The October Monthly deals a winning hand! In our lead article, *Early Round Bluffing in Poker*, "California" Jack Cassidy shows us whether or not early round bluffing is a good idea. October also contains our annual review of the William Lowell Putnam Mathematical Competition. Ever have trouble with the proof of the Lebesgue Decomposition Theorem? Tamás Titkos offers us a simple proof in our Notes Section. Ferebee Tunno reviews *Probability Theory in Finance: A Mathematical Guide to the Black–Scholes Formula (Second edition)* by Séan Dineen. Our Problem Section will keep you busy until Halloween. If you have fond memories of the famous Monthly paper "Can You Hear the Sound of a Drum?," then you will love our lead article for November, *The Sound of Symmetry, *by Zhiqin Lu and Julie Rowlett.- *Scott T. Chapman, Editor*

##### JOURNAL SUBSCRIBERS AND MAA MEMBERS:

To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.

Volume 122, Issue 08, pp. 713 - 812

## Table of Contents

## Articles

### The Seventy-Fifth William Lowell Putnam Mathematical Competition

Leonard F. Klosinski, Gerald L. Alexanderson, and Mark Krusemeyer

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.715

### Early Round Bluffing in Poker

California Jack Cassidy

Using a simplified form of the Von Neumann and Morgenstern poker calculations, the author explores the effect of hand volatility on bluffing strategy, and shows that one should never bluff in the first round of Texas Hold'Em.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.726

### Volume/Surface Area Relations for *n*-Dimensional Spheres, Pseudospheres, and Catenoids

Tom M. Apostol and Mamikon A. Mnatsakanian

Circle and sphere properties are extended to the tractrix and pseudosphere, the catenary and catenoid, and to higher-dimensional analogs.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.745

### Besicovitch, Bisection, and the Normality of 0.(1)(4)(9)(16)(25) · · ·

P. Pollack and J. Vandehey

We revisit Besicovitch's 1935 paper in which he introduced several techniques that have become essential elements of modern combinatorial methods of normality proofs. Despite his paper's influence, the results he inspired are not strong enough to reprove his original result. We provide a new proof of the normality of the constant 0.(1)(4)(9)(16)(25) … formed by concatenating the squares, updating Besicovitch's methods.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.757

### On a Recursion Formula Related to Confl uent Vandermonde

Shui-Hung Hou and Edwin Hou

In this short article, we give a novel and elegant proof of a recursion formula related to confluent Vandermonde determinants. This leads to an easy derivation of the well-known determinant identity due to Schendel.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.766

### On Devaney Chaos and Dense Periodic Points: Period 3 and Higher Implies Chaos

Syahida Che Dzul-Kifl i and Chris Good

We look at density of periodic points and Devaney Chaos. We prove that if *f* is Devaney Chaotic on a compact metric space with no isolated points, then the set of points with prime period at least *n* is dense for each *n*. Conversely, we show that if *f* is a continuous function from a closed interval to itself, for which the set of points with prime period at least *n* is dense for each *n*, then there is a decomposition of the interval into closed subintervals on which either *f* or *f*^{2} is Devaney Chaotic. (In fact, this result holds if the set of points with prime period at least 3 is dense.)

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.773

## Notes

### A Viscosity Proof of the Cauchy–Schwarz Inequality

Tadashi Tokieda

The Cauchy–Schwarz inequality for positive quadratic forms has many proofs. This note gives a new derivation that looks unusual at first, but is natural in retrospect, interpreting the quadratic form as kinetic energy and the inequality as dissipation in a viscous flow.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.781

### A Short Proof of the Fact That the Matrix Trace Is the Expectation of the Numerical Values

Tomasz Kania

Using the fact that the normalized matrix trace is the unique linear functional *f* on the algebra of *n* ×*n* matrices which satisfies *f*(*I*) = 1 and *f*(*AB*) = *f*(*BA*) for all *n* × *n* matrices *A* and *B*, we derive a well-known formula expressing the normalized trace of a complex matrix *A* as the expectation of the numerical values of *A*; that is the function ⟨*Ax*, *x*⟩, where *x* ranges the unit sphere of ℂ*n*.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.782

### van der Waerden and the Primes

Levent Alpoge

In this note we prove the infinitude of the primes via an application of van der Waerden's theorem.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.784

### An Exponential Inequality for Symmetric Random Variables

Raphaël Cerf and Matthias Gorny

We prove a simple exponential inequality which gives a control on the first two empirical moments of a sequence of independent identically distributed symmetric real-valued random variables.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.786

### Pairs of Matrices in *GL*_{2}(R_{≥0}) That Freely Generate

Melvyn B. Nathanson

We present an elementary proof that certain pairs of 2 × 2 matrices with nonnegative real coordinates generate free monoids.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.790

### A Simple Proof of the Lebesgue Decomposition Theorem

Tamás Titkos

Motivated by the notion of operator, a short and simple proof of Lebesgue's decomposition theorem is presented in this note.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.793

### A New Proof of the Change of Variable Theorem for the Riemann Integral

Haryono Tandra

We give a new proof of the complete version of the change of variable theorem for the Riemann integral using several basic properties within the elementary theory.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.795

### I Prefer Pi: Addenda

Jonathan Borwein and Scott Chapman

In the rush to prepare our March 2015 article on Pi [**1**], several infelicities escaped our eye. Herein we repair the damage.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.800

## Problems and Solutions

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.801

## Book Review

**Probability Theory in Finance: A Mathematical Guide to the Black–Scholes Formula (Second edition)** By **Se´an Dineen**

Reviewed by Joseph Malkevitch

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.809

## MathBits

### Funny Forms of the Mean Value Theorem

Cristinel Mortici

### Pythagoras by Integral

Zsolt Lengvárszky

### How to Make Equivalent Measures?

Tamás Titkos

### 100 Years Ago This Month in *the American Mathematical Monthly*

Edited by Vadim Ponomarenko