More ideas (maybe):

  • The mathematics of rotation in n and representing the orientation of a k-dimensional polytope in n (related to the Hyper screen saver)
    Rotating hypercube
  • The most interesting mathematical subject I’ve encountered since graduation: geometric algebra
  • The noncentral t-distribution and the noncentral χ2-distribution
  • Introduction to the most basic concepts of topology and measure theory (leading to continuous functions and measurable functions)
  • Some math related to the problem of unbiased sampling of particulate material (and laboratory subsampling) — For a lot consisting of n particles, the sampling schemes that are guaranteed to be unbiased can be represented as a convex polytope in a high-dimensional real vector space (but how many vertices does this polytope have?)

Problem 1: Tom, Dick, and Harry competed in a track-and-field meet where points were awarded for first, second, and third place in each event. At the end of the meet, Tom had 22 points, Dick and Harry had 9 points each, and nobody else had any points.

If Dick won the javelin throw, who finished second in the 100-yard dash?

One may assume:
  1. The scoring was the same for all events.
  2. More points were awarded for first place than second, and for second place than third.
  3. No fractional points were awarded.
A little logical thinking will give you a possible solution, but you get extra credit for proving that there is only one solution.

Problem 2: The following algorithm calculates the first several terms (0 through n) of a well-known integer sequence. Can you determine which one?

for i = 0 to n do
a[ i ] ← 1;
for m = 2 to n do
for j = m to n do
a[ j ] ← a[ j ] + a[ jm ];