Our May issue is packed with outstanding papers. Not only do we have articles from multiple Ford-Halmos award winning authors (Tom Apostol/Mamikon Mnatsakanian and Harold Boas), but we lead off the issue with Søren Eilers' "The LEGO Counting Problem." Eilers goes into detail in this paper to show us the mistakes that the LEGO group has made over the years in counting the number of different arrangements of six different identically colored 2X4 LEGO bricks. Need an extremely short proof of the Hairy Ball Theorem? You have it in our Notes Section from Peter McGrath. Stay tuned for the June/July issue when Janos Kollar dedicates "Arcology" to the memory of John Nash.

- *Scott T. Chapman, Editor*

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## Table of Contents

### The LEGO Counting Problem

Søren Eilers

We detail the history of the problem of deciding how many ways one may combine n 2 × 4 LEGO bricks, and explain what is known—and not known—about the related question of how these numbers grow with *n*.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.5.415

### Mocposite Functions

Harold P. Boas

Traditional mathematical notation can lead to confusion. Expressions that appear to define composite functions sometimes do not. A particular example with engineering applications is studied in detail.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.5.427

### A New Look at Surfaces of Constant Curvature

Tom M. Apostol and Mamikon A. Mnatsakanian

Equality of zonal areas on a sphere and pseudosphere is extended by elementary geometric methods to surfaces of revolution of constant total (Gaussian) curvature, and constant mean (Delaunay) curvature. Bicycle wheels are used to trace profiles of these surfaces. Surprisingly, cycloids appear as limiting cases of such profiles.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.5.439

### Hardy's Reduction for a Class of Liouville Integrals of Elementary Functions

Jaime Cruz-Sampedro and Margarita Tetlalmatzi-Montiel

This paper is concerned with a class of integrals whose integrands are the product of a rational function times the exponential of a nonconstant rational function. We call these *Liouville integrals*. For these integrals, we provide a student-friendly algorithm producing a two-term decomposition with minimum transcendental and maximum elementary components. This decomposition fulfills the conditions of Hardy's reduction theory, determines whether these integrals are elementary functions, and when in the affirmative, finds them. To achieve our goal, we use partial fraction decomposition, simple notions of linear algebra, and a special case of an 1835 theorem of Liouville that we refer to as *Liouville's criterion on integration*. There is in the literature a complete algorithm to decide if the integral of an elementary function is also elementary. Ours is a gentle alternative for the class of Liouville integrals.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.5.448

### Fast and Simple Modular Interpolation Using Factorial Representation

G. L. Mullen, D. Panario and D. Thomson

We study a representation for polynomial functions over finite rings. This *factorial representation* is particularly useful for fast interpolation, and we show that it is computationally preferable to the Lagrange Interpolation Formula (LIF) and to Newton interpolation over finite fields and rings. Moreover, over arbitrary finite rings the calculation of the factorial representation aborts naturally when a given mapping does not arise as a polynomial function.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.5.471

## Notes

### A Corrigendum to Unreasonable Slightness

Arseniy Sheydvasser

We revisit Bogdan Nica's 2011 paper, “The Unreasonable Slightness of *E*2 over Imaginary Quadratic Rings” and correct an inaccuracy in his proof.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.5.482

### Euler and the Strong Law of Small Numbers

Karl Dilcher and Christophe Vignat

We follow an incorrect entry in a well-known table of series and products through several earlier tables and books, all the way to the relevant (correct) identity in the work of Euler. Along the way we explain what may have led to the error.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.5.486

### Explicit Additive Decomposition of Norms on ℝ2

Iosif PinelisNeel Patel

A well-known result by Lindenstrauss is that any two-dimensional normed space can be isometrically imbedded into *L*1(0, 1). We provide an explicit form of a such an imbedding. The proof is elementary and self-contained.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.5.491

### Parabolas and Archimedes' 2/3-Property

Michael Gaul and Fred Kuczmarski

Archimedes discovered that the area of the region bounded by a parabola and a chord is 2/3 the area of its circumscribing parallelogram with two sides parallel to the parabola's axis. We show that parabolic arcs are the only smooth strictly convex functions with the 2/3-property, where two sides of the circumscribing parallelogram are parallel to the y-axis.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.123.5.497

### An Extremely Short Proof of the Hairy Ball Theorem

Peter McGrath

Using winding numbers, we give an extremely short proof that every continuous field of tangent vectors on *S*2 must vanish somewhere.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.5.502

## Problems and Solutions

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.5.504

## Book Review

*Scientist, Scholar & Scoundrel: A Bibliographical Investigation of the Life and Exploits of Count Guglielmo Libri/Mathematican, Journalist, Patron, Historian of Science, Paleographer, Book Collector, Bibliographer, Antiquarian Bookseller, Forger and Book Thief.* By Jeremy M. Norman

Reviewed by Gerald L. Alexanderson

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.5.512

## MathBits

### An Alternative Approach to the Product Rule

Piotr Josevich

### On Measurable Semigroups in ℝ

Iosif Pinelis

### Rational Nonaxis Points on the Unit Circle Have irrational angles

Tim Hsu, San José

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

Edited by Vadim Ponomarenko