The Wednesday Morning Math Challenge: Week 16

Welcome to Week 16 of the WednesdayMorning Math Challenge. You can find solutions to the Week 15 puzzle here, along with — more importantly — discussions of some ways to approach them. Remember that the real goal here is to think creatively.

This week, we’re going to deal with a seat assignment problem! Consider the following rather general problem:

Suppose that in a university department, there are k committees, each consisting of k faculty members, and that all committees meet in the same room, which has k chairs. Suppose also that at most one person belongs to the intersection of any two committees. Is it possible to assign the committee members to chairs in such a way that each member sits in the same chair for all the different committees to which he or she belongs?

  1. Is this the case for k=2?
  2. Can you show this for k=3?
  3. Is this possible for any choice of k?

Because this is the last week, solutions for this puzzle won’t be coming next week. If you can’t solve #3, don’t feel bad. It’s an open problem in mathematics! However, if you do solve #3, call me. It’s an open problem in mathematics!

I hope you’ve enjoyed this 16 week puzzle season as much as I did. What was your record over the stretch? Hopefully, you learned how to think in different and innovative ways, and carry this skills into your daily life.

Until next time,

John Urschel