Archimedes' Cattle Problem
by Ilan Vardi
This paper presents a solution to a problem posed by Archimedes over two thousand years ago. Don't be fooled by its ancient origins because the problem is actually quite difficult and a complete solution wasn't given until 1880. Even this solution wasn't complete since it merely showed how one could generate the complete answer, which consists of 8 numbers of 206,545 digits (a computer-generated solution was finally given in 1965). In this paper, a streamlined argument is used to find simple explicit formulas for the solution, and one-line expressions for the answer are given. The techniques involve the basic theory of finite fields, elementary number theory, and computer algebra.
Lanczos' Generalized Derivative
by C. W. Groetsch
We examine a curious approximate differentiation rule of Cornelius Lanczos. Lanczos' rule is shown to be a generalized derivative with convergence properties that are reminiscent of the convergence of Fourier partial sums. The effect of data errors on the rule is examined both numerically and analytically and a best possible order of approximation is established.
A Stroll Through the Gaussian Primes
by Ellen Gethner, Stan Wagon, and Brian Wick
email@example.com, firstname.lastname@example.org, email@example.com
One cannot walk to infinity on the real line if one uses steps of bounded length and steps on the prime numbers. This is simply a restatement of the classic result that there are arbitrarily large gaps in the primes. The proof is simple: a gap of size k is given by (k + 1)! + 2, (k + 1)! + 3,..., (k + 1)! + (k,+1).
But the same problem in the complex realm is unsolved. More precisely, an analogous question asks whether one can walk to infinity in Z[i], the Gaussian integers, using the Gaussian primes as stepping stones, and taking steps of bounded length. The Gaussian question is much more complex because of the additional dimension. For example, there are arbitrarily large disks in Z[i] that contain only Gaussian composites; but that has little impact on a trek to infinity for a walker who can, with luck, simply walk around the obstacle.
This problem is sometimes called the Gaussian moat problem, since one way of establishing a walk's nonexistence is to present a sufficiently wide moat (region of composites) that completely surrounds the origin.
In this paper we present two new moats (4 and square root of 18), as well as a computational proof that a square root of 26-moat exists. We also show that there exists a real Gaussian prime that is isolated in a neighborhood of radius k for any positive k.
Galileo's Work on Swiftest Descent from a Circle, and How He Almost Proved the Circle Itself Was the Minimum Time Path
by Herman Erlichson
In Proposition 36 (Third Day) of his Two New Sciences, Galileo proved that descent to the bottom from any point on the lower quadrant of a vertical circle was swifter by a longer two-chord path than it was by the direct one-chord path. In the Scholium to this proposition Galileo proved (not rigorously, as we show) that the swiftest descent using the circle constraint was descent via the circle itself. By the 'circle constraint' we mean that any path to the bottom must consist of a sequence of planes (circle chords), with every segment beginning and ending on the circle (and with the circle itself as the limiting case of an infinite set of infinitesimal chords). The study of this proposition and its Scholium is the subject of this paper. This study provides us with a very interesting example of Galileo's geometrical methods as applied to constrained descent in a uniform gravitational field.
Induction the Hard Way
by Michael Sheard
Suppose you are given P(0) and that for every number n, if P(n) then P(n+1), but you are told that you may not use the Principle of Induction. How many steps would be required to write out a complete proof of P(1,000,000)? Surprisingly, the answer is much less than a million. We explore some strategies for building short proofs, without induction, of P(N) for large values of N.
Good Paths Don't Double Back
by C. Douglas Howard
A Note on the Clarke Subdifferential
by Xianfu Wang
The Number of Conjugacy Classes
by Michael Reid
Which Coronas Are Simple?
by Branko GrÂŸnbaum
Ordinary Differential Equations Texts
Reviewed by David A. SÂ‡nchez
The Schools We Need.
By E. D. Hirsch, Jr.
Reviewed by Owen Thomas
Force and Geometry in Newton's Principia.
By FranÂois De Gandt.
Reviewed by Victor J. Katz