Wednesday, 5 February 2014

Lecture 9

The account of Warren Buffett, Bill Gates and Efron's dice that I gave in lectures you can find here.

The "Names in Boxes Problem", which I talked about today, is a puzzle which I first heard described around a lunch table at AT&T Laboratories, New Jersey, around 2003. The problem was posed in a paper, with a less catchy title:

Anna Gal and Peter Bro Miltersen. The Cell Probe Complexity of Succinct Data Structures.
Proceedings of 30th Intemational Colloquium on Automata. Languages and Programming (ICALP) 2003, 332-344.

It won a best paper award at this ICALP conference. The solution was found by Sven Skyum, a colleague of Miltersen's at the University of Aarhus If you would like to read more about the history of the problem, I recommend this paper: The Locker Problem, (by Max Warshauer, Eucene Curtin, Michael Kleber, Ravi Vakil, The Mathematical Intelligencer, 2006, Volume 28, Issue 1, pp 28-31).

Someone asked me today about whether $1-\log 2\approx 0.30$ is the best possible success probability for the prisoners. I think this question is answered "yes" by the discussion on page 30 of The Locker Problem.

The puzzle is listed as #1 in a collection complied by Peter Winkler callled, "Seven Puzzles You Think You Must Not Have Heard Correctly". I think it is remarkable that we can work out all the needed facts about cycle lengths in a random permutation by using only the simple ideas from probability that we have encountered in our course thus far.

Peter is great collector of good puzzles. He has some nice ones that are geographic. How about these (answer without looking at a map):
  1. If you travel due south from Miami, Florida, what part of the South American continent do you first hit?
  2. What two cities, one in North America and one Africa, are closest?
I have been told that Question 2 in "Seven Puzzles You Think You Must Not Have Heard Correctly" has been used by some colleagues as an entrance interview question. But I think that is rather tough!