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Mathematics Magazine - June 2001

ARTICLES

Elusive Optimality in the Box Problem
D. Marc Kilgour and Nelson M. Blachman
On a game show you are presented with two identical boxes that contain unknown amounts of money, one twice as much as the other. You pick one box and see how much is inside. Now you must decide whether to trade it for the other box, which contains either twice or half what you have already. You might think that a person with a reasonable prior probability distribution plus knowledge of the amount in the box selected could come to a sensible decision about what to do. But, amazingly, there are prior distributions for which your optimal decision seems to be “always trade boxes,” regardless of how much you find in the one you pick first. We explore this situation, explaining why this paradoxical conclusion emerges and illustrating the elusiveness of optimality in the Box Problem with a million-round simulation.

The Anxious Gambler's Ruin
Joseph Bak
Suppose a gambler competes against an opponent until one of them loses all of his money. The famous gambler’s ruin problem determines the probability that the gambler will be ruined in terms of his initial fortune, the intital fortune of his opponent, and his fixed probability of winning each game. In this classic case, an equal amount of money is staked on each game. The anxious gambler’s ruin deals with the probability of ruin for a gambler who bets the maximum possible on each game. That is, the amount bet on each game is the minimum of the complete fortunes of the two opponents. The duration, or expected number of games until one of the players is ruined, is also considered. In addition to deriving formulae for the proabability and the duration, the article examines relations among these variables in both the classic situation and the anxious gambler approach.

Counting Perfect Matchings in Hexagonal Systems Associated with Benzenoids
Fred J. Rispoli
Many hydrocarbons such as benzene have a molecular structure involving hexagonal rings. Chemists call these hydrocarbons benzenoids and have discovered that the number of perfect matchings contained in associated graphs (hexagonal systems) is relevant to the chemistry of benzenoids. Furthermore, standard methods of linear algebra and discrete mathematics can be used to count the perfect matchings. Surprisingly, familiar counting numbers including Fibonacci numbers and binomial coefficients show up.

Using Less Calculus in Teaching Calculus: An Historical Approach
Radoslav M. Dimitri\'c
The routine teaching of calculus in contemporary education practices takes away from the essence and beauty of mathematics. I want to show that shadowed by this frantic "calculus industry" there exists mathematics that is both essential and appealing. To that end, I use examples the great masters tackled: The number e, Bernoulli's inequality, Steiner's problem on the x^th root of x and Reaumur's honeybee cell problem as solved by Boscovich.

NOTES

Strategies for Rolling the Efron Dice
Christopher M. Rump
In this note we consider a two-player competitive game involving the rolling of non-transitive dice. In particular, we analyze games involving selecting from the four dice invented by statistician Bradley Efron. Unlike other non-transitive games such as rock-scissors-paper, the optimal strategy does not randomly choose among the four dice with equal (uniform) probability.

Probabilities of Consecutive Integers in Lotto
Stanley P. Gudder and James N. Hagler
We observe that, quite often, the winning numbers in a lotto game of the form ``pick m numbers between 1 and n'' contains two consecutive integers. Probabilities of this and related events are computed and compared with empirical results. Further, we show the following using only calculus: for a given n, the smallest m which gives a probability greater than or equal to 1/2 that at least two of the numbers are consecutive is about \sqrt{n\ln2}.

Pythagorean Boxes
Raymond A. Beauregard and E. R. Suryanarayan
A Pythagorean rectangle is one with integer sides and integer diagonals. Much has been written about these in the context of Pythagorean triangles which are represented by PTs, that is, integer triples (a, b, c) satisfying a^2 + b^2 = c^2 . A Pythagorean box is a box whose edges and inside diagonals are integers. These are represented by ordered quadruples (x, y, z, w) whose components are integers satisfying the equation x^2+y^2+z^2= w^2 , where w > 0 . We refer to such quadruples as PBs. In this paper we look at the geometric and algebraic properties of PBs with one eye on the many nice properties that PTs exhibit. For example, it is well known that every pair of positive integers n,m determines a PT, namely (n^2-m^2, 2nm, n^2 + m^2) . By analogy we show how every pair of positive rationals determines a PB. The set of PTs can be made into a group in at least two different ways. We show how the set of PBs is a group extension of each of these. In fact, the set of PBs and the set of PTs are made into fields, the former an extension of the latter of degree 2. Pythagorean boxes with a square base are analyzed in terms of a Pell equation, and perfect Pythagorean boxes (diagonal of each side is an integer) are discussed.

A Simple Fact About Eigenvectors That You Probably Don't Know
Warren P. Johnson
The fact alluded to, in its simplest form is this: suppose a and ab are distinct eigenvalues of a 2x2 matrix A, with corresponding eigenvectors u and v. Then both columns of A-aI are multiples of v, and both columns of A-bI are multiples of u. Although this is not difficult to prove, I have never seen it stated anywhere. In our paper it follows from a simple lemma, which also implies the well-known result that real summetric matrices have orthogonal eigenvectors.

A Generalized General Associative Law
William P. Wardlaw
An overview of the many different treatments of the General Associative Law in the literature is given. Then the following generalization of the General Associative Law is presented: A groupoid G (set with a single binary operation) is n-associative if the product of any n elements is independent of how they are associated, that is, if a1a2 . . . an denotes unambiguously an element of G, which is independent of the way the product is parenthesized. The “Generalized General Associative Law” asserts that if n> 3 and G is n-associative, then G is (n+1)-associative. This result is proved and several examples are given to show that it is actually a generalization of the General Associative Law.

An Application of the Marriage Lemma
Andrew Lenard
It is possible to provide a list of ordered arrangements of n letters in such a manner that every subset of those letters, when suitabley ordered, appears as an initial segment of at least one of those arrangements. This proposition is proved with a significant application of the “Marrriage Lemma,” and a construction is provided for a shortest possible such list.

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