January 2011 Contents
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Chutes and Ladders for the Impatient
Michael Jones, Leslie Cheteyan, Stewart Hengeveld
After reviewing the rules for Chutes and Ladders, we deﬁne a Markov chain model for the game and use properties of Markov chains to determine what spinner range (15) minimizes the expected number of turns to complete the game. Because the Markov chain of the full game consists of 101 states, we demonstrate our analysis with a 10-state variation with a single chute and single ladder. We conclude with an unsolved problem about expected lengths for generalized Chutes and Ladders games.
Reginald Koo and Martin Jones
Quite a number of interesting problems in probability feature an event with probability equal to 1/e. This article discusses three such problems and attempts to explain why this probability occurs with such frequency.
The Band around a Convex Body
We give elementary proofs of formulas for the area and perimeter of a planar convex body surrounded by a band of uniform thickness. The primary tool is a integral formula for the perimeter of a convex body which describes the perimeter in terms of the projections of the body onto lines in the plane.
Two-Person Pie-Cutting: The Fairest Cuts
Julius Barbanel and Steven B. Brams
Barbanel, Brams, and Stromquist (in 2009) asked whether there exists a two-person moving-knife procedure that yields an envy-free, undominated, and equitable division of a pie. We present two procedures: One yields an envy-free, almost undominated, and almost equitable allocation, whereas the second yields an allocation without the two ‘almosts.’ The latter, however, requires broadening the definition of a procedure, raising philosophical, as opposed to mathematical, issues. An analogous approach for cakes fails because of problems in eliciting truthful preferences.
Augustus De Morgan: Champion of Hamilton, Boole, Gompertz, and Ramchundra
Augustus De Morgan’s support was crucial to the achievements of the four mathematicians: Hamilton, Boole, Gompertz, and Ramchundra, some of whose work is considered greater than his own. This article explores the contributions De Morgan thus made to mathematics from behind the scenes.
Computing Determinants by Double-Crossing
Eve Torrence, Deanna Leggett, John Perry
Charles Dodgson’s method of computing determinants is attractive, but fails if an interior entry of an intermediate matrix is zero. This paper reviews Dodgson’s method and introduces a generalization, the double-crossing method, that provides a workaround for many interesting cases.
A poem about the nineteenth-century mathematician Sophie Germain.
An Elementary Treatment of General Inner Products
A typical first course on linear algebra is usually restricted to vector spaces over the real numbers and the usual positive-definite inner product. Hence, the proof that dim(S) + dim(S^) = dim(V) is not presented in a way that generalizes to non positive-definite inner products or to vector spaces over other fields. In this note we give such a proof.
The Cantor subset of the unit interval [0, 1) is large in cardinality and also large algebraically, that is, the smallest subgroup of [0, 1) generated by the Cantor set (using addition mod 1 as the group operation) is the whole of [0, 1). In this paper, we show how to construct Cantor-like sets which are large in cardinality but small algebraically. In fact for the set we construct, the subgroup of [0, 1) that it generates is, like the Cantor set itself, nowhere dense in [0, 1).