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**Central Force Laws, Hodographs, and Polar Reciprocals **

by G. Don Chakerian

There is no question that Newton gave an impeccable derivation of his inverse square law of gravitation from Kepler's laws of planetary motion; but there is a continuing controversy as to whether the Principia provides adequate justification for the converse. The hodograph of an orbit, produced by translating all the velocity vectors to the origin, proves very useful in analyzing the connections among these ideas. One striking result is this: if an orbit satisfies KeplerÕs second law (equal areas in equal times) then its hodograph is similar to the polar reciprocal of the orbit. When the orbit is a conic section with the sun at one focus, this leads directly to the inverse square law.

**Folding Quartic Roots **

by B. Carter Edwards and Jerry Shurman

Recent articles show how certain geometric systems---origami or the "mira"---go beyond straight-edge and compass to solve cubic equations. The key is to construct the common tangents to two parabolas. While this idea is not new, origami and the mira share the novel feature of constructing the tangents from the parabolas' foci and directrices rather than from the parabolas themselves. Thus these methods really use only points and lines.

This note shows how to solve certain quartic equations similarly, using the common tangents to a parabola and a circle. The geometric construction requires origami-type folds and a compass construction, thus using points, lines, and one circle.

**Cardan Polynomials and the Reduction of Radicals**

by Thomas J. Osler

In this paper a method is presented for examining and simplifying certain nested radical expressions. For this purpose, we present a new family of polynomials, useful in acheiving the simplifications. These are called Cardan polynomials because the formula for their roots resembles Cardan's solution for the cubic equation. The Cardan polynomials are a normalized form of the Chebyshev polynomials.

**Avoiding Your Spouse at a Bridge Party **

by Barbara Haas Margolius

The Bridge Couples Problem: Suppose n married couples (2n people) are invited to a bridge party. Bridge partners are chosen at random, without regard to gender. What is the probability that no one will be paired with his or her spouse?

In this Note, I prove that this probability converges very slowly to e^{{-1/2}}. In fact, to achieve ten decimal places of accuracy in an estimate for e^{{-1/2}} based on our bridge couples problem probabilities, we would have to invite more than 2 billion couples to our party.

**The Cwatset of a Graph **

by Nancy-Elizabeth Bush and Paul A. Isihara

From Sherman & Wattenberg's groundbreaking 1994 introduction in this Magazine, to D.K. Biss winning the Morgan Prize for excellence in undergraduate research four years later, **cwatset theory **still continues to be a hot topic for undergraduate research. That a cwatset may be used to describe the differences between a graph and its isomorphic equivalents is yet another example of a nice undergraduate research discovery. A host of interesting theorems, conjectures and open problems arise in conjunction with this fundamental connection between cwatsets and graphs.

**Period-3 Orbits via Sylvester's Theorem and Resultants**

by Jacqueline Burm and Paul Fishback

A common problem in discrete dynamical systems involves using analytic methods to determine exact bifurcation values. In this Note, we show how the resultant is a useful tool for doing so when the system involves polynomial mappings. A method using resultants is constructed and then applied to the logistic function and Henon mapping.

**Writing Numbers in Base 3, the Hard Way**

by Gary E. Michalek

In expressing a number in base ten, the numbers 0 through 9 are used as the coefficients of powers of ten. In base two, the coefficients of the powers of two are 0 and 1. Negative numbers are expressible in base two if we use -1 as a coefficient as well. We say that the set consisting of 0, 1 and -1 is a base two coefficient system. In this note we consider coefficients for base three. The usual coefficient system consists of 0, 1, 2, and their opposites. We consider other sets that could be used in place of this. We begin by describing a method for determining whether a set is a coefficient system. Some necessary conditions are examined, and this provides motivation for a general result about the set of differences of the elements in a list of representatives of the congruence classes modulo 3. This set is a coefficient system if and only if the representatives of the congruence classes are relatively prime.

**Proof Without Words: Look Mom, No Substitution! **

by Marc Chamberland

A definite integral that most students would probably solve by trigonometric substitution is evaluated by a simple picture.

**Permutations and Coin-Tossing Sequences**

by David Callan

The answers to two problems, one involving permutations and the other involving coin-tossing sequences, unexpectedly turn out to be the same. This note gives a combinatorial explanation of the phenomenon.

**An Easy Solution to Mini Lights Out **

by Jennie Missigman and Richard A. Weida

Mini Lights Out is a puzzle played on a four-by-four array of lighted buttons. Each button is either on or off. When a button is pressed, it and its vertical horizontal neighbors, including "wrapping" around the edges of the array, change state: off to on, or on to off. One is given a configuration of buttons some on, some off, and the goal is to turn all the light off by pressing the correct buttons. Using undergraduate linear algebra, we show that every possible configuration is solvable, and show an easy way to determine the solution.

**Proof Without Words: Logarithm of a Number and Its Reciprocal **

by Vincent J. Ferlini

The area interpretation of the integral illustrates geometrically that the natural logarithm of a number has the negative value of the logarithm of its reciprocal.

**The Classification of Groups of Order 2p**

by Joseph A. Gallian

In this note we give a simple proof of the well-known theorem that the only groups of order 2p, where p is an odd prime, are the cyclic group and the dihedral group. Our argument uses only Lagrange's Theorem and properties of cosets.

**A Characterization of Infinite Cyclic Groups**

by Charles P. Lanski

We prove a nontrivial characterization of infinite cyclic groups, using only basic notions about groups: orders of elements, cyclic groups, cosets, commutators, and quotients. When a group has a subgroup of infinite index, that group is surely not cyclic. We show that an infinite group must be cyclic if each of its nonidentity subgroups has finite index.

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61st Annual William Lowell Putnam Mathematical Competition