Can pure mathematics harness citizen science? Having just finished Michael Nielsen's book "Reinventing Discovery", I find myself wondering if there are ways that pure mathematics research can engage the public in the way that GalaxyZoo or Foldit have in astronomy and protein science respectively. In other words, to break up large-scale research problems into manageable chunks which can be tackled by massed ranks of amateur enuthusiasts (citizen scientists).
Let me give you three examples of what I have in mind (which each have their own drawbacks)


*

*One obvious historically relevant possibility which comes to mind would have been getting people to check cases of the four-colour theorem. This could obviously be done by computer, but perhaps if it were done by hand(s) the proof would be less controversial. The main drawback is that the proof has already been done (by computer) and people would maybe feel like they were wasting their time because they could be replaced by a computer. A good citizen science project would make crucial use of the fact that its citizens were human.

*Perhaps, given a suitably programmed piece of software for manipulating Kirby diagrams, amateurs could be let loose on checking new potential counterexamples to the smooth Poincare conjecture in dimension 4 (like the recent Nash ones which were found to not be counterexamples by Akbulut). The disadvantage would be that such software would be pretty hard to write (I imagine).

*Donaldson's Lefschetz fibration theorem effectively reduces symplectic geometry in 4-dimensions to study of the mapping class groups of surfaces. In particular it would be useful to find new factorisations of the identity in mapping class groups into words of right-handed Dehn twists (these correspond to Lefschetz fibrations on closed symplectic manifolds). A piece of software allowing people to play with mapping class groups might help find such interesting factorisations. The drawback would be that this definitely feels like searching for a needle in a haystack. The beauty of GalaxyZoo is that each click you make is contributing positively to the project, even if you're not finding a new type of galaxy. With a needle-in-a-haystack problem, users would get frustrated very quickly. It's also possible that a computer would perform better.
So my question is:
Can we collectively come up with some reasonable (proto)-propositions for research projects in pure mathematics which would be amenable to citizen science? Ideally these would be both useful for pure mathematics, and intellectually engaging and rewarding for the citizens (preferably not searching for a needle in a haystack).
Edit: Just to be clear, I am interested in specific suggestions for mathematical problems which could be amenable to solution by 'citizen science' and I don't want to open up a discussion about whether this is desirable or sensible: that's not part of MathOverflow's remit. A priori, it's not even clear to me that such mathematical problems exist and I would be interested to hear if more imaginative people than me can come up with suggestions.
 A: abcathome.com
was an effort of Leiden University to involve citizen scientist in finding new 'good' abc-triples.
A: 1.Harnessing the existing public interest on major math problems such as Turbulence and Riemann hypothesis. Provide people with software to study toy models from these problems and search for interesting structures. For example, in turbulence we have the rise of persistent structures (called coherent structures) such as the hairpin.

(Attracting (red) and repelling (blue) Lagrangian CSs extracted from a two-dimensional turbulence experiment)


*Analytic Number theory is amenable to public science like having people search for particular kinds of primes by providing them with some software. 

A: One might find an analogy useful.  Can one harness citizen volunteerism in building a bridge?
Indeed, many might find a bridge more useful, and any experience in bridge building, materials
acquisition, aesthetic design, or even fund raising can be used.  However, to do a good job, the
primary work should be the domain of those trained in the discipline of bridge-building.  Even
the advice of experienced but retired bridgebuilders should not be taken verbatim, but should always
be considered and possibly dismissed against the situation of the present bridge being built.
Mathematics is not bridge building, but if the goal is to involve citizens (and not just their
computers), then for a project to be successful, it must not try to get citizens to do things for
which they are inadequately trained.  There are many ways to run a computer program the
wrong way; I would trust the masses to find a number of bugs in a program, but not to
verify its correctness.  So the citizens could participate in testing certain grapsable aspects of
a theory, assuming there are parts that can be made accessible.
I would also turn to the masses for inspiration and for pedagogical testing.  If I give a lecture
to a group of people, I am interested in the feedback of those who did not understand it, or
those who had a different perspective.  I would appreciate any helpful (to me) efforts made to improve or
broaden the scope of my communication, be it written or otherwise.  I would also appreciate
reasonable attempts at communicating a different perspective of the issue, so that I could
"steal the idea" and use it elsewhere.
I have more than once had the fantasy of making an adventure-based computer game
where the goal and steps to arrive it could be mapped either to a soution to an optimization problem
or proof attempts at some interesting conjecture.  Software design systems have evolved
to the point where the fantasy is being realized, if only in early stages.  That would be another
avenue for non-trained participants to contribute.
Gerhard "Ask Me About System Design" Paseman, 2012.11.18 
A: I have a suggestion.  In fact, I've had this idea on the backburner for some time.

