The Monthly helps March 2015 come in like a lion! Join our celebration of the "BIG" Pi Day with our second annual Pi Day paper " I Prefer Pi: A Brief History and Anthology of Articles in the American Mathematical Monthly," by Jonathan Borwein and Monthly Editor Scott Chapman. Have a look at the large role that Pi played in the development of the Monthly during its 120 year long history. Also, our celebration continues when we honor in our lead article the MAA's 2015 Gung and Hu Award winner W. James Lewis. Rueben Hersch reviews David Tall's "How Humans Learn to Think Mathematically. Exploring the Three Worlds of Mathematics" and if all this is not enough, the Problem Section will keep you more than busy. Happy Pi Day, and don't forget the ice cream! - *Scott T. Chapman, Editor*

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Volume 122, Issue 03, pp. 189 - 296

## Table of Contents

## Articles

### Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2015 to W. James Lewis for Distinguished Service to Mathematics

Jennifer J. Quinn

The Mathematical Association of America presented the 2015 Gung and Hu Award for Distinguished Service to Mathematics to W. James Lewis for his outstanding contributions to the mathematics education of teachers, for his leadership in the mathematics profession and in academia at all levels, for his work increasing the visibility and participation of women in mathematics, for his exemplary work serving the state of Nebraska, and especially for his vision and ability to bring together diverse stakeholders in support of positive change in mathematics.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.191

### I Prefer Pi: A Brief History and Anthology of Articles in the *American Mathematical Monthly*

Jonathan M. Borwein and Scott T. Chapman

In celebration of both a special “big” *π* Day (3/14/15) and the 2015 centennial of the Mathematical Association of America, we review the illustrious history of the constant *π* in the pages of the *American Mathematical Monthly*.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.195

### Max Dehn and the Origins of Topology and Infinite Group Theory

David Peifer

This article provides a brief history of the life, work, and legacy of Max Dehn. The emphasis is put on Dehn’s papers from 1910 and 1911. Some of the main ideas from these papers are investigated, including Dehn surgery, the word problem, the conjugacy problem, the Dehn algorithm, and Dehn diagrams. A few examples are included to illustrate the impact that Dehn’s work has had on subsequent research in logic, topology, and geometric group theory.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.217

### A Simple Answer to Gelfand’s Question

Jaap Eising, David Radcliffe and Jaap Top

Using elementary techniques, a question named after the famous Russian mathematician I. M. Gelfand is answered. This concerns the leading (i.e., most significant) digit in the decimal expansion of integers *2n, 3n, . . . , 9n* . The history of this question, some of which is very recent, is reviewed.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.234

### Using Undetermined Coefficients to Solve Certain Classes of Variable-Coefficient Equations

Doreen De Leon

This paper considers the question of when undetermined coefficients may be applied to compute the particular solution of certain second order nonhomogeneous differential equations with variable coefficients. A few examples illustrate the results of this investigation.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.246

### Variations on a Generating-Function Theme: Enumerating Compositions with Parts Avoiding an Arithmetic Sequence

Matthias Beck and Neville Robbins

A composition of a positive integer n is a k-tuple *(λ1, λ2, . . . , λk ) ∈ Zk >0* such that *n = λ1 + λ2 + ・ ・ ・ + λk* . Our goal is to enumerate those compositions whose parts *λ1, λ2, . . . , λk* avoid a fixed arithmetic sequence. When this sequence is given by the even integers (i.e., all parts of the compositions must be odd), it is well known that the number of compositions is given by the Fibonacci sequence. A much more recent theorem says that when the parts are required to avoid all multiples of a given integer *k*, the resulting compositions are counted by a sequence given by a Fibonacci-type recursion of depth *k*. We extend this result to arbitrary arithmetic sequences. Our main tool is a lemma on generating functions that is no secret among experts but deserves to be more widely known.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.256

## Notes

### Area Minimizing Polyhedral Surfaces are Saddle

Anton Petrunin

We show that area minimizing minimal polyhedral surfaces are saddle.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.264

### Open Discontinuous Maps from R^{n} to R^{n}

Bill Goldbloom Bloch

In 1962 Spira found an example of open map from R*n* onto R*n* that is discontinuous at a Cantor set. Subsequent published examples of everywhere discontinuous open maps from Rn onto Rn are either nonmeasurable, not computable, or difficult to visualize. We provide a simple example. Published examples of open discontinuous maps from Rn onto Rn are discontinuous at infinitely many points and are infinite-to-one. We give two more examples of open discontinuous maps from R*n* onto R*n* ; the first is discontinuous at only one point, and the second is at most two-to-one.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.268

### An Application of Modular Hyperbolas to Quadratic Residues

Mizan R. Khan and Richard Magner

There are many elementary proofs of the classical result that −1 is a quadratic residue of an odd prime p if and only if *p* ≡ 1 (mod 4). In this note we prove this result by using the symmetries of a modular hyperbola. Consequently, our proof has a more geometric flavor than many of the other proofs.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.272

### Finite Groups with a Certain Number of Cyclic Subgroups

Marius Tărnăuceanu

In this short note we describe the finite groups *G *having |*G*| − 1 cyclic subgroups. This leads to a nice characterization of the symmetric group *S*3.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.275

### Powers of Matrices and Algebraic Centrosymmetry

John Konvalina

We prove an unexpected hidden symmetry of arbitrary square matrices. The powers of any square matrix have an inherent symmetry property that we call *algebraic centrosymmetry*. Using the multivariable polynomial representation for the entries in a power matrix derived from the definition of matrix multiplication, we show the bottom half of the matrix is effectively the algebraic “reverse” of the top half of the matrix.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.277

### Partitioning R^{n} into Connected Components

Peter Horak

Intuition is one of the crucial parts in mathematical research. However, it might turn out to be a deceiving part. Statements that seem to be obvious are very hard to prove or they are even not valid. In this note we describe an example that has both features. Although it is “obvious,” its proof in R2 is based on a deep result and it is not true in R3.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.280

## Problems and Solutions

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.284

## Book Review

*How Humans Learn to Think Mathematically. Exploring the Three Worlds of Mathematics* by David Tall

Reviewed by Reuben Hersh

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.03.292

## MathBits

### A Geometric Proof of Morrie’s Law

Samuel G. Moreno and Ester M. García-Caballero

### An Explicit Formula for the Prime Counting Function Which is Valid Infinitely Often

Konstantinos Gaitanas

### 100 Years Ago in this American Mathematical Monthly

Edited by Vadim Ponomarenko