We welcome you to the March issue of the *Monthly* where Sam Vandervelde shows how a counting problem involving “polite” placements of arrows is related to the combinatorial numbers named after Julius Worpitzky; Russell Gordon and José Santos introduce a class of geometric construction problems and give conditions for their classical (i.e., straightedge and compass) constructability; Gilbert Strang shows us some of the more interesting ways to multiply and factor matrices; David Salwinski studies continuous binomial coefficients, culminating in a continuous analogue of the binomial theorem; and Maxim Gilula gives a countable family of permutations of the natural numbers for which the sums of rearrangements of conditionally convergent series of real numbers can be computed explicitly.

In the Notes Leo Goldmakher and Paul Pollack refine Lagrange’s four-square theorem, Peter Shiu offers a refinement to the three gaps theorem, Andrew Leahy compares Torricelli’s 1644 proof of Archimedes’ quadrature formula with earlier ones, and Marcin Mazur establishes a new inequality relating the volume and edge lengths of a tetrahedron.

If it isn’t spring yet where you are, then ward off the cold by solving problems. Or perhaps you might curl up with a good book, such as *The Joy of SET* by Liz McMahon, Gary Gordon, Hannah Gordon, and Rebecca Gordon, which is reviewed by Darren Glass.

— Susan Jane Colley, Editor

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## Table of Contents

### Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2018 to David Bressoud for Distinguished Service to Mathematics

p. 195.

Annalisa Crannell

DOI: 10.1080/00029890.2018.1408288

### The Worpitzky Numbers Revisited

p. 198.

Sam Vandervelde

We relate how the Worpitzky number triangle emerges in the setting of a novel counting problem that involves placing up- and right-arrows in a rectangular grid in a polite manner. We next survey various properties, relying on a mixture of combinatorial argument and the functional equation for Worpitzky polynomials. We conclude with a short proof of the Von Staudt–Clausen theorem, a context in which Worpitzky numbers arose.

DOI: 10.1080/00029890.2018.1408347

### An Interesting Construction Problem

p. 207.

Russell A. Gordon & José Carlos Santos

We present a collection of simple construction problems and determine conditions for which the construction can be carried out with compass and straightedge. Since the problems reduce to finding the roots of quartic polynomials, it is not sufficient to show that the polynomials are irreducible to conclude that the roots are not constructible. Some interesting results appear as we discover general conditions that determine when the solutions are or are not constructible.

DOI: 10.1080/00029890.2018.1408357

### Multiplying and Factoring Matrices

p. 223.

Gilbert Strang

All of us learn and teach matrix multiplication using rows times columns. Those inner products are the entries of *AB*. But to go backward—to factor a matrix into triangular or orthogonal or diagonal matrices—outer products are much better. Now *AB* is the sum of columns of *A* times rows of *B*: rank one matrices. Our goal is to produce those columns and rows as simply as possible for *A* = *LU* (elimination) and *A* = *CE* (echelon form) and *A* = *QR* (Gram–Schmidt). Diagonalization by eigenvectors and by singular vectors is also expressed by columns times rows.

DOI: 10.1080/00029890.2018.1408378

### The Continuous Binomial Coefficient: An Elementary Approach

p. 231.

David Salwinski

In this article, we use elementary methods to investigate continuous binomial coefficients: functions *yx* of the real variable *x* defined by way of the gamma function with *y* a fixed real number. We begin with a brief qualitative description of these functions and then derive several interesting representations of them including an infinite product and Taylor series. We also prove various integral formulas involving continuous binomial coefficients, many of which remarkably mirror summation formulas of the familiar binomial coefficients. We conclude by proving a continuous analog of the binomial theorem.

DOI: 10.1080/00029890.2017.1409570

### A Class of Simple Rearrangements of the Alternating Harmonic Series

p. 245.

Maxim Gilula

We present an easily defined countable family of permutations of the natural numbers for which explicit rearrangements (i.e., the sums induced by the permutations) can be computed. The digamma function proves to be the key tool for the computations found here for the alternating harmonic series. The permutations φ under consideration are simple in the sense that φ○φ is the identity function. We show that the countable set of rearrangements obtained from the permutations considered below is dense in the reals.

DOI: 10.1080/00029890.2017.1409571

## NOTES

### Refinements of Lagrange’s Four-Square Theorem

p. 258.

Leo Goldmakher & Paul Pollack

A well-known theorem of Lagrange asserts that every nonnegative integer n can be written in the form *a*^{2} + *b*^{2} + *c*^{2} + *d*^{2}, where *a*, *b*, *c*, *d* ∈ ℤ. We characterize the values assumed by *a* + *b* + *c* + *d* as we range over all such representations of *n*.

DOI: 10.1080/00029890.2018.1409575

### A Footnote to the Three Gaps Theorem

p. 264.

Peter Shiu

Let α > 0 be an irrational number and *n* > 1. The three gaps theorem states that, when the fractional parts of α, 2α, …, *n*α are arranged in an ascending order, the gaps between successive terms may take at most three distinct values. We give a criterion for the exact number of distinct values being taken.

DOI: 10.1080/00029890.2018.1412210

### The Method of Archimedes in the Seventeenth Century

p. 267.

Andrew Leahy

In his *Quadrature of the Parabola Solved by Many Methods through the New Geometry of Indivisibles*, Evangelista Torricelli presented a proof of Archimedes’ quadrature formula that closely resembles Proposition I of Archimedes’ *Method of Mechanical Theorems*. We look at the historical precursors to this rediscovery, and compare the two proofs.

DOI: 10.1080/00029890.2018.1413857

### An Inequality for the Volume of a Tetrahedron

p. 273.

Marcin Mazur

In this note we prove a curious inequality involving the sides and volume of a tetrahedron.

DOI: 10.1080/00029890.2018.1411741

## Problems and Solutions

p. 276.

DOI: 10.1080/00029890.2018.1424478

## Book Review

p. 284.

*The Joy of SET* by Liz McMahon, Gary Gordon, Hannah Gordon, and Rebecca Gordon

Reviewed by Darren Glass

DOI: 10.1080/00029890.2018.1412661

## MathBits

### The Harmonic Series Diverges

p. 222.

### AdaM and GrahaM Play the Stock Market

p. 257.

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

p. 272.