Some Light Math with Tom and Hans

Today we had the BSM orientation where we listened to the director of the program talk about BSM, the facility, Budapest, and the courses.  We had a reception with some of the professors and all of the students on the program, allowing us to meet new students and talk to the professors.


I spoke with the Combinatorics 2A professor who was handing out problems for the first extra credit assignment:

1. In a group of 70 students, for every choice of distinct students A, B, student A knows a language which student B does not know.  At least how many languages do they know together.

2. I am in a 36-story building. I have with me two glass balls. I know that if I throw the ball out of the window, it won't ever break if the floor number is less than X, and it will always break if the floor number is equal to or greater than X. Assuming that I can reuse the balls which don't break, find X in the minimum number of throws.  (Copied without permission from http://www.tanyakhovanova.com/Puzzles/).

Looking at these, I wondered if he had heard the one with the chessboard and the mathematicians (if you haven't heard this story, suffice it to say that it was one of my triumphs of fall semester).  He hadn't:

3. The Devil kidnaps 2 mathematicians: A and B.  He tells them both that he has a chessboard in another room; on each square is a coin that is either heads or tails independently of all the other coins.  He will take A into the room and designate a specific square on the board.  Mathematician A will then have to flip exactly 1 coin on the board (it can be any coin on any square) and then leave the room.  Mathematician B must then enter the room and indicate which square the Devil originally designated, knowing nothing except the configuration of the board after A flipped one coin (which one is a mystery).  The two have time to discuss a strategy beforehand.  How do they succeed?

When I got home, my roommates and I discussed the classes we're thinking about taking.  I'm considering Conjecture & Proof, Topics in Geometry, Combinatorics 2A, Theory of Computing, and Topics in Analysis (the full list of classes can be found here).  We then decided to solve problems 1 and 2, which we did, at which point Tom related another problem he had heard from a professor:

4. You're on a boat 1 mile from a perfectly straight shoreline.  It's too cold for you too go above deck, so you cant actually see anything, and you don't know which way land is.  What is the best path to take to minimize the upper-bound on the time it takes you to find the shore.

We think we have the answer to this too.  Feel free to comment your thoughts on the puzzles.  If these have been too easy, I'm still stuck on one Erez gave me.


Webpages that may or may not be relevant:

http://wikitravel.org/en/Szentendre
http://www.wimp.com/glasstransparent/
http://www.wimp.com/hawaiianukulele/