Vol. 121, No. 6, pp.471-560.

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## ARTICLES

### Rational Numbers, Finite Cyclic Monoids, Divisibility Rules, and Numbers of Type 99 . . . 900 . . . 0

Alberto Facchini and Giulia Simonetta

There is a curious connection between decimal representations of rational numbers, the structure of finite cyclic monoids, divisibility rules between integers, and divisors of the numbers of the form 99 . . . 900 . . . 0. In all of these cases, we find not only periodicity from some point on, but also the same type of periodicity.

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### On the Solution of Linear Mean Recurrences

David Borwein, Jonathan M. Borwein, and Brailey Sims

Motivated by questions of algorithm analysis, we provide several distinct approaches to determining convergence and limit values for a class of linear iterations.

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### MATHBIT: Rejection of Laplace’s Demon

Josef Rukavicka

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### Vertical Projection in a Resisting Medium: Reflections on Observations of Mersenne

C. W. Groetsch and S. A. Yost

This article, inspired by a 17th-century woodcut, validates empirical observations of Marin Mersenne (1588–1648) on timing of vertically-launched projectiles for a general mathematical model of resistance.

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### Three Dimensions of Knot Coloring

J. Scott Carter, Daniel S. Silver, and Susan G. Williams

The 1926 paper of J. W. Alexander and G. B. Briggs suggests a simple combinatorial invariant by coloring the crossings of a knot diagram. It is equivalent to the well-known Fox *n*-coloring of arcs and lesser-known Dehn *n*-coloring of regions. The equivalence of the three approaches to knot coloring is presented.

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### MATHBIT: An Alternate Proof of Sury’s Fibonacci–Lucas Relation

Harris Kwong

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### On Hamilton’s Nearly-Forgotten Early Work on the Relation between Rotations and Quaternions and on the Composition of Rotations

Jose Pujol

In 1844, barely a year after inventing quaternions, Hamilton presented a paper to the Royal Irish Academy in which he introduced the scalar and vector products as used today (except for the sign of the former). He used them to solve the problem of the composition of any number of rotations about successive axes through any angles. Hamilton’s solution is surprisingly simple and includes the extremely important relation between rotations and quaternions. This work, published in 1847, was largely ignored, even to this date. On the other hand, the rotation-quaternion relation was published in 1845 by Cayley, using results derived by Rodrigues in 1840. As a consequence, Hamilton’s 1847 contribution to this problem has been overlooked, to the point that it has been claimed that he did not understand that relation, which is clearly not correct. This article settles this matter by going over Hamilton’s derivation, which is simpler than the Cayley–Rodrigues analysis, and does not require any advanced mathematics. Modern derivations of the equation for the rotation of a vector about an axis are based on a vector decomposition similar to that introduced by Hamilton. To appreciate the significance of Hamilton’s results, they are placed in a historical context.

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### MATHBIT: A Heuristic Argument for Hua’s Identity Using Geometric Series

B. Sury

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### Lagrange’s Identity and Congressional Apportionment

Tommy Wright

This note makes use of Lagrange’s Identity to provide a bridge between an insightful motivation and an elementary derivation of the *method of equal proportions*. The method of equal proportions is the current method for apportioning the 435 seats in the U.S. House of Representatives among the 50 states, following each decennial census.

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## NOTES

### Pigeonholes and Repunits

Chai Wah Wu

It is well known that any integer $$k$$ has a multiple consisting of only the digits 1 and 0. As an extension of this result, we study integers of the form $$111\cdots000$$ or $$111\cdots111$$ that are a multiple of $$k$$. We show that if $$k > 2$$ and $$k$$ is not a power of 3, then the multiple can be chosen to have at most $$k-1$$ digits.

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### Record Crossword Puzzles

Kevin K. Ferland

The maximum number of clues possible for a $$15\times15$$ daily *New York Times *crossword puzzle is shown to be 96, and all possible puzzle grids with 96 clues are presented. Moreover, a crossword puzzle with 96 clues is given, in the theme of this result.

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### A Combinatorial Proof for the Alternating Convolution of the Central Binomial Coefficients

Michael Z. Spivey

We give a combinatorial proof of the identity for the alternating convolution of the central binomial coefficients. Our proof entails applying an involution to certain colored permutations and showing that only permutations containing cycles of even length remain.

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### A Converse of the Gauss–Lucas Theorem

Nikolai Nikolov and Blagovest Sendov

All linear operators $$L:C[z]\rightarrow C[z]$$ that decrease the diameter of the zero set of any $$p\in C[z]$$ are found.

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### Finding Your Seat Versus Tossing a Coin

Yared Nigussie

In a classroom of *n *seats and *n *students, the first student sits at random, whereas every other student must sit at her/his seat, but may sit randomly if her/his seat is already taken. The probability that a student finds her/his seat is given by a simple formula. Two entertaining proofs are given.

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### A Simple Direct Proof of Marden’s Theorem

Erich Badertscher

Marden’s theorem characterizes the critical points of complex polynomials of degree 3 in a nice geometrical way. Our proof of the theorem is based directly on the defining property of ellipses.

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## PROBLEMS AND SOLUTIONS

Problems 11782-117888

Solutions 11651, 11653, 11654, 11661

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## REVIEWS

*Which Numbers Are Real?* By Michael Henle

Reviewed by James A. Swenson

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