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

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In this issue we learn from the masters. Frank J. Swetz tells about ancient Chinese treatments of similar triangles, Russell A. Gordon tells us about integer-sided triangles with 60-degree angles, Greg N. Frederickson tells us about dissecting a pizza, and Kenneth A. Ross tells about distributions of first digits. There's much more, including solutions to the 2011 Putnam Exam. —Walter Stromquist, Editor

**Similarity vs. the "In-and-Out Complementary Principle": A Cultural Faux Pa**

Frank J. Swetz

Modern investigators of early Chinese mathematical classics have often attributed a recognition and application of the principles of geometric similarity to the authors and commentators of these works. In problem situations involving the use of sighting poles and the determination of a remote distance, Chinese mathematicians frequently employed proportionality relations involving the sides of relevant pairs of right triangles. In such situations the modern western observer sees similarity; however, the Chinese employed a concept now called “the In-and-Out-Complementary-Principle,” IOCP. They did not use geometric similarity. This article identifies the issues of confusion and examines the concept and application of IOCP.

**Properties of Eisenstein Triples**

Russell A. Gordo

An Eisenstein triple consists of three positive integers such that . These triples share a number of properties with the more familiar Pythagorean triples, but there are some noticeable differences. In addition, new approaches are sometimes required to discover and prove these properties. This paper presents an introduction to Eisenstein triples and considers some of their elementary properties. One of the primary goals is to determine Eisenstein triples for which a given positive integer x appears as one of the quantities , or . It turns out that all of the resulting triples depend on the positive divisors of the integer and that the numbers of such triples satisfy some interesting and unexpected equations.

**The Proof Is in the Pizza**

Greg N. Frederickson

Symmetry, circumcircles, and angles of intercepted chords are some of the ingredients, as mathematicians give dissection proofs of the pizza theorem to show that they can get their fair share, even when the original pizza cuts are off-center. And with just a slight variation in the recipe, we also get a dissection proof of the calzone theorem.

**A Galois Connection in the Social Network**

James Propp

This article shows that if Â“knowingÂ” is a symmetric relation, then the set of people who know all of the people who know all of the people who know everyone in S coincides with the set of people who know everyone in S. In symbols this becomes . This fact fits into the broader context of Galois connections between partially ordered sets.

**First Digits of Squares and Cubes**

Kenneth A. Ross

Benford’s Law is the observation that in many lists of numbers that arise in the real world, for , the likelihood that the first digit is d is . A similar phenomenon is known for essentially all geometric sequences , namely the relative frequencies of the first digit d of the first N terms in such a sequence tend to In this note, it is shown that the analogous statements for the sequence of squares and the sequence of cubes do not hold. However, in these cases, interesting subsequences of do converge, but not to.

**The Triple Angle Sine and Cosine Formulas**

Claudi Alsina and Roger B. Nelsen

**Cyclotomic Polynomials, Symmetric Polynomials, and a Generalization of Euler’s Totient Function**

Julian Freed-Brown, Matthew Holden, Michael E. Orrison, and Michael Vrable

We introduce a generalization of Euler’s totient function that, when applied to an integer , can be written as a polynomial in the prime factors of n. We then show how cyclotomic and complete homogeneous symmetric polynomials appear as factors of these polynomials when n has at most two distinct prime divisors.

**Math Bit e: Finding e in Pascal’s Triangle**

Halan J. Brothersa

If s_{n} is the product of the entries in row n of Pascal’s triangle then , which has the limiting value e.

**Plausible and Genuine Extensions of L’Hospital’s Rule**

J. Marshall Ash, Allan Berele, and Stefan Catoiu

Let* f* and *g* be differentiable real valued functions. Motivated by L’Hospital’s Rule, we might expect the convergence of to imply the convergence of when all have limit 0 as x tends to infinity and also when all four functions have infinite limits at infinity. We find this to be true, subject to some very mild additional conditions, when the four functions have limit zero, but not necessarily to be true in the infinity case. For limits, the discrete analogue of L’Hospital’s Rule is the Stolz-Cesàro Theorem. We also find a result for series that is in the spirit of the Stolz-Cesàro Theorem.

72nd Annual William Lowell Putnam Mathematical Competition