Lesson 5

Today's Topics

1. More pigeonhole principle:
   1. Consider a round-robin tournament with 4 teams, where every team plays every team once. What are the different possible outcomes of the tournament?
   2. Is it possible to have more than one undefeated team?
   3. Is it possible to have more than one winless team?
   4. In all the outcomes where there are no winless teams (most of them,right?), there are two teams that have exactly the same record. Why?
   5. Is that true if there are a different number of teams in the tournament?
2. The "Process":
   1. Experiment
   2. Notice
   3. Conjecture
   4. Check
   5. Visualize
   6. Prove / Disprove / Counterexample
   7. Refine

A guess is a fine place to start, but it is a terrible place to stop.