###
ARTICLES

**Ellipse to Hyperbola: “With This String I Thee Wed”**

Tom M. Apostol and Mamikon A. Mnatsakanian

pp. 83–97

We introduce a string mechanism that traces both elliptic and hyperbolic arcs having the same foci. This suggests replacing each focus by a focal circle centered at that focus, a simple step that leads to new characteristic properties of central conics that also extend to the parabola.

The classical description of an ellipse and hyperbola as the locus of a point whose sum or absolute difference of focal distances is constant, is generalized to a common bifocal property, in which the sum or absolute difference of the distances to the focal circles is constant. Surprisingly, each of the sum or difference can be constant on both the ellipse and hyperbola. When the radius of one focal circle is infinite, the bifocal property becomes a new property of the parabola.

We also introduce special focal circles, called circular directrices, which provide equidistance properties for central conics analogous to the classical focus-directrix property of the parabola. Those familiar with paperfolding activities for constructing an ellipse or hyperbola using a circle as a guide, will be pleased to learn that the guiding circle is, in fact, a circular directrix.

**The Bhāskarā-Āryabhata Approximation to the Sine Function**

Shailesh A. Shirali

pp. 98–107

In the seventh century AD the Indian mathematician Bhāskarā; I gave a curious rational approximation to the sine function; he stated that if 0≤*x*≤180 then sin *x* deg is approximately equal to 4*x*(180-*x*)/(40500-*x*(180-*x*)). He stated this in verse form, in the style of the day, and attributed it to his illustrious predecessor Āryabhata (fifth century AD); however there is no trace of such a formula in Āryabhata’s known works. Considering the simplicity of the formula it turns out to be astonishingly accurate. Bhāskarā did not give any justification for the formula, nor did he qualify it in any way. In this paper we examine the formula from an empirical point of view, measuring its goodness of fit against various criteria. We find that the formula measures well, and indeed that these different criteria yield formulas that are very close to the one given by Bhāskarā.

**Integrals Don’t Have Anything to Do with Discrete Math, Do They?**

P. Mark Kayll

pp. 108–119

To students just beginning their study of mathematics, the discipline appears to come in two distinct flavours: continuous and discrete. This article attempts to bridge the apparent divide by describing a surprising connection between these ostensible opposites. Various inhabitants from both worlds make appearances: rook polynomials, Euler’s gamma function, derangements, and the Gaussian density. Uncloaking combinatorial proof of an integral identity serves as a thread tying these notions together.

###
NOTES

**Positively Prodigious Powers or How Dudeney Done It?**

Andrew Bremner

pp. 120–125

Dudeney’s puzzles of a hundred years ago included writing integers (specifically 9 and 17) as sums of two cubes of positive rational numbers (where in the former case, a solution other than 1, 2 is required). We study the corresponding equations *x*^{3} + *y*^{3 }= 9 and *x*^{3}+*y*^{3} = 17 as examples of specific elliptic curves. The group structure is introduced, and the smallest solutions found for Dudeney’s puzzles. Generalization to *x*^{3} + *y*^{3}=*n* reveals that sometimes the smallest rational solution can be very large, for example when *n* = 94 and *n* = 4981: the latter solution involves fractions with numerator and denominator having almost 17 million digits.

**The Quadratic Character of 2**

Rafael Jakimczuk

pp. 126–127

The number 2 is a quadratic residue mod *p* if *p *= 8*K* + 1 or *p* = 8*k* + 7, but not *p *= 8*k *+ 3 or *p *= 8*k *+ 5. This is proved by a simple counting argument, assuming the existence of a primitive root mod *p*.

**Two Generalizations of the 5/8 Bound on Commutativity in Nonabelian Finite Groups**

Thomas Langley, David Levitt, and Joseph Rower

pp. 128–136

The probability that two elements in a nonabelian finite group commute is at most 5/8, and this bound is realized exactly when the center of the group is one fourth of the group. We generalize this result by finding similar bounds on the probability that a product of several group elements is equal to its reverse, and the probability that a product is equal to at least one cyclic rearrangement of itself. Both of these naturally extend the 5/8 bound.

**How Commutative Are Direct Products of Dihedral Groups?**

Cody Clifton, David Guichard, and Patrick Keef

pp. 137–140

If *G* is a finite group, then Pr(*G*) is the probability that two randomly selected elements of *G* commute. So *G* is abelian iff Pr(*G*) = 1. For any positive integer *m*, we show that there is a group *G* which is a direct product of dihedral groups such that Pr(*G*) = 1/*m*.We also show that there is a dihedral group *G* such that Pr(*G*) = *m / m* ^{'}, where *m* ^{'} is relatively prime to *m. *

**Crossword Word Count**

Matthew Duchnowski

p. 141

What is the smallest number of inch marks on a ruler that allow us to measure all integral distances? This question motivates our survey of Golomb rulers, perfect rulers, and minimal rulers—different types of rulers for allowing the most measurements with the smallest number of marks.

**Parity Party with Picture Proofs: An Odd Checkerboard Problem**

Jason I. Brown, Erick Knight, and David Wolfe

pp. 142–149

How many ways can checkers be placed on an *m *x *n* board so that each square (whether or not it is occupied) is orthogonally adjacent to an odd number of checkers? After connecting the problem to graph theory and linear algebra, we provide an answer to this problem. The solution depends not only on the parity of *m* and *n*, but also, surprisingly, on the number of trailing 1’s in their binary expansions.