Given a positive integer \(m\), the authors exhibit a group with the...

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**Designing a Pleasing Sound Mathematically**

by Erich Neuwirth

Fourier series and waveshapes are fundamental in analyzing musical tones. As an application of this theory, a special parameterized waveform with simple mathematical and pleasing musical qualities is constructed. Computer programs for Mathematica and for Microsoft Excel are given which allow experimenting with the parameters of this waveform.

**A History of Lagrange's Theorem on Groups**

by Richard L. Roth

Lagrange's theorem for groups states that the order of a subgroup of a finite group G divides the order of G. It is one of the first theorems of group theory we study in an abstract algebra class. But the original form as stated by Lagrange in 1771 was quite different and predated the invention of group theory. It arose in an attempt to solve the general polynomial of degree 5 or higher, and its relation to symmetric functions. We will trace the theorem as it changed form over the years and developed into the theorem we know today.

**The Euler-Maclaurin and Taylor Formulas: Twin, Elementary Derivations**

by Vito Lampret

Computing technology still relies on the calculus, contrary to what many observers believe. For example, calculators can sum only finitely manyÑperhaps relatively few--terms of a series by brute force. To calculate numerical integrals and sums accurately, computers need methods of reducing the set of required operations.

The Euler-Maclaurin formula enables such a reduction; thus new technologies and calculus ideas complement each other. Using repeated integration by parts we derive in elementary fashion both Taylor's formula for approximating functions by Taylor polynomials and the Euler-Maclaurin formula, which expresses the difference between an integral and an appropriate Riemann sum approximation.

We illustrate, including examples, how the Euler-Maclaurin formula can be applied to numerical summation, to numerical integration, and to the connection improper integrals and infinite series.

**Location of Incenters and Fermat Points in Variable**

by Anthony Varilly

In 1765, Euler proved that several important centers of a triangle are collinear; the line containing these points is named after him. The incenter I of a triangle, however, is generally not on this line. Less than twenty years ago, Guinand discovered that although I is not necessarily on the Euler line, it is always fairly close toit. Guinand's theorem states that for any non-equilateral triangle, the incenter lies inside, and the excenters lie outside, the circle whose diameter joins the centroid to the orthocenter; henceforth the orthocentroidal circle. Furthermore, Guinand constructed a family of locus curves for I which cover the interior of this circle twice, showing that there are no other restrictions on the positions of the incenter with respect to the Euler line.

Here we show that the Fermat point also lies inside the orthocentroidal circle; we also construct a simpler family of curves for I, covering the interior once only except for the nine-point center N, which corresponds to the limiting case of an equilateral triangle.

**Enumerating Row Arrangements of Three Species**

by Michael Richard Avidon, Richard D. Mabry, and Paul D. Sisson

In how many ways can C cats, D dogs, and E emus be seated in a line, with no two animals of the same species adjacent? We give a complete elementary solution of this problem in enumerative combinatorics and mention several nonelementary methods.

**Proof Without Words: The Triple-Angle Formulas for Sine and Cosine **

by Shingo Okuda

Scrutinizing this diagram leads to a proof of the triple-angle formulas for sine and cosine.

**The Disadvantage of Too Much Success**

by Martin L. Jones and Reginald Koo

In this note we consider coin-tossing models in which "too much success" is defined by the occurrence of success runs of length r, where r is a positive integer. When a success run occurs, play terminates and the player has no further opportunities to earn rewards. We consider three different types of stopping criteria and calculate the best success probability p for maximizing the expected reward.

**The Wily Hunting of the Proof, a Poem**

by Jasper D. Memory

**Means to an End**

by Richard P. Kubelka

The Arithmetic-Geometric Mean Inequality guarantees that GM/AM cannot exceed 1. But, except in the cases where GM=AM, it gives us no clue as to the actual value of GM/AM. In this paper we find the rather surprising limiting value of that ratio as we take the arithmetic and geometric means of more and more terms from the sequence 1^{s}, 2^{s}, 3^{s}, . . . .

**Visualizing Leibniz's Rule**

by Marc Frantz

Leibniz's rule for differentiating under the integral sign results in a formula that contains two types of terms: an integral and a product of two functions. I show that there is a simple geometric interpretation of these terms as, respectively, the area between two curves and the area of a rectangle. The geometric picture, in turn, allows an informal derivation of the rule that should appeal to visually oriented students.

**A Celestial Cubic**

by Charles W. Groetsch

The search for extra-solar planets is one oldest scientific quests and one of the hottest topics in contemporary astronomy. In this note a refinement of a simple mathematical model for a single extra-solar planet is used to introduce an interesting cubic equation. This equation is then becomes a vehicle for a classroom discussion of convergence of fixed point iteration and an assessment of the accuracy of a widely used simpler model for the mass of an alien planet. Teachers interested in integrating mathematical analysis, physical science and computation in their lessons may find the material of this note useful in a number of undergraduate settings.

**Proof Without Words: Pythagorean Theorem**

by Jose A. Gomez

A proof of the Pythagorean theorem by dissection.

edited by Elgin H. Johnston

edited by Paul J. Campbell

29th USA Mathematical Olympiad Problems and Solutions