This issue has diverse articles that should offer something for every reader. Charles Groetsch uses calculus and elementary differential equations to examine the relationship between mass and fall time under two models of resistance treated in Newton's *Principia*. Also with a historical bent, Dave Richeson defines a trisectrix—a curve that can be used to trisect an angle—based on ideas from 1928 when Henry Scudder described how to use a carpenter's square to trisect an angle. Gaston Brouwer generalizes the double-angle formulas for trigonometric functions to generate identities in the spirit of Morrie's law. Other items in the issue include David Treeby's use of centers of mass to generate some combinatorial identities and Bernhard Klaassen's definition of a spiral tiling. Manuel Ricardo Falcão Moreira and S. Muralidharan each examine well-known problems from new perspectives. Moreira uses symmetry to study Kaprekar's map on four digits to show that 6174 is the unique fixed point. Muralidharan applies divide and conquer to solve the 15 puzzle. Robert Foote provides a formula that unifies the Pythagorean theorem for Euclidean, spherical, and hyperbolic geometries. Besides proofs without words, the Problems, and the Reviews, Tracy Bennett offers a crossword puzzle, Mathematics in Love, a not-so-obvious nod to Valentine's day.

The solutions to the 77th William Lowell Putnam Competition that was held in December 2016 will appear in the April 2017 issue. Because of the new production time table and the desire to include student solutions, it is not possible to provide the solutions in the February issue any longer.

—*Michael A. Jones, Editor*

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Vol. 90, No. 1, pp. 1 – 84

## Articles

### Hammer and Feather: Some Calculus of Mass and Fall Time

p. 3.

C. W. Groetsch

Techniques from calculus and elementary differential equations are used to explore the relationship between mass and fall time in the two models of resistance treated by Newton in his *Principia*. The material can be used to enrich various undergraduate classes and to acquaint, and perhaps interest, students in an aspect of the fascinating story of mathematization of nature.

To purchase from JSTOR: 10.4169/math.mag.90.1.3

### A Trisectrix from a Carpenter's Square

p. 8.

David Richeson

In 1928, Henry Scudder described how to use a carpenter's square to trisect an angle. We use the ideas behind Scudder's technique to define a trisectrix—a curve that can be used to trisect an angle. We also describe a compass that could be used to draw the curve.

To purchase from JSTOR: 10.4169/math.mag.90.1.8

### A Generalization of the Angle Doubling Formulas for Trigonometric Functions

p. 12.

Gaston Brouwer

The angle doubling formula sin 2*θ* = 2sin *θ *cos *θ* for the sine function is well known. By replacing the cosine in this formula with sin (*π*/2 - *θ*), we see that sin 2*θ* can be written as the product of two sine functions where the second sine function is obtained from the basic sine function by only using a phase shift of the angle *θ* and a reflection about the horizontal axis. In this paper, we will show that, for any natural number *n*, sin *nθ* can be written as the product of *n* sine functions involving only phase shifts of the angle *θ* and a possible reflection about the horizontal axis. Similar formulas will be derived for the cosine and tangent functions.

To purchase from JSTOR: 10.4169/math.mag.90.1.12

### A Moment's Thought: Centers of Mass and Combinatorial Identities

p. 19.

David Treeby

We provide proofs of well-known formulae using physical arguments. Specifically, we locate the center of mass of a configuration of masses two different ways, then equate the results. Most notably, we show how this idea leads to a new proof without words for the sum of squares of consecutive natural numbers. We also demonstrate how the method can be profitably applied to certain combinatorial identities, and Fibonacci summations.

To purchase from JSTOR: 10.4169/math.mag.90.1.19

### How to Define a Spiral Tiling?

p. 26.

Bernhard Klaassen

A precise mathematical definition is given for spiral plane tilings. It is not restricted to monohedral tilings and is tested on a series of examples from literature. Unwanted cases from regular tilings can be excluded. In case of one single arm a modified definition can be applied. Also the special case of locally infinite tilings with one singular point can be treated with any number of spiral arms. The question whether such a definition in mathematical terms could be given was posed by Grünbaum and Shephard in the late 1970s.

To purchase from JSTOR: 10.4169/math.mag.90.1.26

### Dihedral Symmetry in Kaprekar's Problem

p. 39.

Manuel R. F. Moreira

In a problem involving an operation on four digit integers, introduced more than sixty years ago, a dihedral type of symmetry is used to prove the existence of a unique fixed point and to assess the speed of convergence towards this fixed point. Additional symmetry points the way to a formula to compute how far an integer is from the fixed point in terms of this operation.

To purchase from JSTOR: 10.4169/math.mag.90.1.39

### The Fifteen Puzzle-A New Approach

p. 48.

S. Muralidharan

We give an elementary, nongroup theoretic proof tat exactly half of the 15! arrangements of the fifteen puzzle can be restored to the natural order.

To purchase from JSTOR: 10.4169/math.mag.90.1.48

### Proof Without Words: ℓ^{1}(ℝ) is a Subset of ℓ^{2}(ℝ)

p. 58.

Juan Luis Varona

We prove that ℓ^{1}(ℝ) is a subset of ℓ^{2}(ℝ) by means of a spiral.

To purchase from JSTOR: 10.4169/math.mag.90.1.58

### A Unified Pythagorean Theorem in Euclidean, Spherical, and Hyperbolic Geometries

p. 59.

Robert L. Foote

We state a formula for the Pythagorean theorem that is valid in Euclidean, spherical, and hyperbolic geometries and give a proof using only properties the geometries have in common.

To purchase from JSTOR: 10.4169/math.mag.90.1.59

### Euler's Favorite Proof Meets a Theorem of Vantieghem

p. 70.

Konstantinos Gaitanas

This article may be regarded as a connection between the past and the present. In 2008 E. Vantieghem proved a new primality criterion whose proof requires the use of cyclotomic polynomials. Almost two and a half centuries ago, Euler used a not-so-well-known method of proof to prove Fermat's little theorem. In this article we show that Euler's method can be adoted in order to prove Vantieghem's result in a more elementary way.

To purchase from JSTOR: 10.4169/math.mag.90.1.70

### Mathematics in Love

p. 73.

Tracy Bennett

To purchase from JSTOR: 10.4169/math.mag.90.1.73

## Problems and Solutions

p. 75.

Proposals, 2011-2015

Quickies, 1067-1068

Solutions, 1981-1985

Answers, 1067-1068

To purchase from JSTOR: 10.4169/math.mag.90.1.75

## Reviews

p. 83.

Simple math problems no one can solve; the peril of p-values; going for 2

To purchase from JSTOR: 10.4169/math.mag.90.1.83