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About a year ago, as an experiment, I set up on this blog a “mini-polymath” project, in which readers were invited to collaboratively solve the sixth question on the 2009 International Mathematical Olympiad (aka “the grasshopper problem”). After two or three days of somewhat chaotic activity, multiple solutions to this problem were obtained, and have been archived on this page.
In the post-mortem discussion of this experiment, it became clear that the project could have benefited from some more planning and organisation, for instance by setting up a wiki page early on to try to collect strategies, insights, partial results, etc. Also, the project was opened without any advance warning or prior discussion, which led to an interesting but chaotic opening few hours to the project.
About a month from now, the 51st International Mathematical Olympiad will be held in Kazahkstan, with the actual problems being released on July 7 and 8. Traditionally, the sixth problem of the Olympiad (which would thus be made public on July 8) is the hardest, and often the most interesting to solve. So in the interest of getting another data point for the polymath process, I am thinking of setting up another mini-polymath for this question (though I of course do not know in advance what this question will be!). But this time, I would like to try to plan things out well in advance, to see if this makes much of a difference in how the project unfolds.
So I would like to open a discussion among those readers who might be interested in such a project, regarding the logistics of such a project. Some basic issues include:
- Date and time. Clearly, the project cannot start any earlier than July 8. One could either try to fix a specific time (to within an hour, say), to officially start the project, or one could open the thread in advance of the IMO organisers releasing the questions, and just let the first person to find the questions post them to the thread, and start the clock from there. I assume one can rely on the honour code to refrain from privately trying to solve the question before any official starting time.
- Location. In addition to this blog here, there is now also a dedicated polymath blog for these projects, which has some minor advantages over this one (e.g. numbered and threaded comments with wide margins). It has fairly low levels of activity right now (though we are just starting to write up some modest progress from the ongoing Polymath4 “finding primes” project), but this may actually be a plus when running the project, to minimise cross-traffic. Also, another benefit of the other blog is that the project can be co-administered by several people, and not just by myself. This blog here admittedly has significantly higher traffic than the polymath blog at present, but I would certainly post a crosslink to the polymath blog if the project started.
- Ground rules. The rules for the first mini-polymath project can be found here. Basically the spirit of the rules is that the objective is not to be the first to produce an individual solution, but instead to contribute to a collective solution by sharing one’s insights, even if they are small or end up being inconclusive. (See also Tim Gowers’ original post regarding polymath projects.) But perhaps some tweaking to the rules may be needed. (For instance, we may want to have some semi-official role for moderators to organise the discussion. Ideally I would prefer not to be the sole moderator, in part because I want to see the extent to which such projects can flourish independently of one key person.)
- Set up. It seems clear that we should have an official wiki page (probably a subpage from the polymath wiki) well in advance of the project actually starting (which could also be used to advertise the project beyond the usual readership of this blog). Is there anything else which it might be useful to have in place before the project starts?
- Contingency planning. It may happen that for one reason or another, 2010 IMO Q6 will not turn out to be that good of a polymath problem. I suppose it may be sensible to reserve the right to switch to, say, 2010 IMO Q5, if need be. This might be one place where I might exercise some unilateral judgement, as it may be difficult to quickly get consensus on these sorts of issues. I don’t know if it’s worth discussing these sorts of possibilities in advance; it may simply be better to improvise when and if some course corrections need to be made.
Anyway, I hope that this project will be interesting, and am hoping to get some good suggestions as to how make it an instructive and enjoyable experience for all.
A few months ago, I gave a talk at the IMO in Bremen on “Structure and randomness in the prime numbers”. I have now converted the slides from that talk into a more traditional paper (7 pages in length), for submission to a Festschrift for the Bremen Olympiad. The content is much the same as the slides, but some references have been added.
The International Mathematical Olympiad (IMO) consists of a set of six problems, to be solved in two sessions of four and a half hours each. Traditionally, the last problem (Problem 6) is significantly harder than the others. Problem 6 of the 2009 IMO, which was given out last Wednesday, reads as follows:
Problem 6. Let be distinct positive integers and let be a set of positive integers not containing . A grasshopper is to jump along the real axis, starting at the point and making jumps to the right with lengths in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in .
Of the 500-odd participants in the Olympiad, only a half-dozen or so managed to solve this problem completely (I don’t have precise statistics yet). I myself worked it out about seven hours after first hearing about the problem, though I was preoccupied with other things for most of that time period.
I thought that this problem might make a nice “mini-Polymath” project to be solved collaboratively; it is significantly simpler than an unsolved research problem (in particular, being an IMO problem, it is already known that there is a solution, which uses only elementary methods), and the problem is receptive to the incremental, one-trivial-observation-at-a-time polymath approach. So I would like to invite people to try solving the problem collaboratively on this blog, by posting one’s own comments, thoughts, and partial progress on the problem here.
To keep with the spirit of the polymath approach, I would however like to impose some ground rules:
- Everyone who does not already know the solution, and has not worked on the problem already, is welcome to jump in and participate, regardless of mathematical level.
- However, in order not to spoil the experiment, I would ask that those of you who have already found a solution not to give any hint of the solution here until after the collaborative effort has found its solution. (At that point, everyone is welcome to give out their solutions here.) For instance, I will not be participating in the project except as a moderator.
- For similar reasons, I would ask that competitors at the actual 2009 IMO refrain from commenting substantively on the problem on this thread until after the collaborative effort has succeeded. (I know this may require some significant restraint, but I suspect the problem will become too easy if we get comments along the lines of “This was a tough problem! I tried X and Y and Z, and they didn’t work; I tried W also but ran out of time. I hear that someone solved the problem using U, though.” Of course, after the collaborative effort has succeeded, you are more than welcome to share your own experiences with the problem.)
- Participants should avoid explicitly searching for solutions to this problem on the internet (I would imagine spoilers would become available in a few days). If you do accidentally find such a solution online, I would ask that you recuse yourself from the rest of the collaboration, until after they have found a solution also. (In particular, posting links to a solution is strongly discouraged until after the collaborative effort has succeeded.)
- In a similar vein, extensive searching of the mathematical literature should only be undertaken if there is a consensus to do so on this thread.
- Participants are also discouraged from working too hard on this problem “offline”; if you have a potentially useful observation, one should share it with the other collaborators here, rather than develop it further in private, unless it is “obvious” how to carry the observation further.
- Actually, even “frivolous” observations can (and should) be posted on this thread, if there is even a small chance that some other participant may be able to find it helpful for solving the problem.
- Similarly, “failed” attempts at a solution are also worth posting; another participant may be able to salvage the argument, or else the failure can be used as a data point to eliminate some approaches to the problem, and to isolate more promising ones.
- Participants should view themselves as contributing to a team effort, rather than competing with each other (in contrast to the actual IMO). The point is not to obtain bragging rights for being the first or quickest to solve the problem (which has, after all, already been solved), but instead to experimentally test the hypothesis that a mathematical problem can be solved by a massive collaboration, without requiring serious effort on behalf of any one of the participants. (See Tim Gowers’ essay “is massively collaborative mathematics possible?” for more discussion.)
- To make it easier to reference comments in this thread, I would ask commenters to number their comments (so that the first comment be labeled 1., the second comment be labeled 2., and so forth.)
- Unlike the actual IMO, there is no artificial time limit on this exercise, though if there is insufficient participation, or the collaborative effort grinds to a halt, I may at my discretion close the experiment and give out solutions after a reasonable time period.