# Mathematics Magazine - October 2002

### ARTICLES

A Lotka-Volterra Three-species Food Chain
Erica Chauvet, Joseph E. Paullet, Joseph P. Previte, and Zac Walls

In this paper, we completely characterize the qualitative behavior of a linear three species food chain where the dynamics are given by classic (non-logistic) Lotka-Volterra type equations. The dynamics of the associated system are quite complicated, as the model exhibits degeneracies, which make it an excellent instructional tool whose analysis involves advanced topics such as: trapping regions, nonlinear analysis, non-isolated equilibria, invariant sets, and Lyapunov-type functions.

Proof by Poem: The RSA Encryption Algorithm
Daniel G. Treat

Take two large prime numbers, q and p. Find the product n, and the totient φ. If e and φ have GCD one and d is e's inverse, then you're done! For sending m raised to the e reduced mod n gives secre-c.

Teaching Mathematics in the Seventeenth and Twenty-first Centuries
Dennis C. Smolarski, S. J.

The 1599 Ratio Studiorum of the Jesuit Order guided the educational practice in Jesuit Schools for several centuries and provided for a role for mathematics unusually prominent for sixteenth century Italy. It included suggestions for student involvement in the learning of mathematics that today might be called "student colloquia" and "interactive review sessions." The primary driving force behind the norms pertaining to mathematics in the Ratio was Christopher Clavius, a Jesuit mathematician at the Roman College, who was convinced about the importance of including mathematics in the curriculum.

Counting Trash in Poker
Joel E. Iiams

When poker was invented there was no such thing as a standard deck. Today a standard poker deck contains fifty-two cards and normally a poker hand consists of five cards. Winning hands are determined using a hierarchy based on frequency. Ties may be broken using a sub-hierarchy. The invention of high-low, split-pot games, appears to be a natural consequence of this sub-hierarchy. A single occurrence of a small perturbation from the norm led us to investigate the uniqueness of poker as we know it today.

4 or 4? Mathematics or Accident?
Ana Luzón and Manuel A. Morón

With this note we want to point out some coincidences between divisibility properties of numbers and topological properties of their figures. From now on, we call number both the concept and the figure. We say also that a number p ≠ 1 is prime if its only divisors are 1 and p. Given a number n, a proper prime divisor of n is a prime number, different from n, dividing n. [Note: This shows that the numeral 4 is often printed incorrectly, missing the desired opening at the top.]

### NOTES

Equitable, Envy free, and Efficient Cake Cutting for Two People and Its Application to Divisible Goods
Michael A. Jones

By recasting the Dubins Spanier procedure geometrically, I demonstrate the existence of an optimal planar cut: a division of cake between two people that satisfies three important criteria, allowing the two players to "have their cake, and eat it, too." These criteria are described in the note. Such a division is proved to exist using geometric properties of monotonic, continuous functions.

When applied to discrete goods, the optimal planar cut theorem provides an alternate, geometric proof that Brams and Taylor's Adjusted Winner procedure provides an allocation of discrete goods to two people that satisfies the three criteria.

Four Ways to Evaluate the Poisson Integral
Hongwei Chen

In general, it is difficult to decide whether or not a given function can be integrated in elementary ways. In light of this, it is quite surprising that the value of the Poisson integral can be determined precisely. This note shows four different elementary methods of evaluating this integral.

Cycles in the Generalized Fibonacci Sequence modulo a Prime
Dominic Vella and Alfred Vella

What link could there be between the primes p for which Sophie Germain proved Fermat's Last Theorem nearly two hundred years ago and the length of periodic cycles that occur in the generalized Fibonacci sequence when reduced modulo p? We investigate cycles in these Fibonacci sequences and show that we can (under certain conditions) determine the length of these cycles where p is what we call a reverse Sophie Germain prime.

Products of Chord Lengths of an Ellipse
Thomas E. Price

Suppose the unit circle is divided into n equal arcs and points are placed at the ends of the arcs. Choose one of these points to be the base point and draw chords connecting each of the other points to the base point. It is known that the product of the lengths of these chords equals n. In this note it is shown this result reduces to the calculation of derivatives of polynomials with carefully chosen roots. We introduce a more general family of polynomials and generalize the circle result to chords of an ellipse.

Surprisingly Accurate Rational Approximations
Tom M. Apostol and Mamikon A. Mnatsakanian

It is remarkable that some famous irrational numbers such as π, e, the square root of 2, and the natural logarithm of 2 can be approximated very closely by rational numbers with relatively small denominators. For example, the rational number 22/7 gives two decimal accuracy for π, and 355/113 gives six decimals. The fraction 19/7 gives two decimal accuracy for e, and 99/70 gives four decimal accuracy for the square root of 2, whereas 21306/70777 gives ten decimal accuracy for log 2. In these examples, the number of correct decimals is twice the number of digits in the denominator of the approximating rational. This property is not merely a coincidence, nor is it unique to these numbers. This note reveals the surprising fact that every irrational can be approximated very closely by a rational whose denominator has a number of digits very nearly equal to half the number of decimal digits secured by the approximation.

Running with Rover
R. Bruce Crofoot

Runners beware! You may be lured into the land of differential geometry. Some interesting mathematics emerges very naturally from a consideration of the distances traveled by two objects moving along curved, parallel paths in a plane, as when two runners remain side by side.

### PROOFS WITHOUT WORDS

Lunes and the Regular Hexagon
Roger Nelsen

If a regular hexagon is inscribed in a circle and six semicircles contstructed on its sides, then the area of the hexagon equals the area of the six lunes plus the area of a circle whose diameter is equal in length to one of the sides of the hexagon. [Hippocrates of Chios, ca. 440 B.C.E.]

Dividing a Frosted Cake
Nicholaus Sanford

How do you cut a frosted rectangular cake into n pieces so that each person gets the same amount of cake and frosting?

Semidirect Products: x ‌→ ax+b as a First Example
Shreeram S. Abhyankar and Chris Christensen

After precalculus, students usually do not look very closely at the familiar transformation x ‌→ ax + b. Our goal is to show that this familiar transformation is a pathway to some interesting and important ideas in group theory. By building on this accessible example, it is possible to introduce the semidirect product (a topic which is usually first seen at the graduate level) in an undergraduate abstract algebra course. After being introduced to the semidirect product, students are able to better understand the structures of some of the groups of small order that are typically discussed in a first course in abstract algebra.

### POEM

There Are Three Methods for Solving Problems
Carol Le Guennec

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