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Mathematics Magazine Contents - December 2014

We hope you'll find that this issue has been worth waiting for. Do you like solving equations by radicals? Iterating functions, whether of integers or real numbers? Extending patterns from Pascal's triangle into four dimensions? "Straightedge" and compass constructions on a sphere? You'll find all of these attractions here, along with our Reviews and Problems Sections and a crossword puzzle.

This is the issue in which we acknowledge our referees. If you find your friends on the list, why not do something nice for them today? —Walter Stromquist, editor


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Vol. 87, No. 5, pp 321 – 409

Letter from Editor


Conway's Subprime Fibonacci Sequences

Richard K. Guy, Tanya Khovanova, Julian Salazar

It’s the age-old recurrence with a twist: sum the last two terms and if the result is composite, divide by its smallest prime divisor to get the next term (e.g., 0, 1, 1, 2, 3, 5, 4, 3, 7, …). These sequences exhibit pseudo-random behavior and generally terminate in a handful of cycles, properties reminiscent of 3x C 1 and related sequences. We examine the elementary properties of these “subprime” Fibonacci sequences.

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Iteration of Sine and Related Power Series

Christopher Towse

We consider the nth iterate of the sine function and related functions by looking at the growth and form of the coefficients in the resulting power series. By controlling for the growth of the terms, as functions of n, we find a surprising relationship to a family of algebraic functions.

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Proof Without Words: Rapid Construction of Altitudes of Triangles

Larry Hoehn

The altitudes of triangles are constructed using a circle and three lines.

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Straightedge and Compass Constructions in Spherical Geometry

Daniel J. Heath

We examine straightedge and compass constructions in spherical geometry. We show via examples that the starting conditions affect the set of constructible points. Although current tools do not allow for a complete solution, we take a tour through group theory and real analysis to show that, in general, the set of constructible points is dense on the sphere. However, we conclude with more questions than answers.

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Award Winners

Brendan W. Sullivan

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Mathematics, Models, and Magz, Part II: Investigating Patterns in Pascal's Simplex

Victor Garcia, Jean Pedersen

This paper is a sequel to Mathematics, models, and Magz, Part 1: Investigating patterns in Pascal’striangle and tetrahedron. In the current paper the authors use a set of magnetic toys to extend results of Part I about patterns concerning binomial and tetranomial coefficients. In Pascal’s tetrahedron they study vertex truncations and edge truncations. Then they extend those ideas to obtain analogous results in Pascal’s simplex.

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Cardano, Casus Irreducibilis, and Finite Fields

Matt D. Lunsford

For centuries, mathematicians were puzzled by the fact that Cardano’s formula requires an excursion into the complex numbers to express in radicals the roots of an irreducible cubic polynomial with rational coefficients having only real roots. In this paper, we consider an analogous case for irreducible cubics over finite fields. In this simpler setting, we explore whether the roots of an irreducible cubic polynomial are expressible in terms of irreducible radicals present in its splitting field. The key theorem gives a simple divisibility condition that answers this question completely. Thus there is a situation that mirrors the historic “casus irreducibilis,” in that both cases require the presence of elements not in the splitting field in order to express the roots of an irreducible cubic in terms of irreducible radicals.

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A Triangle Theorem

Lee Sallows

The medians of a triangle divide it into six parts, which can be reassembled into three congruent triangles.

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Multiplicative Subgroups of $$\mathbb{C}$$ that Contain Regular Jordan Curves

Wijarn Sodsiri

In this paper, we characterize subgroups of the multiplicative group $$\mathbb{C}$$ of nonzero complex numbers that contain Jordan curves of area zero.

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One Picture, All the Conics

Óscar Ciaurri

In this note we obtain the envelopes of some families of lines and the envelope of the family of envelopes. All the conics appear.

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The Pill Problem, Lattice Paths, and Catalan Numbers

Margaret Bayer, Keith Brandt

We define the pill tree, which is a rooted, binary tree consisting of different sequences of whole and half pills from a problem posed by Knuth and McCarthy. We observe some connections among the pill tree, lattice paths, and Catalan numbers, and give an explicit formula for the number of nodes in the tree.

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How to study math; improbability; math in movies, unsettling news

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