What do twins do at twin conventions? Martin Griffiths doesn't know, but he has an idea for what a combinatorialist could do. Of course, it involves Stirling Numbers.

Also in this issue: flight plans, infinite products, and power series, and a proposal to save Pi Day. —*Walter Stromquist*

Vol. 86, No. 1, pp.3-80.

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Michelle L. Ghrist and Eric E. Lane

We explore one aspect of a multivariable calculus project: parametrizing an elliptical path for an airplane that travels at constant speed. We find that parametrizing a constant speed elliptical path is significantly more complicated than a circular path and involves taking the inverse of the elliptic integral of the second kind. We compare our constant-speed parametrization to the standard ellipse parametrization $$(x(t)=a\cos(wt), y(t)=b\sin(wt))$$ and generalize to parametrizing other constant-speed curves.

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Samuel G. Moreno and Esther M. García

In this article, the authors show that Viète's formula is only the tip of the iceberg. Under the surface, they search for curious and interesting Viète-like infinite products, rare species made of products of nested square roots of 2, but here with some minus signs occurring inside. To explore this fascinating world, they only use the simple trigonometric identity $$\cos x=2\cos((\pi+ 2x)/4)\cos((\pi-2x)/4)$$, combined with a recent formula by L. D. Servi.

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Martin Griffiths

In this article we consider a particular combinatorial scenario involving $$n$$ sets of identical twins. We show how, subject to various assumptions and conditions on the possible groupings, formulas may be obtained in terms of $$n$$ for the number of ways in which these $$2n$$ individuals can be seated at $$k$$ tables for any fixed value of $$k$$. This is achieved by deriving recurrence relations and subsequently employing exponential generating functions.

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Siad Daboul, Jan Mangaldan, Michael Z. Spivey, and Peter J. Taylor

We give five proofs that the coefficients in the $$n$$th derivative of $$e^{1/x}$$ are the Lah numbers, a triangle of integers whose best-known applications are in combinatorics and finite difference calculus. Our proofs use tools from several areas of mathematics, including binomial coefficients, Faà di Bruno's formula, set partitions, Maclaurin series, factorial powers, the Poisson probability distribution, and hypergeometric functions.

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A. F. Beardon

We discuss solutions of the equation $$f(2x)=2f(x)f'(x)$$, which is essentially a delay differential equation, with the boundary condition $$f'(0)=1$$, on the interval $$[0,\infty)$$. In particular, we note that the only known solutions of this type that are positive when $$x$$ is positive are the functions $$c^{-1}\sinh(cx)$$, where $$c>0$$, and the function $$x$$.

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Mark Lynch

In 1932, Stanislaw Golab proved that, for a large class of metrics in the plane, the perimeter of the unit disk can vary from 6 to 8. Hence, the ratio corresponding to pi can vary from 3 to 4. We illustrate this result by defining a family of metrics that can be understood easily by any student of linear algebra.

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Angel Plaza and Hans R. Walser

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Richard Beals

This note examines the application of the power series method to the solution of second order homogeneous linear ODEs, and shows that it works in a straightforward way only under restrictive conditions—in cases that reduce to the hypergeometric equation or the confluent hypergeometric equation. On the other hand, it is noted that these equations account for most "special functions."

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Proposals 1911-1915

Quickies Q1025-Q1028

Solutions 1886-1890

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73rd Annual William Lowell Putnam Examination

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