Stopping Strategies and Gambler's Ruin
James D. Harper and Kenneth A. Ross
Consider a gambler who repeatedly plays a game risking $1 to win $1, with a fixed probability of success each game. If that probability is 1/2, as in coin flipping, the game is fair. A typical gambling strategy, called a stopping strategy, is the decision in advance to start with some amount, say $5, and play until the fortune is doubled, say, or until it is all lost (that is, the gambler is ruined). Such strategies have been studied for 300 years. Similar stopping strategies, with unequal potential risks and gains each game, are analyzed. An example is the fair game where the gambler risks $1 to win $2 with probability 1/3. The analogous stopping strategy would be to play until the gambler has lost $5 or gained $5 or $6. Recurrence relations are used to determine the probability of ruin and both probabilities of success. The expected durations of such strategies are also determined. Similar questions are addressed for other games, including unfair games.
Arthur Cayley and the Abstract Group Concept
Sujoy Chakraborty and Munibur Rahman Chowdhury
The group concept has proved to be a most fruitful and unifying idea in mathematics and in diverse applications. Initially, the concept was applied to a set of permutations by E. Galois (1811-1832) in his quest for an answer to the question: when and why is an algebraic equation solvable by radicals? A. L. Cauchy (1789-1857) wrote extensively on groups of permutations during 1844-46, establishing them as independent objects of study. The group concept attracted the attention of A. Cayley (1821-1895) when Â– in late 1853 Â– he came across a nonabelian group of six transformations which leave a certain equation, appearing in a physical context, unchanged. This led Cayley to generalize the group concept not only to (finite) groups of transformations, but to any finite set of symbols. We show that in his very first paper (1854) on group theory, Cayley was not only in full and conscious possession of the abstract group concept and had classified abstract groups of orders up to six, but had also clearly hinted at the construction of the group algebra of a group. He later (1859) extended the classification to groups of order eight.
Dirichlet: His Life, His Principle, and His Problem
Pamela Gorkin and Joshua H. Smith
If a function is continuous on the boundary of a disk, is there a harmonic function that is continuous on the closure of the disk and equal to the original on the boundary? This question is a famous one, and it is now known as the Dirichlet problem. According to notes from Dirichlet's lectures, the problem was solved using something now called the Dirichlet principle. But there was a gap in the proof of the principle, and Weierstrass presented a clear and simple example criticizing the principle. In this paper, we discuss Dirichlet's history as well as the history of his problem and principle. Once we establish the existence of a solution to the problem, how do we find it? We conclude our paper with an algorithm for finding the solution to the Dirichlet problem for the unit disk when the boundary data is a rational function that is continuous on the boundary, and we provide several examples.
Heads Up: No Teamwork Required
Martin J. Erickson
A team competes by first deciding individually whether or not to participate, and then the active participants flip individual coins. The team wins if at least one coin is flipped and all flipped coins come up heads. Without communication, there is a strategy by which the team wins with probability greater than 1/4, regardless of the number of team members.
The Humble Sum of Remainders Function
Michael Z. Spivey
The sum of divisors function is one of the fundamental functions in elementary number theory. In this article we discuss one of its lesser known relatives, the sum of remainders function. We do this by illustrating how straightforward variations of the sum of remainders can 1) provide an alternative characterization for perfect numbers, and 2) help provide a formula for sums of powers of the first $n$ positive integers. Finally, we give a brief discussion of some nice arithmetical function properties that the sum of remainders function does not have Â— helping explain perhaps why it is less well known.
On Tiling the n-Dimensional Cube
A square may be tiled with k squares for every integer k larger than 5. We show that a cube may be tiled with k cubes for every integer k larger than 47. More generally, we show that for every dimensionn there is anf(n) with the property that an n-dimensional cube may be tiled with kn-dimensional cubes for every k bigger than f(n).
The Magic Mirror Property of the Cube Corner
Juan A. Acebón and Renato Spigler
We show that a ray of light, successively reflected from three mutually perpendicular planes, in ordinary three-space, comes back in the same direction where it came from. This fact has applications in laser resonators and in space measurements and communications, but also in the simple case of bicycle and cars retroreflectors.
Spherical Triangles of Area π and Isosceles Tetrahedra
Jeff Brooks and John Strantzen
A beautiful theorem of spherical geometry says that four copies of a spherical triangle of area $\pi$ tile the unit sphere. A classical theorem of euclidean geometry says that a tetrahedron whose four triangular faces are congruent exists if and only if the faces are congruent acute angle triangles. A necessary and sufficient condition for this is that the opposite edges of the tetrahedron are congruent. Such a tetrahedron is known as an isosceles tetrahedron. Our result asserts that any tiling of the sphere by triangles of area π can be obtained as the projection from the center of an inscribed isosceles tetrahedron.
A Butterfly Theorem or Quadrilaterals
Sidney H. Kung
Using an Â“area methodÂ” we present a proof of a butterfly theorem for quadrilaterals. Our result shows that a butterfly inscribed in a convex quadrilateral satisfies the same relation as a butterfly inscribed in a circle. The method may be applied to obtain identical results for concave quadrilaterals and triangles as well.
Row Rank Equals Column Rank
William P. Wardlaw
This note presents a short (perhaps shortest?) elementary proof that the row rank of a matrix is equal to its column rank. The methods introduced are then used to derive several other properties of the ranks of matrices over fields, and these results are extended to matrices over arbitrary commutative rings.
A Modern Approach to a Medieval Problem
Traditional pursuit problems have the prey running away from the predator. Here, the author has shown that even when they are moving toward each other, the results are startling. Their motion has been analyzed and conditions for their meeting have been worked out.