From: greg@manifold.berkeley.edu (Greg Kuperberg) Newsgroups: rec.puzzles,sci.math Subject: Math problems with \$6700 in prizes!!!! Date: 17 Jun 92 00:39:31 GMT Great rewards are available to problem solvers worldwide! Here is a list of math problems with \$6700 in prizes! If you are the first person to answer one of these questions, you get the prize! Warning: The poser of each question is the sole and final arbiter of what constitutes a completely correct solution, who is the first to solve it, how much money a putative solution deserves, and all other terms of the offer. The wording of the problems given here is due to me, and the wording preferred by the problem posers may differ. If you have ideas for one or more of these problems, you can send me mail at greg@math.berkeley.edu. If the ideas are interesting and especially if you crack one of the problems, I will try to get you in touch with the relevant problem poser or posers. In addition, if you have your own math problem (or problems) with a prize attached, please contact me. New contributions are always welcome. I can't promise that I will include your problem in my list, but I will give it serious attention. The problems are listed by the size of the award, with the person offering the prize and the amount wagered for a completely correct solution. In the future there may be problems with a non-monetary prize like a bottle of wine, a live goose, or tickets to the opera, as well as problems for which the prize depends on the answer to the question, for example \$1000 for a yes and three lemons for a no. All problems so far offer the same prize independent of the answer to the question. And now, what you've all been waiting for: The problems! ---------------------------------------------------------------------------- Paul Erdos: \$3000 If the sums of the reciprocals of a set of positive integers is infinite, does the set contain arbitrarily long finite arithmetic progressions? John Conway: \$1000 A thrackle is a graph drawn in the plane with straight or curvy edges in such a way that any two edges either cross each other exactly once or share one endpoint, but not both. No other kinds of incidence between edges or vertices or self-intersections of an edge are allowed. Is there a thrackle with more edges than vertices? Ron Graham: \$1000 If you 2-color the integers from 1 to 2^2^...^2 (k times), is there a monochromatic arithmetic progression of length k? David Gale: \$500 Are there infinitely many positive integers n such that 2^n does not contain a 7 in its decimal expansion? Ron Graham: \$500 What is the shortest curve (not necessarily closed) that does not fit in an equilateral triangle? Ron Graham: \$500 Are there infinitely many positive integers n such that 2n choose n is not divisible by 3,5, or 7? David Gale: \$200 In the game of chomp, players alternate stating triples of non-negative integers, and once a triple (a,b,c) is named, then for ever after neither player can name a triple (d,e,f) with d>=a, e>=b, and f>=c. A player who names (0,0,0) loses. Does the first or second player have a winning strategy?