# Mathematics Magazine June 2002

### Articles

Golden, √2, and Pi Flowers: A Spiral Story
Michael Naylor
Flowers and seed pods can be neatly simulated mathematically by placing points so that consecutive points differ by a constant angle. In many spirals in nature, this angle, measured in revolutions, is equal to the golden ratio. This paper examines what happens when that number is replaced by some other famous irrationals, and, in doing so, gives us some insight into rational approximations of irrationals and continued fractions.

Pillow Chess
Grant Cairns
In an old chess variant called cylindrical chess, one imagines that the left and right hand edges are joined together, so that pieces can pass off one side and reappear on the other. One can also play chess on the torus by additionally identifying the top and bottom edges, but now one has to use unorthodox starting positions for the pieces.

This paper introduces a way of playing chess on a sphere, with the pieces in their traditional starting positions, and with rules very close to the traditional rules of movement. We revisit two classic mathematical problems in this setting: the knight's tour problem (can a knight visit all the squares of the board exactly once and return to its starting Position?), and the n-queens problem (can one place n queens on an n x n board such that no pair is attacking each other?). We find that, unlike the traditional chessboard, both problems have a solution on the pillow board, regardless of its size.

Doubly Recursive Multivariate Automatic Differentiation
Dan Kalman
Automatic Differentiation is a capability that can be built into computational systems to provide for the automatic generation of numerical values for derivatives of expressions. It can be viewed as an extension of the normal evaluation process for scientific computing languages. Thus, execution of the statements

x = 3
ƒ = x2 + 5

produces two results: the value of ƒ (namely 14), as well as the value of ƒ (namely 6). Neither the user of the system, nor the system itself, formulates a symbolic expression for the derivative, however. Instead, the sequence of operations that is implicit in the formula for ƒ is translated directly into a set of corresponding operations for ƒ. This article formulates an automatic differentiation system that computes all partial derivatives of an expression up to an arbitrary order with respect to an arbitrary number of variables. By using recursion on both the number of variables and the number of derivatives, the formulation retains the simplicity observed for automatic generation of a single derivative with respect to a single variable.

### NOTES

Marc Brodie
Suppose n families of four (4n people) are invited to a bridge party. Bridge partners are chosen at random, without regard to gender or generation. What is the probability that no one will be paired with a member of his or her family? After discussing the connection between this problem and the card game War, we show that as n grows without bound, the probability approaches e-3/2.

Bernoulli on Arc Length
Victor Moll, Judith Nowalsky, Gined Roa, and Leonardo Solanilla
We present Johnann Bernoulli's general theorem for rectifying regular curves and its relation to the theory of integral transformations.

Finite Groups That Have Exactly n Elements of Order n
Carrie E. Finch, Richard M. Foote, Lenny Jones, and Donald Spickler, Jr.
A finite group G with exactly n elements of order n is called minimal if no proper subgroup of G has exactly n elements of order n. The authors determine all minimal finite groups that have exactly n elements of order n.

Rootless Matrices
Bertram Yood
We say that an n x n matrix A is rootless if there is no matrix S and no positive integer r > 1 such that Sr = A. Our main theorem says that A is rootless if An = 0, but A(n-1) is nonzero. However, this does not characterize rootless matrices. We offer questions that could lead the reader to give a complete description of all rootless matrices.

Lots of Smiths
Patrick Costello and Kathy Lewis
A Smith number is a composite number where the sum of the digits equals the sum of the digits in its prime factors. Wayne McDaniel was able to show that in fact there are infinitely many Smith numbers by constructing a sequence of them. In this Note, we construct a slightly different sequence and indicate that even more sequences can be found. Hence there are lots of Smiths.

### PROOFS WITHOUT WORDS

A Line through the Incenter of a Triangle
Sidney H. Kung
A Proof Without Words: Dissection is used to show that a line passing through the incenter of a triangle bisects the perimeter if and only if it bisects the area.

### PROBLEMS

edited by Elgin H. Johnston

### REVIEWS

edited by Paul J. Campbell