### ARTICLES

**Differential EquationsÂ—Not Just a Bag of Tricks!**

Shirley Llamado Yap

3-14

Differential equations often seems like a hodgepodge of techniques designed to solve very specific equations. There is, however, a unifying theme for many of those techniques, which is to find the right coordinate system with which to express the equation. In the mid-nineteenth century, Sophus Lie discovered how to unify and extend many of the techniques that existed at his time by using the symmetries of differential equations to find good coordinate systems. This paper explains the symmetry method at a level suitable for undergraduates who have taken vector calculus and differential equations.

**f (f (x)) = -x, Windmills, and Beyond**

Martin Griffiths

15-23

In this article we consider the solutions of a particular functional equation and some generalizations. This equation possesses non-trivial yet accessible solutions, and is rather appealing because of the way that the enforced graphical symmetry of any solution implies that it must possess a certain non-obvious analytic property. Indeed, a central theme here is this interplay of the visual with the analytic aspects of the problem. After giving a brief historical overview of functional equations, we go on to study the general properties shared by functions satisfying our particular equation, as well as specific solutions. Finally, solutions to a generalization of the equation are obtained.

**Galileo and Oresme: Who Is Modern? Who Is Medieval?**

Olympia Nicodemi

24-32

If a body is constantly accelerated from rest to a final velocity *v*, then the distance it goes in time *t* is the same as if it had travelled at constant velocity *v*/2. This fact, the Mean Velocity Law, was proved by Galileo on his way to proving the Law of Free Fall, a milestone in the evolution of modern physics. But the Mean Velocity Law itself was well known as far back as the Middle Ages. In this article, we compare Galileo’s mathematical approach to that of the medieval philosopher and mathematician, Nicole Oresme, and we also look at the scientific contextÂ—just what information the law gaveÂ—in each era, with some surprising results.

**Repeating Decimals: A Period Piece**

Kenneth A. Ross

33-45

Consider the repeating decimal of a reduced fraction *t/n* between 0 and 1, where none of the primes 2, 3 or 5 is a factor of *n*. The fraction satisfies the *m*-block property if *m* divides the period ? and, when the repeating portion is broken into *m* blocks of equal length, the sum of the *m* blocks is a string of nines or an integer multiple of a string of nines. An easily-verified sufficient condition is given that implies *t/n* has the *m*-block property. The condition is not necessary. For *n* equal to a prime power *pa* (*p* >= 7), the condition is shown to be sufficient as well. Moreover, *t/pa* has the *m*-block property if and only if *m* is not a power of *p*. (These general results for prime powers seem to be new.) The 2-block property for primes is known as Midy’s theorem (1836). It holds for all prime powers, but not in general. Many examples and a brief history are included.

### NOTES

**Gergonne’s Card Trick, Positional Notation, and Radix Sort**

Ethan D. Bolker

46-49

Gergonne’s three pile card trick has been a favorite of mathematicians for nearly two centuries. This new exposition uses the radix sorting algorithm well known to computer scientists to explain why the trick works, and to explore generalizations. The presentation suggests strategies for introducing the trick and base three arithmetic to elementary school students.

**A GM-AM Ratio**

Conway Xu

49-50

If *s* > -1, then the limit of the ratio of the geometric mean of 1^{s}, . . . , 2^{s},/sup> to their arithmetic mean, as *n* increases to infinity, is (*s* + 1)/*e*^{s}. This is proved using Riemann sums. Similar limits are also established for the arithmetic and geometric means separately.

Finbarr Holland

51-54

Two proofs are given of the limit relation , a result due to Michael Spivey. One is elementary, and suitable for discussion in an introductory course on Analysis; the other is more sophisticated, and uses the machinery of the Lebesgue integral. Generalizations of the result are left as exercises for the reader.

**Letter to the Editor**

Michael Z. Spivey

54-55

**Integer-Coefficient Polynomials Have Prime-Rich Images**

Bryan Bischof, Javier Gomez-Calderon, Andrew Perriello

55-57

It is well known that a nonconstant polynomial *f* (*x*) with integer coefficients produces, for integer values of *x*, at least one composite image. In this note, we use Taylor expansions to improve this elementary result, showing that *f* (*x*) takes an infinite number of composite values. Given a positive integer *n*, we show that *f* (*x*) takes an infinite number of values that are divisible by at least *n* distinct primes, and an infinite number of values that are divisible by *pn* for some prime *p*.

### Proof Without Words:

**Sums of Octagonal Numbers**

Hasan Unal

58

**Yet Another Elementary Solution of the Brachistochrone Problem**

Gary Brookfield

59-63

This note provides an elementary solution of the brachistochrone problem. This problem is to find the curve connecting two given points so that an object slides without friction along the curve from one point to the other point in the least possible time. The key is to introduce a coordinate system where the expected cycloid solutions are built in.

**A Property Characterizing the Catenary**

Edward Parker

63-64

We show that the area under a catenary curve is proportional to its length in the following sense: given a catenary curve, we can take any horizontal interval and examine the ratio of the area under the curve to the length of the curve on that interval, and we find that the resulting ratio is independent of the chosen interval. This property extends to the three-dimensional case as well: the volume contained by a horizontal interval of a catenoid surface is proportional to its surface area in the same sense. We also show from this property that the centroid of the area under an interval of a catenary is the midpoint of the segment connecting the centroid of the catenary and the* x*-axis.

**Problems Section**

65-60

**Reviews Section**

71-72

**Guidelines for Authors**

73-74

**70th Annual William Lowell Putnam Mathematical Competition**

75-80