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The mini-polymath project to find solutions to Problem 6 of the 2009 IMO is still ongoing, but I thought that, while the memories of the experience are still fresh, it would be a good time to open a parallel thread to collect the impressions that participants and observers had of how the project was conducted, how successful it was, and how it (or future projects) could be made to run more smoothly.
Just to get the ball rolling, here are some impressions I got as a (rather passive) moderator:
- There is no shortage of potential interest in polymath projects. I was impressed by how the project could round up a dozen interested and qualified participants in a matter of hours; this is one particular strength of the polymath paradigm. Of course, it helped that this particular project was elementary, and was guaranteed to have an elementary (and relatively short) solution. Nevertheless, the availability of volunteers does bode well for future projects of this type.
- A wiki needs to be set up as soon as possible. The wiki for polymath1 was an enormously valuable resource, once it was set up. I had naively thought that the mini-polymath1 project would be short enough that a wiki was not necessary, but now I see that it would have come in handy for organising and storing the arguments, strategies, insights, and ideas that arose through the linear blog thread format, but which was difficult to summarise in that format. (I have belatedly set a wiki for this project up here.) For the next polymath project (I have none planned yet, but can imagine that one would eventually arise), I will try to ensure a wiki is available early on.
- There is an increasing temptation to work offline as the project develops. In the rules of the polymath projects to date, the idea is for participants to avoid working “offline” for too long, instead reporting all partial progress and thoughts on the blog and/or the wiki as it occurs. This ideal seems to be adhered to well in the first phases of the project, when the “easy” but essential observations are being made, and the various “low-hanging fruits” are harvested, but at some point it seems that one needs to do more non-trivial amounts of computation and thought, which is still much easier to do offline than online. It is possible that future technological advances (e.g. the concurrent editing capabilities of platforms such as Google Wave) may change this, though; also a culture and etiquette of collaborative thinking might also evolve over time, much as how mathematical research has already adapted to happily absorb new modes of communication, such as email. In the meantime, though, I think one has accommodate both online and offline modes of thinking to make a polymath project as successful as possible, avoiding degeneration into a mass of low-quality observations on one hand, and a fracturing into isolated research efforts on the other.
- Without leadership or organisation, the big picture can be obscured by chaos. As I was distracted by other tasks (for instance, flying from Bremen back to Los Angeles), and had already known of a solution to the problem, I adopted a laissez faire attitude to task of moderating the project. This worked to some extent, and there was certainly no shortage of ideas being tossed back and forth, arguments being checked and repaired, etc., but I think that with more active moderation, one could have had a bit more focus on longer-term strategy and vision than there was. Perhaps in future projects one could be more explicit in the rules about encouraging this sort of perspective (for instance, in encouraging periodic summaries of the situation either on the blog or on the wiki).
- Polymath projects tend to generate multiple solutions to a problem, rather than a single solution. A single researcher will tend to focus on only one idea at a time, and is thus generally led to just a single solution (if that idea ends up being successful); but a polymath project is more capable of pursuing several independent lines of attack simultaneously, and so often when the breakthrough comes, one gets multiple solutions as a result. This makes it harder to do direct comparison of success between polymath projects and individual efforts; from the (limited) data points available, I tentatively hypothesise that polymath projects tend to be slower, but obtain broader and deeper results, than what a dedicated individual effort would accomplish.
- Polymath progress is both very fast and very slow. I’ve noticed something paradoxical about these projects. On the one hand, progress can be very fast in the sense that ideas get tossed out there at a rapid rate; also, with all the proofreaders, errors in arguments get picked up much quicker than when only one mathematician is involved. On the other hand, it can take a while for an idea or insight obtained by one participant to be fully absorbed by the others, and sometimes the key observation can be drowned out by a large number of less important observations. The process seems somewhat analogous to that of evolution and natural selection in biology; consider for instance how the meme of “try using induction”, which was the ultimately successful approach, had to first fight among competing memes such as “try using contradiction”, “try counting arguments”, “try topological arguments on the cube”, etc., before gaining widespread acceptance. In contrast, an individual might through luck (or experience) hit upon the right approach (in this case, induction) very early on and end up with a solution far quicker than a polymath effort; conversely, he or she may select the wrong approach and end up wasting far more time than a polymath would.
- The wordpress blog format is adequate, but far from ideal. Technical problems (most notably, the spam filter, the inability to preview or edit comments [except by myself], and the (temporary) lack of nesting and automatic comment numbering) made things more frustrating and clunky than they should be. Adding the wiki helps some of the problems, but definitely not all, especially since there is no integration between the blog and the wiki. But the LaTeX support included in the WordPress blog is valuable, even if it does act up sometimes. Hopefully future technologies will provide better platforms for this sort of thing. (As a temporary fix, one might set up some dedicated blog (or other forum) for polymath projects with customised code, rather than relying on hosts.)
Well, participation in the IMO 2009 Q6 polymath project has exceeded my expectations; it appears that the collaborative effort has scored some partial successes toward a solution in the first 24 hours of its existence, but is not quite there yet.
As the thread has become overly long, I am following established polymath practice and starting a new thread here, hopefully to try to impose more order onto the chaos. In order to assist this effort, some of the participants may wish to summarise some of the “state of play” so far. (I am unable to do this due to being biased by the knowledge of a solution.)
The comment numbering system has not worked as smoothly as I had hoped, but it is still better than nothing. I propose that comments in this thread start at 200, so as not to collide with the previous thread.
Of course, all the rules of the polymath exercise (as discussed in the previous thread) continue to apply. I am pleased to see that virtually everyone participating has adhered to the collaborative spirit of the exercise, for instance in keeping criticism constructive and technical rather than pejorative and personal.
As an experiment, I have enabled one level of comment threading; thus one can reply to a specific comment in the thread below. If one does so, I would request that you number the comment by subsection, for instance the first response to comment 234 would be numbered 234.1, the next one 234.2, and so forth. There is a tradeoff here; the threading adds more structure to the comments, but the flip side of this is that it makes it harder to read off at a glance all the comments made since the last time one checks the thread. Hopefully participants will be able to use their discretion when to reply and when not to. [Update, Jul 21: Threading should work properly now, I didn't set it properly before.]
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.