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.
