If February is making you shiver, then warm up with this month's *American Mathematical Monthly*. Heading to Las Vegas to warm up? Then you might wonder if you can play a fair game of craps with a loaded pair of dice. Ian Morrison and David Swinarski answer this question in their paper of the same name.

Does the winter weather have you thinking about snowflake curves? Then check out our note titled "Helge von Koch's Snowflake Curve Revisited" by E. W. Dekking and F. M. Dekking. Jeff Nunemacher reviews "Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem" by Al Cuoco and Joseph J. Rotman, and as always our Problem Section will give you something to do by the fire.

Stay tuned for the March issue when we announce the 2016 Yueh-Gin Gung and Dr. Charles Y. Hu Award winner. - *Scott T. Chapman, Editor*

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## Table of Contents

### Optimally Topologically Transitive Orbits in Discrete Dynamical Systems

Francis C. Motta, Patrick D. Shipman and Bethany D. Springer

Every orbit of a rigid rotation of a circle by a fixed irrational angle is dense. However, the apparent uniformity of the distribution of iterates after a finite number of iterations appears strikingly different for various choices of a rotation angle. Motivated by this observation, we introduce a scalar function on the orbits of a discrete dynamical system defined on a bounded metric space, called the linear limit density, which we interpret as a measure of an orbit′s approach to density. Utilizing the three-distance theorem, we compute the exact value of the linear limit density of orbits of rigid rotations by irrational rotation angles with period-1 continued fraction expansions. We further show that any discrete dynamical system defined by an orientation-preserving diffeomorphism of the circle has an orbit with a larger linear limit density than any orbit of the rigid rotation by the golden number. Bernoulli shift maps acting on sequences over a finite alphabet provide another illustrative class of dynamical systems with dense orbits. Our study of the efficiency of an orbit′s approach to density leads us to demonstrate the existence of a class of infinite sequences with finite linear limit density constructed by recursively extending finite de Bruijn sequences.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.2.115

### Can You Play a Fair Game of Craps with a Loaded Pair of Dice?

Ian Morrison and David Swinarski

The parts-to-totals map sends the distributions of a set of independent random variables on a finite set of probability spaces to the total distribution of their sum on the product space. We study, in special cases modeled by dice, the geometry of the extension to complex pseudoprobabilities of this map, arithmetic questions about the existence of real points in certain fibers, and, when these exist, of strict points, having all coordinates in the unit interval.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.2.136

### A Geometric Solution of a Cubic by Omar Khayyam … in Which Colored Diagrams Are Used Instead of Letters for the Greater Ease of Learners

Deborah A. Kent and David J. Muraki

The visual language employed by Oliver Byrne in his 1847 edition of Euclid′s Elements provides a natural syntax for communicating the geometrical spirit of Omar Khayyam′s 11th-century constructions for solving cubic equations. Inspired by the subtitle (co-opted for this article) from Byrne′s The Elements of Euclid, we rework one of these constructions by adapting his distinct pictographic style. This graphical presentation removes the modern reliance on algebraic notation and focuses instead on a visualization that emphasizes Khayyam′s use of ratios, conic sections, and dimensional reasoning.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.2.149

### Old Friends in Unexpected Places: Pascal (and Other) Matrices in GLn(ℂ)

Josh Hiller

One of the most beautiful results in early linear algebra is that every matrix over an algebraically closed field is similar to a Jordan matrix. This, of course, immediately proves that the Borel subgroup (the subgroup consisting of invertible* n* × *n* upper triangular matrices) is conjugate dense in GL_{n}(ℂ). However, other matrices are also conjugate dense in GL_{n}(ℂ). In this article, we show that a variation of Pascal matrices are one such class of conjugate dense matrices. We then use this fact to reduce finding matrix powers, roots, and inverses down to the corresponding problem for an appropriately chosen diagonal matrix. Having done this, we explore how one might create a wide array of other conjugate dense subgroups of GL_{n}(ℂ).

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.2.161

### Using Conic Sections to Factor Integers

John Brillhart, Richard Blecksmith and Mike Decaro

This paper explores the factorization of an odd, composite integer* N* that has been expressed in two different ways as *mx*^{2} ±* ny*^{2}. The negative case *mx*^{2} − *ny*^{2} = *N* turns out to be quite different from the positive case *mx*^{2} − *ny*^{2} = *N* because it deals with a hyperbola instead of an ellipse. Of particular interest in the negative case is that Pell-connected representations produce trivial factorizations.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.2.168

## Notes

### Probabilistic Proofs of a Binomial Identity, Its Inverse, and Generalizations

Michael Z. Spivey

We give elementary probabilistic proofs of a binomial identity, its binomial inverse, and generalizations of both of these. The proofs are obtained by interpreting the sides of each identity as the probability of an event in two different ways. Each proof uses a classic balls-and-jars scenario.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.2.175

### Helge von Koch′s Snowflake Curve Revisited

E. W. Dekking and F. M. Dekking

We try to solve the problem of filling in von Koch′s snowflake curve by a recursively defined curve. Our solutions are “reducible,” a property that gives rise to some new issues in the mathematics of plane-filling curves.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.2.181

### Strong Convexity Does Not Imply Radial Unboundedness

Jonathan Baker

We demonstrate that not all strongly convex functionals are radially unbounded. Moreover, the only exceptions must be unbounded below in every neighborhood; hence, such exceptions are interesting in optimization only in that they have no local minima.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.2.185

### On Sums of Three Pentagonal Numbers

Xin-Jie Sun

Pentagonal numbers are integers of the form *p*_{5}(*x*) =* x*(3*x* − 1)/2 with *x* ∈ ℤ. In 1994, R. K. Guy observed that any nonnegative integer can be expressed as the sum of three pentagonal numbers. In this note, we use the generating function method to show that any nonnegative integer can be written as *p*_{5}(x) + *p*_{5}(y) + *p*_{5}(z) with x, y, z ∈ ℤ and x + y + z ≡ ⌊√2⌋ (mod 2).

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.2.189

### A Combinatorial Proof of an Alternating Convolution Identity for Multichoose Numbers

Jeffrey Lutgen

We give a combinatorial proof for an identity involving an alternating sum of products of multichoose numbers. In our proof, we construct a sign-reversing involution on certain ordered pairs of words, then count the leftovers. We also outline two alternative proofs, one using hypergeometric functions and the other using mechanical summation.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.2.192/p>

## Problems and Solutions

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.2.197

## Book Review

*Learning Modern Algebra: From Early Attempts to Prove Fermat′s Last Theorem* By Al Cuoco and Joseph J. Rotman

Reviewed by Jeffrey Nunemacher

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.2.205

## MathBits

### Cyclic Extensions Are Radical

Mariano Suárez-Álvarez

### 100 Years Ago This Month in the American Mathematical Monthly

Edited by Vadim Ponomarenko