instances of disease in a large population of blood samples was first considered more than 70 years ago. Since then, several procedures have been used to lower the expected number of tests required. The algorithm presented here, in contrast to previous ones, takes a constructive rather than a top-down approach. As far as could be verified, it offers the first proven solution to the problem of finding a predetermined procedure that guarantees the minimum expected number of tests. Computer results strongly suggest that the algorithm has a Fibonacci-based pattern.

To purchase from JSTOR: 10.4169/math.mag.90.3.167

### Proof Without Words: Arctangent of Two and the Golden Ratio

p. 179.

Ángel Plaza

It is proved without words that the golden ratio, φ, and the arctangent of 2 are related.

To purchase from JSTOR: 10.4169/math.mag.90.3.179

### A Proof of the Law of Sines Using the Law of Cosines

p. 180.

Patrik Nystedt

We give a proof of the law of sines using the law of cosines.

To purchase from JSTOR: 10.4169/math.mag.90.3.108

### Extending Two Classic Proofs of the Pythagorean Theorem to the Law of Cosines

p. 182.

Kevin K. Ferland

Two well-known proofs of the Pythagorean theorem are generalized to prove the law of cosines in a geometrically elegant way by computing areas.

To purchase from JSTOR: 10.4169/math.mag.90.3.182

### Proof Without Words: Pythagorean Theorem via Ptolemy’s Theorem

p. 187.

Nam Gu Heo

We provide a visual proof of Pythagorean theorem. The main idea of the proof is to compute (a + b)2 in two different ways: one with aid of Ptolemy's theorem and the other one by dissecting a square.

To purchase from JSTOR: 10.4169/math.mag.90.3.187

### Serving Strategy in Tennis: Accuracy versus Power

p. 188.

Yigal Gerchak and D. Marc Kilgour

In tennis, the server has an advantage—the opportunity to serve again without penalty, if the attempt results in a fault. A common strategy is to hit a powerful or “tricky” first serve followed, if necessary, by a weaker second serve that has a lower probability of faulting even if it is easier to return. Recently, commentators have argued that this standard strategy is flawed and that the second serve should be as difficult to return as the first. This advice contradicts Gale's theorem, which we reformulate and provide with a new (analytic) proof. Then we extend it with a model of the rally that follows the successful return of a serve, providing additional insight into the relative effectiveness of the “two first serves” strategy. The only tools we use are basic probability and introductory calculus.

To purchase from JSTOR: 10.4169/math.mag.90.3.188

### The M&M Game: From Morsels to Modern Mathematics

p. 197.

Ivan Badinski, Christopher Huffaker, Nathan McCue, Cameron N. Miller, Kayla S. Miller, Steven J. Miller, and Michael Stone

To an adult, it's obvious that the day of someone's death is not precisely determined by the day of birth, but it's a very different story for a child. We invented what we call the M&M Game to help explain randomness: Given k people, each simultaneously flips a fair coin, with each eating an M&M on a head and not eating on a tail. The process then continues until all M&M'S are consumed, and two people are deemed to die at the same time if they run out of M&M'S together. We analyze the game and highlight connections to the memoryless process, combinatorics, statistical inference, and hypergeometric functions.

To purchase from JSTOR: 10.4169/math.mag.90.3.197

### Fontaine’s Forgotten Method for Inexact Differential Equations

p. 208.

Aaron E. Leanhardt and Adam E. Parker

In 1739 Alexis-Claude Clairaut published the modern integrating factor method of solving inexact ordinary differential equations (ODEs). He was motivated by a 1738 Alexis Fontaine paper with a different method which requires solving a difficult partial differential equation (PDE). Here we revisit Fontaine's method, examine his modest attempt to solve the PDE, and utilize a different technique to give (we believe) the first family of ODEs solvable by Fontaine's method with no obvious solution using the modern technique.

To purchase from JSTOR: 10.4169/math.mag.90.3.208

### Proof Without Words: An Elegant Property of a Triangle Having an Angle of 60 Degrees

p. 220.

Victor Oxman and Moshe Stupel

In a triangle ABC in which angle A measures 60 degrees, the bisectors of angles B and C are used to construct a cyclic quadrilateral with two congruent sides.

To purchase from JSTOR: 10.4169/math.mag.90.3.220

### Some Probabilistic Interpretations of the Multinomial Theorem

p. 221.

Kuldeep Kumar Kataria

We give a simple probabilistic proof of an important combinatorial identity. In the process, we show via probabilistic arguments that there are exactly  terms in the multinomial expansion of (λ1 + λ2 + ⋯ + λm)n. Also, an alternate probabilistic proof of the multinomial theorem is obtained using the convolution property of the Poisson distribution.

To purchase from JSTOR: 10.4169/math.mag.90.3.221

### Is This the Easiest Proof That nth Roots are Always Integers or Irrational?

p. 225.

Jeffrey Bergen

Most proofs that nth roots of positive integers are always integers or irrational involve concepts such as prime numbers, relative primeness, or the fundamental theorem of arithmetic. We provide an argument which uses only the well ordering principle and the division algorithm. You can decide if it is the easiest proof possible.

To purchase from JSTOR: 10.4169/math.mag.90.3.225

### MathFest 2017

p. 226.

Brendan Sullivan

To purchase from JSTOR: 10.4169/math.mag.90.3.226

### The Maclaurin Inequality and Rectangular Boxes

p. 228.

Claudi Alsina and Roger B. Nelsen

To purchase from JSTOR: 10.4169/math.mag.90.3.228

## Problems and Solutions

p. 231.

Proposals, 2021-2025

Quickies, 1071-1072

Solutions, 1991-1995