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American Mathematical Monthly -June-July 2008

June/July 2008

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Periodic Orbits of Billiards on an Equilateral Triangle
By: Andrew M. Baxter and Ron Umble,
How many ways can one set a billiard ball in motion on a frictionless triangular equilateral table so that the ball retraces the same path after n bounces? Such a path is called a periodic orbit of period n. When n is odd there is at most one such orbit, but when n is even there are uncountably many. Fortunately there is a natural equivalence relation on orbits of even period. Using techniques from plane geometry, number theory, and combinatorics we construct a bijection between equivalence classes of these orbits and a new type of integer partition. This allows us to count equivalence classes containing orbits of a given period by counting partitions.


The Freshman's Approach to Conway's Napkin Problem
By: Niklas Eriksen
In the March 2006 issue of the MONTHLY, Claesson and Petersen gave a thorough solution to Conway's napkin problem. The problem is the following: Assume that n mathematicians arrive in random order at a conference dinner with a circular table, and that the napkins are placed exactly halfway between the plates so that the guests do not know whether they are supposed to use the right or the left napkin. Each guest prefers these napkins with probabilities p and 1-p, respectively, and tries her preferred alternative before trying the other, if the preferred napkin has been taken. Which proportion of guests is expected to sit down at a place where both adjacent napkins have been taken and thus be without a napkin? Claesson and Petersen use a system of generating functions to compute both the expectation and the variance of this proportion and to address similar questions, for instance regarding the number of guests who get a napkin though not the preferred one. However, these expectations can also be computed using purely elementary methods, such as the binomial theorem. We present the freshman's approach to the napkin problem and related problems, for instance the one with French diners mentioned, but not solved, by Claesson and Petersen.


Edge Detection Using Fourier Coefficients
By: Shlomo Engelberg
Many applications call for the determination of the points at which a function changes values in a discontinuous fashion and require knowledge of the change in the function's value at such points. We present methods of determining the locations of discontinuities by considering the Fourier coefficients of a function. We take the Fourier coefficients and return a function that tends to zero at points of continuity of the original function and that tends to the height of the jump at the location of a jump. We make use of many elementary results from analysis and many properties of Fourier series. In order to understand the material presented, a reasonable knowledge of Fourier series and a good understanding of the properties of infinite series are required.


The mth Ratio Test: New Convergence Tests for Series
By: Sayel A. Ali,
The failure of the ordinary ratio test for convergence of series, when the limit of the ratio is 1, motivated mathematician to analyze this ratio and search for sharper tests than the ratio test. As a result, the world of mathematics was enriched with delicate tests like Kummer’s, Raabe’s, and Gauss’s. In this paper we give the mth ratio test. It succeeds on a wide range of examples where the ratio test fails. Also, it succeeds on series that require the test of Raabe or Gauss.


Summing a Curious, Slowly Convergent Series
By: Thomas Schmelzer and Robert Baillie,
If we delete from the harmonic series all terms whose denominators contain any pattern of digits such as "314159", the remaining terms form a convergent series. Ever since Kempner proved the first such surprising result in 1914, it has been difficult to compute the sums of these series. The convergence is so slow that even direct summation using all the computers in the world would be of no use. We present a method for efficiently computing these sums. For example, the sum of the series whose denominators omit "314159" is approximately 2302582.33386. We also explain why this sum is so close to 106 log(10).


Arithmetic in the Ring of Formal Power Series with Integer Coefficients
By: Daniel Birmajer and Juan B. Gil,
In this article we discuss arithmetic (units, primes, unique factorization) in the ring of formal power series over the integers. Power series are the natural extension of the ring of polynomials, so both rings share similar properties but also have some major differences. For instance, in contrast to polynomials, there are many invertible power series: the necessary and sufficient condition for a power series to be invertible is that its constant term be invertible. On the other hand, as for polynomials, it is in general difficult to determine whether an integral power series is irreducible. In order to detect irreducibility, we develop criteria similar to Eisenstein's criterion for polynomials and propose an easy argument that provides us with an infinite class of irreducible power series over the integers. As in the case of Eisenstein's criterion, our criteria give only sufficient conditions, and the question of whether or not a given power series is irreducible remains open in a vast array of cases, including quadratic polynomials.



A Shortened Classical Proof of the Quadratic Reciprocity Law
By: Wouter Castryck

Infinite Divisibility of GCD Matrices
By: Rajendra Bhatia and J. A. Dias da Silva,

Borsuk-Ulam Implies Brouwer: A Direct Construction Revisited
By: Alexey Yu. Volovikov

On the Radon-Nikodym Theorem
By: George Koumoullis

Cycles in Graphs and Groups
By: William M. Kantor

A Letter from Fitzroy House
By: Michael D. Hirschhorn


An Invitation to Modern Number Theory
By: Steven J. Miller and Bamin Takloo-Bighash
Reviewed by: Jeffrey D. Vaaler