I was reading the New York Times Numberplay blog, where they had a cute little puzzle: There are 64 mathematicians at a conference. Each of them knows his (or her) own unique theorem. At social time, the 64 pair up and exchange all of their knowledge, which takes an hour. At the end of the hour, they pair up again, exchange knowledge, etc. The question is -- how long does it take before every mathematician knows every theorem (presumably with an optimal selection of pairings)? There is really nothing special about 64. One can do this for an integer n >=2 (n=1 is uninteresting). If n is odd, there is just an "odd man out" for each pairing. This question is pretty easy (though I'll leave it to you to work out the solution), but a comment on the blog lead me to give the following variants: 1) Suppose that each pairing is chosen from all possible ones uniformly at random. The time until all mathematicians know all theorems, T_n, is now a random variable. What is its distribution? In particular, what is its expectation? 2) Suppose that there are m regular mathematicians and n forgetful mathematicians. The forgetful know nothing, and promptly forget everything they are told. However, there is no way to distinguish the regular from the forgetful by outward appearance. One can ask the same two questions: what is an optimal deterministic choice of pairings, and, in the random pairing case, what is the distribution of the waiting time until all m regular mathematicians know all m theorems? 3) Now suppose that each mathematician has a faulty memory. At first I suggested that there was a fixed probability, p, so that after exchanging the information, each piece of knowledge (theorem) contained in the memory of each mathematician is deleted with probability p, independently. However, after a discussion with my colleague Fred Kochman, I realized that there was some probability > 0 that a particular theorem might be deleted from everyone's memory, and so the expectation of the waiting time would be infinite. I thought of two possible ways to repair this: a) Every mathematician always remember his/her own theorem (i.e. it can't be deleted). b) More symmetrically, after mathematicians A and B exchange information, and before any deleting, they have the same information. For each theorem i that they know, with probability 1-p, they both remember it, with probability p/2 A remembers it and B forgets it, and similarly with probability p/2 A forgets it and B remembers. Thus, someone always will remember a theorem. Victor