Presented are solutions to variations of a combinatorics problem from a recent International Mathematics Olympiad. In particular, the matrix algebra solution illustrates an interaction among the undergraduate areas of geometry, combinatorics, linear algebra, and group theory. (JJK)

This paper first defines the bridge club scheduling problem that was presented to the author and then explores the meaning of an optimal solution. Next, an analytical solution is sought based on the classification of the problem as a resolvable partially balanced incomplete block design. Finally, four increasingly sophisticated techniques of…

Knot theory has been inspirational to algebraic and geometric topology. The principal problem has been to ascertain whether two links are equivalent. New methods have been discovered which are effective and simple. Considered are background information; the oriented polynomial; the Jones polynomial; the semioriented polynomial; and calculations,…

Presented are 44 examples in which students are invited to guess what pattern of numbers is emerging and to decide whether the pattern will persist. Topics of examples include Pascal's triangle, integers, vertices, Fibonacci numbers, power series, partition functions, and Euler's theorem. The answers to all problems are included. (KR)

Some appearances of pi in a wide variety of problems are presented. Sections focus on some history, the first analytical expressions for pi, Euler's summation formula, Euler and Bernoulli, approximations to pi, two and three series for the arctangent, more analytical expressions for pi, and arctangent formulas for calculating pi. (MNS)

Presented is a proof where special attention is accorded to rigor and detail in proving the lemma that relates ruler-and-compass constructions to arithmetical operations. The idea that some angles cannot be trisected by a ruler and compass is proved using three different cases. (KR)

Presented is the theorem proposed by Volterra based on the idea that there is no function continuous at each rational point and discontinuous at each irrational point. Discussed are the two conclusions that were drawn by Volterra based on his solution to this problem. (KR)

Described are the variations found between the western and African versions of the same logic puzzle. It is demonstrated that mathematical ideas are of concern in traditional non-Western cultures as well as in the West. (KR)

Explored are the distributions of residual components in two model systems. A system of components with exponentially distributed lifetimes and the two-dimensional "leaf model" in which objects fall on a plane with positions independent and normally distributed are discussed. Included are the definition, application, computations, and theorem. (KR)

The problem of determining the most energy-efficient strategy to use in approaching a traffic light that is sighted at an unknown phase in its cycle is discussed. Included are calculations, results, and conclusions. (KR)

Described are some ways in which individual states code driver's license numbers. The encoding of month, date of birth, year of birth, sex, and others involving elementary mathematics and the use of an elaborate system of hashing functions in describing the first, middle, and last names are discussed. (KR)

Presented is a combinatorial problem based on the Hash House Harriers rule which states that the route of the run should not have previously been traversed by the club. Discovered is how many weeks the club can meet before the rule has to be broken. (KR)

Durer's method for drawing an ellipse is used to explain why some people think an ellipse is egg shaped and to show how this method can be used to derive the Cartesian form of the ellipse. Historical background and suggestions for further reading are included. (KR)

Described is a way that elemental mathematics can be applied to explain an astronomical phenomenon. The fact that the extreme of sunrise and sunset do not occur on the shortest or longest days of the year is analyzed using graphs and elementary calculus. (KR)

Presented is a scheduling algorithm that uses all the busses at each step for any rectangular array. Included are two lemmas, proofs, a theorem, the solution, and variations on this problem. (KR)