Rick’s Tricky Six Puzzle: S5 Sits Specially in S6
by Alex Fink and Richard Guy
Rick Wilson identified a sliding block puzzle, the Tricky Six puzzle, in which a uniquely small fraction of the possible scrambled arrangements of the six moving pieces can be restored to the solved state. The permutations one can perform form the abstract group S5, the symmetric group on five letters, but surprisingly they aren't any of the Â“obviousÂ” copies of S5 in S6 that fix a single point and allow the other five to be permuted arbitrarily. This special S5 comes from the outer automorphism of S6, a remarkable group-theoretic map whose presence is felt in several combinatorial objects. We track down this outer automorphism in the Tricky Six puzzle as well as the projective plane of order 4, the Hoffman-Singleton graph, the Steiner system S(5,6,12), and a couple of error-correcting codes.
Proof Without Words
Steiner’s Problem on the Number e
by Roger B. Nelsen
The problem in the title is: For what positive number x is the xth root of x the greatest? We answer by wordlessly proving that for positive x , the xth root ofx is less than or equal to the eth root of e.
The Geometry of Paradoxes of Voting Power
by Michael A. Jones
Geometry of 3-voter simple-weighted voting games is used to explain and to classify the causes of voting power and to demonstrate that all power indices are susceptible to paradoxical behavior. The voting rule defines hyperplanes that partition the simplex of all possible voters’ weights. Counterintuitive outcomes may occur when changes are made to the game. Changes may result in three types of geometric phenomena: a change in the players' weights may cause a point to pass a hyperplane, a change in a voting rule may affect the size and number of parts in the hyperplane partition, and a change in the number of voters results in a projection to or from a boundary of the space of all games.
Proof Without Words
The Inequality of the Arithmetic, Geometric, and Harmonic Means
by C.L. Frenzen
Geometry (a right triangle inscribed on the diameter of a semicircle) is used to establish the inequality of the arithmetic, geometric, and harmonic means in a visually transparent way.
Counting on Chebyshev Polynomials
by Arthur T. Benjamin and Daniel Walton
Chebyshev polynomials have several elegant combinatorial interpretations. Specificially, the Chebyshev polynomials of the first kind are defined by T0(x) = 1, T1(x) = x, and Tn(x) = 2x Tn-1(x) - Tn-2}(x). Chebyshev polynomials of the second kind Un(x) are defined the same way, except U1(x) = 2x. Tn and Un are shown to count tilings of length n strips with squares and dominoes, where the tiles are given weights and sometimes color. Using these interpretations, many identities satisfied by Chebyshev polynomials can be given combinatorial proofs.
Two Sums of Sines and Cosines
by Judy A. Holdener
This illustration of two sums of sines and cosines shows a finger supporting a tray of glasses. The center of mass of the tray, on which the glasses are placed at the Nth roots of unity, clearly lies at the origin. This shows that the sums of real and imaginary parts are zero and these are the sums of sines and cosines of the title.
Long Days on the Fibonacci Clock
by Edward Dunne
An elementary school mathematics teacher posed a problem concerning Fibonacci sequences and modular arithmetic, which led the author into a popular and old problem: finding the longest possible periods of Fibonacci-type sequences in the integers modulo a prime. The methods here rely on linear algebra and the interpretation of the problem as a matrix acting on a finite set. Along the way, familiar topics in number theory and abstract algebra make appearances. The problem, in one guise or another, dates back to Lagrange and Gauss. In particular settings, it is related to the generation of pseudorandom numbers.
Fooling Newton as Much as One Can
by Jorma K. Merikoski and Timo Tossavainen
Consider the equation f(x)=0 where f is a differentiable real function of one real variable. We give an example where Newton's method converges but the limit does not satisfy the equation. We also show that such an example does not exist if f is continuously differentiable or, more generally, if fis bounded near the limit. This paper is a sequel to P. Horton's paper in the December~2007 issue of the Magazine. In his examples f is not differentiable.
Ramanujan’s 6-8-10 Equation and Beyond
by Marc A. Chamberland
Ramanujan's 6-8-10 formula has been described as one of his most beautiful results. Attempts to generalize this polynomial formula have been very limited. This note demonstrates how the computer can be used to find many similar formulas. A recurrence relationship is also found which may be used to prove various identities.
ClassifyingÎ¬ -Almost Squares
by Bobbe Cooper
In the first semester of calculus, we solve lots of fence problems, maximizing area and minimizing the amount of fence. For variety, we solve problems with different costs of fence sides, lakes in the field, or corrals with extra fence dividers in the middle. The answers are usually not integers, unless the professor has carefully chosen the parameters! What happens if we insist that the answer should be an integer? Can we find a pattern for the solutions of these varied fence problems?