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American Mathematical Monthly -June-July 2000


June-July 2000

When Close Enough is Close Enough
by Edward R. Scheinerman
We are given two real numbers that we believe to be equal. It is, of course, not sufficient simply to calculate these numbers to a large number of decimal places in order to declare the equality proven. Or is it?


Loops of Regular Polygons
by K. Robin McLean
Imagine that you have a large supply of identical flagstones, each in the form of a regular octagon. You wish to lay out a path by placing some of the octagons edge-to-edge in a complete loop. What shapes of path are possible? Is there a path with exactly (say) 9 octagons? This mathematical investigation designed for children leads to a new application of field theory to elementary geometry that is quite different from classical results on ruler and compasses constructions.

When the octagons are replaced by regular n-gons (and are allowed to overlap, but not to coincide) there is a fascinating interplay between arithmetic and geometry. For example, when n is even, its smallest odd prime factor influences the number of n-gons in a loop. When n is odd, the orientation of the n-gons (vertex up, vertex down) alternates as we go round the loop. In this case, the existence of a loop in which two n-gons with opposite orientations share the same center depends on whether n is the power of a prime. Regular n-gons capable of tessellating the plane (n = 3, 4, 6) can be characterized by their loops.


Random Approaches to Fibonacci Identities
by Arthur T. Benjamin, Gregory M. Levin, Karl Mahlburg, and Jennifer J. Quinn,,,
By combining combinatorics and probability we offer combinatorial proofs of classical Fibonacci Identities. Among these are several proofs of Binet’s Formula, which express Fibonacci numbers in terms of the Golden Ratio.


Surprise Maximization
by D. Borwein, J. M. Borwein, P. Maréchal,,
The Surprise Examination or Unexpected Hanging Paradox has long fascinated mathematicians and philosophers, as the number of publications devoted to it attests. For an exhaustive bibliography on the subject, see We examine and solve the optimization problems arising from an information theoretic avoidance of the Paradox. These problems provide a very satisfactory application of both the Kuhn-Tucker theory and of various classical inequalities and estimation techniques.


Polygons and Numerical Ranges
by Pei Yuan Wu
Some gems from the nineteenth-century projective geometry, namely, those concerning inscribing ellipses in the plane, have recently been found to have a close connection with the numerical ranges of complex matrices, definitely a twentieth-century subject. The connection lies in the rather superficial fact that numerical ranges of 2-by-2 matrices are elliptical discs. In this article, we illustrate this with three ellipse-inscribing results: (1) Poncelet's closure theorem on the existence of infinitely many n-gons interscribed between two ellipses when one such n-gon exists, (2) a consequence of theorems of Brianchon and Ceva on a condition for the existence of an ellipse inscribed in a triangle with given tangent points, and (3) Siebeck's theorem on a condition for the existence of an inscribing ellipse in a triangle with given foci. These are generalized to the higher-dimensional case, in which the role of the inscribing ellipse is played by the boundary of the numerical range of a matrix in a certain special class.


Proofs as Games
by Pavel Pudlák
Many mathematicians would like to learn more about mathematical logic, but are repelled by the formalism used there. This paper shows that some problems in logic can be translated into a more comprehensible language. In particular, we show an interpretation of propositional resolution proofs in terms of a two player game. Thus proofs become strategies of the first player, while to prove a lower bound on the lengths of proofs one has to devise a 'superstrategy' for the second player. We present the famous lower bound on the lengths of resolution proofs of Armin Haken in these terms. After reading the paper, the reader should be able to attack some open problems concerning the lengths of resolution proofs.

For other papers related to this subject, see the author’s bibliography at



Holey Coronas
by Wlodzimierz Kuperberg

Stirling's Approximation for n!: the Ultimate Short Proof?
by Dan Romik

The Hankel Determinant of Exponential Polynomials
by Richard Ehrenborg

What's Best?
by Arthur T. Benjamin and Matthew T. Fluet,

Unsolved Problems

A Celtic Magnecurve Problem
by Doug Engel

Problems and Solutions


Taking Chances: Winning with Probability
By John Haigh

Reviewed by David Aldous

The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation
By Gary William Flake

Reviewed by Michael Frame