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 ];