Here is a rough copy of what we believe are the TFC/101, 200-level and 300-level questions for Thursday. Printed versions available from department.

TFC/101: The Triangle

What the is perimeter of this triangle?

(Those squares are meant to be right angles, the 12-long thing the altitude, and the 20-long thing the hypotenuse.)

200-level

The inhabitants of a vegetarian-prone community agree on the following two rules:
  1. No one will eat meat two days in a row.
  2. A person who eats no meat one day will flip a coin and eat meat the next day if and only if a head appears (on the coin).
With these assumptions, the proportion of people who eat meat on a given day tends towards some constant which does not depend on the initial proportion of people who eat meat in the community. What is the constant? Explain your reasoning.

300-level

Let S be a circle. For any two points x,y in S let lambda(x,y) be the length of the shortest arc of S that connects these two points. Does there exist a mapping phi:S ----> R^3 (where R^3 is "R cubed") such that for every x,y in S

lambda(x,y) = rho( phi(x), phi(y)),

where rho(A,B) is the distance between A and B in R^3?