The widespread popularity of Monopoly means that many students know the rules of this game and have developed an intuitive feel for the frequencies of visits to the various places on the board. An analysis of Monopoly can take advantage of students' interest in this game to introduce them to Markov chain theory. The transition matrix for a typical roll can be found as the product of four easily understood matrices representing the transitions due to rolling the dice, going to Jail, drawing a Chance card, and drawing a Community Chest card. The long-run frequencies for landing on the 40 squares of the board are then readily computed. Careful analysis reveals how to adjust the model to take into account features of the Monopoly game such as the special rules for going to and leaving Jail. The final model yields tables showing not only the frequencies for landing on the different properties, but also the expected value per roll of the different property groups, and the expected number of rolls required to recoup the investment needed to buy each property group and equip it with hotels.
If it is possible to cover the plane with copies of tiles chosen from a given finite set of square tiles with colored edges, so that the colors on the common edges of juxtaposed tiles match, this set of tiles is said to tile the plane. In 1961, Hao Wang conjectured that any set of tiles that tile the plane can be used to tile the plane periodically. That is, one can tile a square block of some size so that its top and bottom rows of tiles are alike and its left and right columns are alike. Clearly the plane could then be covered with the square blocks obtained by removing the right edges and bottom rows of such a block. In 1971, Raphael Robinson showed found a set of 56 square tiles with colored edges that tiles the plane, but that cannot tile it periodically, thus providing a relatively simple counterexample to Wang's conjecture. By replacing the colored edges with other markings that determine the matching rules to govern the admissible placements of the tiles, Robinson's tiles can be reduced to a set of 28 square tiles. It is not difficult to see that these tile the plane, and to understand why no such tiling can be periodic. The argument provides an attractive introduction to the study of nonperiodic tilings of the plane and of space, a field currently of pure and applied interest.
If a, b, and c are the sides of a triangle and h is the altitude to side c, then it is shown that the inequality a + b > c + h is true for all triangles in which the angle opposite side c is less than arctan(24/7), and is false for all triangles in which this angle is greater than or equal to a right angle. The solution to a classical problem that asks which of two squares inscribed in a right triangle has larger area is shown to be equivalent to establishing that the inequality is false for right triangles. A complete analysis is given to determine the direction of the inequality for any triangle, involving an interesting mix of analysis and computer graphics.
If a pair of points is chosen at random from a given convex figure, how does the average distance between them depend on the diameter of the figure? This question arose in an investigation of the optimal size of territory for various species that maintain an exclusive two or three dimensional territory for feeding or nesting. Dimensional analysis is used to determine the form of the functional dependence of the average distance on the diameter, and simulation yields approximate values of the single functional parameter. But the determination of the exact value of this parameter for all but the simplest figures leads quickly to intractable multiple integrals. By appealing to Crofton's theorem, a nineteenth century generalization of Leibniz's rule for differentiating an integral that is often useful in problems of geometric probability, evaluation of the integrals is replaced by the easy task of solving an appropriate first-order linear differential equation.
What Is the Margin of Error of a Poll?
In the 1996 presidential race the difference between the percentage for Clinton and the percentage for Dole frequently drew headlines. On one occasion a poll found this difference to be 20 percentage points, while another poll taken the same day reported the difference to be only 10 points. If each poll claimed a margin of error of 3.5% for its estimates of the percentages in favor of the candidates, does this discrepancy indicate that one or more of the polls was probably flawed? A careful analysis shows that a difference of as much as 10 percentage points in two independent polls in such a case will in fact not be a rare event.
Paul Halmos, in his recent book Problems for Mathematicians Young and Old, gives an intriguing problem about sailors dividing a pile of coconuts, with a solution that uses a fixed point theorem and eigenvalues. Here two elementary solutions of the problem are presented that would be suitable for presentation to a freshman class for prospective teachers or liberal arts students.
Many calculus textbooks contain a problem that asks students to find the optimal shape for a rectangular pen that a farmer can build with t feet of fencing, using all or part of a barn wall of length b feet to save fence. Straightforward arguments show that for t ³ 3b the pen of maximal area is square, if t ² 2b the ratio of length to width of is 2:1, and for 2b ² t ² 3b this ratio is 2b/(t-b).Exploiting a Factorization of x^n - y^n
The factorization of x^n - y^n generalizes the key step in summing a finite geometric series. Here it is used to prove several identities and inequalities, and to derive the formula for the derivative of the nth power and nth root functions.
The title of this short note explains the content.
The set of all smooth maps from the half-open interval [0, 2pi) into real 3- space is a topological space; knots are the one-to-one functions in this space and the other maps are called singular knots. Visualizing this function space is not easy, but picturing just the Lissajous maps, three dimensional analogues of Lissajous curves, is a manageable task. The Lissajous maps with coordinate functions having given periods can be identified with the points on the surface of a torus, and the student is led by a series of computer explorations and exercises to discover which points on this torus correspond to singular knots. This results in a decomposition of the torus into disjoint connected regions, in each of which all the points correspond to knots with the same topological type. Students with a strong interest in computer graphics and topology should find this research project appealing.
Applications of differential equations in physics and engineering have too often been presented by writing down a differential equation with little more than a diagram as derivation, integrating, and then declaring the problem solved when the explicit solutions appear. Here, for two problems in optics: reflection and refraction, students experiment with simple interactive computer pages to become familiar with the physical laws that will later be expressed by differential equations, and to discover qualitative features of their integral curves even before the equations have been formulated. Having thus developed their intuition, students are ready to appreciate derivations of the differential equations and formulas for exact solutions. This method of presentation can be adapted to other applied problems having nothing to do with optics or differential equations. The point is to have students better appreciate the many approaches available to them as they struggle with realistic problems.