THE METHOD OF AURAS

In this course we will identify a series of prototypic problems (prototypes). To each of these prototypes we associate a related group of problems which we call the aura of that prototype. We shall discover that, although we always solve the prototype in class, the other problems in the aura can rapidly become quite difficult. Even though we can't always solve all of the problems in the aura of a given prototype, we shall discover that just describing and thinking about them will be a valuable learning experience. Aura #3 below is particularly amazing in this regard. Here are three of the most interesting examples from CSE 21, Fall 1997:

(1) Aura #1 (material in Schaum)

• P1 (the prototype): Problem 3.16 of Schaum's Outline asks for the probability that a coin of diameter 1/2 tossed onto the integral lattice ZxZ covers a point of the lattice.
• P2 (second problem in the aura): What is the probability that an equilateral triangle of side 1/2 tossed onto the integral lattice covers a lattice point?
• P3 (third problem in the aura): What is the probability that a star-shaped coin, having area A and cut from a coin of diameter less than 1, tossed on the integral lattice ZxZ covers a lattice point?
• P4 (fourth problem in the aura): What is the probability that a star-shaped coin of area A (no restrictions) tossed on the integral lattice ZxZ covers a lattice point?

(2) Aura #2 (material in Schaum)

• P1 (the prototype): Problem 3.17 of Schaum's Outline asks for the probability that three points selected at random from the circumference of a circle all lie on a semicircle.
• P2 (second problem in the aura): What is the probability that k points selected at random from the circumference of a circle all lie on a semicircle?
• P3 (third problem in the aura): What is the probability that four points selected at random from the surface of a sphere all lie on a hemisphere?
• P4 (fourth problem in the aura): What is the probability that k points selected at random from the surface of a sphere all lie on the same hemisphere?

(3) Aura #3 (material in Epp, Chapter 11 - directed graphs)

• P1 (the prototype lattice-exit model): Given a directed graph on the nonnegative lattice NxN, we look at its restriction to a finite region D and label the vertices in D to help us in the task of "escaping the lattice" with minimal cost. We call this model the "prototype of the lattice-exit models." We prove a theorem in class about large-scale regularities in any such labeling scheme.
• P2 (second problem in the aura, the terminal model): We vary the prototype slightly in terms of the rule for computing the cost of exiting the lattice. We believe that an almost identical result to that of P1 is "true," but we have no proof analogous to that of the prototype. We can, however, "prove" this result using a recent result of the logician Harvey Friedman (Ohio State) called the "Jump-Free Theorem." Sadly, the only way to prove the Jump-Free Theorem is to go beyond the realm of modern mathematics into the world of strange axioms added to the usual axioms that mathematicians the world over use and accept.
• P3 (third problem in the aura, the bureaucratic model): Again, we vary the prototype in what seems like a minor way and, just as before, we are stuck and have no standard (ZFC) proof of the result analogous to that of the prototype. Once more, however, the result we are after follows from the mysterious Jump-Free Theorem!
• P4 (fourth problem in the aura, the committee model): Same story as P2 and P3. An extension of P3 is still provable using the Jump-Free Theorem, and we have no other proof. But now we have a good excuse for our failure to find a proof. Professor Friedman, who by now has taken an interest in our class and its web site, has proved that this problem, P4, is itself independent of mathematics (as known and practiced) just like the Jump-Free Theorem. Sorry, but this may be the hardest problem ever assigned to a class at any university in history! You don't have to work it to pass the course...