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American Mathematical Monthly - May 2007

Random-Turn Hex and Other Selection Games
By: Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson

The game of Hex has two players who take turns placing stones of their respective colors on the hexagons of a rhombus-shaped hexagonal grid. Black wins by completing a crossing between two opposite edges, while White wins by completing a crossing between the other pair of opposite edges. Although ordinary Hex is famously difficult to analyze, Random-Turn Hex-in which players toss a coin before each turn to decide who gets to place the next stone-has a simple optimal strategy. It belongs to a general class of random-turn games-called selection games-in which the expected payoff when both players play the random-turn game optimally is the same as when both players play randomly. We also describe the optimal strategy and study the expected length of the game under optimal play for Random-Turn Hex and several other selection games.

Unwrapping Curves from Cylinders and Cones
By: Tom M. Apostol and Mamikon Mnatsakanian
apostol@caltech.edu, mamikon@caltech.edu

What happens to the shape of a curve lying on the surface of a circular cylinder when the cylinder is unwrapped onto a plane? Conversely, draw a plane curve on transparent plastic, and roll it into cylinders of different radii. What shapes does the curve take on these cylinders? How do they appear when viewed from different directions? Similar questions are investigated for space curves unwrapped from the surface of a right circular cone, including conic sections, spirals, and geodesics. Unwrapped conic sections produce a new class of plane curves called generalized conics.

This paper formulates these somewhat vague questions in terms of equations, and analyzes them with surprisingly simple two-dimensional geometric transformations that lead to many unexpected results. The methods for analyzing cones and cylinders differ substantially, but both use the fact that unwrapping a developable surface preserves arclength. Applications are given to diverse fields such as descriptive geometry, computer graphics, sheet metal construction, and educational hands-on activities.

Primes Generated by Recurrence Sequences
By: Graham Everest, Shaun Stevens, Duncan Tamsett, and Tom Ward
g.everest@uea.ac.uk, shaun.stevens@uea.ac.uk, duncan.tamsetts.freeserve.co.uk, t.ward@uea.ac.uk

The notorious "Mersenne prime problem," which asks if infinitely many terms of the sequence 1,3,7,15,31,63,... are prime, remains open. However, a closely related problem has a complete answer. Beyond the term 63, every number in the sequence has a primitive divisor (that is, a prime factor that does not divide any earlier term). We investigate the appearance of primitive divisors in sequences defined by quadratic polynomials, finding asymptotic estimates for the number of terms with primitive divisors. Along the way, we discuss how mathematicians use a mixture of heuristic and rigorous arguments to inform their expectations about prime appearance and primitive divisors in several natural recurrence sequences.

Notes

Continuous Newton's Method, Inverse Functions and Nash-Moser
By: John Neuberger
jwn@unt.edu

A Note on Alternating Permutations
By: Anthony Mendes
aamendes@calpoly.edu

The Circle and the Cycloid: A Surprising Link
By: Gary Lawlor and Michael Kerckhove
lawlor@mathed.byu.edu, mkerckho@richmond.edu

A Matrix Approach to Zero-Divisors in R[x]
By: Jack Ohm
veebc@earthlink.net Problems and Solutions

Reviews

Making Transcendence Transparent.
By Edward B. Burger and Robert Tubbs
Reviewed by: David M. Bressoud
bressoud@macalester.edu

Conics.
By Keith Kendig
Reviewed by: Mark Schwartz
mdschwar@owu.edu

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