Question:  Given a triple of permutations $\theta=(\alpha,\beta,\gamma)$, with $\alpha,\beta,\gamma \in S_n$, does there exist a Latin square that admits $\theta$ as an autotopism?

(If you're an algebraist, take the same question and replace "Latin square" with "quasigroup".)
I nearly went bonkers answering this question up to $n=17$ for this paper.
While algorithmic methods would take big chunks out of this problem, there would always be some cases that wouldn't work.  Backtracking algorithms would sometimes paint themselves into a corner early on, and take virtually forever to escape.  And, even if they did work, as soon as I resolve all cases for some value of $n$, it left open the $n+1$ case.

Why this is suitable for crowd computing:

*

*Answering an instance of this question is much like solving a Sudoku problem.  All the user has to do is input numbers in a matrix and the computer can check that there's no clashes.


*Humans have an advantage over computers: they will be able to see that they painted themselves into a corner early on.


*An individual question is not that hard (but there's a lot of them).


*Once you have a solution, it's straightforward to check that it's correct, and can act forever as a "certificate" for a given $\theta$.

I foresee implementing this as a puzzle, where the user is presented with a $n \times n$ matrix, with some boundaries (representing the cycles of $\alpha$ and $\beta$) and they place in a symbol from $\{1,2,\ldots,n\}$ into any empty cell.  Given that entry, the computer generates the orbit under the action of $\langle \theta \rangle$, thereby filling in some more cells.  From the user's point of view, it looks like the numbers "wrap around" and orbits also "pass through" walls in the matrix.
A: There are many open bijective problems in algebraic combinatorics,
that could be attacked by amateurs.
It is basically about finding a rule to explain expansions. The data could be generated easily, or posted by a mathematician, and the public can try to explain it.
Examples:


*

*The $e$-expansion of certain chromatic symmetric functions. This is about assigning a partition to acyclic orientations of certain graphs, to match some data.

*The qt-Catalan symmetry problem. Basically finding an involution on Dyck paths interchanging two quite elementary statistics.

*Inventing statistics on semi-standard tableaux explaining the qt-Kostka numbers related to modified Macdonald polynomials.
A: *

*I have no proposal, but only want to mention a historical example of what can be called
"Citizen science" in mathematics. http://www.computer.org/portal/web/csdl/doi/10.1109/85.707573.
This is how the book of Abramowitz and Stegun, Handbook of Mathematical Functions was created.
During the Great Depression, the National Bureau of Standards hired jobless people
(not professional mathematicians)
to compute tables of special functions. The result was a good and useful book. Perhaps nobody
is using tables nowadays but the book is still useful.

*Suppose that the 4-color was checked by 1000 amateurs instead of a computer. Would the proof be
more reliable, or more convincing?

*In astronomy, there is a whole area which is mostly done as "Citizen's Science". It is the discovery
of new comets. Only amateurs can afford just to look at random places in the sky. However, with
improvement of computers speed and software, I predict that even this will be soon "automatized".

*And one more question: it is somehow taken for granted that "citizen's science" is something
desirable, so this part of the question is not even discussed. (I am not so sure).
A: There are two obvious classes of problems that are amenable to this sort of thing.


*

*Converting human-readable proofs into machine-checkable proofs that are verifiable using something like Mizar, HOL Light, Isabelle, Coq, etc.  This is a large amount of work that a professional mathematician might find boring but that amateurs could find interesting.

*Large-scale searches and computations that can be farmed out, like SETI@home and GIMPS.  The existence of a finite projective plane of order 12 would be one example, but it would be easy to come up with a lot of other problems of this sort.
