Mathematical Puzzles, Martin Gardner, and Peter Winkler

I am certainly not alone when I say that interest in mathematics was sparked by Martin Gardner’s Mathematical Games column in Scientific American. I have a strong memory of many boring physics classes in high school which I whiled away reading through the stack of Scientific Americans in the corner. Those columns led to mathematics in university (where I probably not coincidently received a “D” in physics) and onward through my interest in discrete optimization.

While Gardner has long since stopped writing the column, his influence remains strong. Every year, a group of mathemeticians, magicians, jugglers, puzzle makers and so on meet at a “Gathering for Gardner“, a get-together that looks to be a blast!

Last year, Peter Winkler, formerly of Lucent and now at Darmouth, put together a set of puzzles, cleverly and correctly entitled “7 Puzzles You Think You Must Not Have Heard Correctly”. Here is one I particularly like:

The names of 100 prisoners are placed in 100 wooden boxes, one name to a box, and the boxes are lined up on a table in a room. One by one, the prisoners are led into the room; each may look in at most 50 boxes, but must leave the room exactly as he found it and is permitted no further communication with the others.

The prisoners have a chance to plot their strategy in advance, and they are going to need it, because unless every single prisoner finds his own name all will subsequently be executed. Find a strategy for them which which has probability of success exceeding 30%. Comment: If each prisoner examines a random set of 50 boxes, their probability of survival is an unenviable 1/2^100 = 000000000000000000000000000008. They could do worse if they all look in the same 50 boxes, their chances drop to zero. 30% seems ridiculously out of reach but yes, you heard the problem correctly.

Is this really possible? Check out Peter’s writeup for the solution (and seven other problems). Also check out his book for more.