Have you created a problem that you couldn't solve in math or logic but that someone else could?
That's actually one of my happiest achievements in mathematics.
Some years ago I came up with a conjecture based on a combination of wishful thinking, minuscule amounts of data and extremely vague intuition. I made some progress on one half of my own conjecture but wasn't able to resolve it completely, and I had no idea how to tackle the other half. The conjecture apparently spread around and eventually both halves were solved and published by people I don't personally know at all.
In some ways, this made me more proud than my own published results. I asked a good question!
Background
Given a finite group G and an expression such as XY^2Z^{-1}X^{-1}ZY^{-1}, one may ask about the values of the expression if random elements of G are substituted into the variables X,Y,Z. Officially, the "expression" is an element of the free group with some finite number of generators, and it's pretty natural to ask about what you get when you evaluate such words when group elements are assigned to the generators.
Specifically, if you plug in uniform, random elements of a finite group into some such expression, what is the probability that you get 1, the identity element? Is this probability at least some positive constant, perhaps depending on G but not on the expression?
It is not so strange to guess that the identity should come up often when you evaluate such expressions. In fact, it's easy to see that this must be the case when the group is abelian, for example. It's also reasonable to assume that given that the expression is not allowed to contain any constants.
The Conjecture
After lots of experimentation, partial results, staring at pages full of calculations, false proofs and wrong turns, I proposed the following conjecture:
A finite group G is solvable if and only if there exists some positive constant c>0 (depending on G) such that, for every r \in \mathbb{N} and every w \in F_r (the free group on r generators), the probability that w(g_1,\ldots,g_r)=1 is at least c, where the g_i are uniformly, independently chosen elements of G.
As promised, this conjecture has two halves:
- Showing that for solvable groups you can never contrive a word that pushes the probability down below some fixed positive constant (the probability of getting 1 is "bounded away from 0").
- Showing that for non-solvable groups you can always come up with a sequence of contrived words that do push the probability as close to 0 as you wish.
This conjecture seemed attractive to me for several reasons. First, it's quite elementary, requiring nothing more than the definition of a group (even the notion of a free group is not really needed here). Second, there's no obvious reason why "solvability" is the right dividing line here, and it's quite pleasing that it is: the condition of solvability originally came about from the relationship between groups and solvability of polynomials by radicals, and here we are re-interpreting it as "ease of solving equations inside the group". Indeed, the conjecture can be regarded as saying that equations like XY^2Z^{-1}X^{-1}ZY^{-1} = 1 are easy to solve in solvable groups and possibly hard to solve in non-solvable groups. Doesn't that sound quaint?
The Resolution
I made no explicit effort to publicize the conjecture. I never published it, nor my proof of the first half in the case that G is nilpotent. Somehow it made the rounds; some friends from grad school heard me discuss it and told their friends, my thesis advisor mentioned it to a few people, and eventually people smart enough to tackle it actually found it interesting enough to tackle, and solved it. It took about 4 years after the conjecture came into being.
The second half was solved first, by Miklós Abért in 2005: On the probability of satisfying a word in a group. The first half was solved by Nikolay Nikolov and Dan Segal, who were thus able to state the result as A characterization of finite soluble groups, published in the Bulletin of the LMS in 2006.
While looking up the links for this answer I noticed that more people are aware of those results and are finding them interesting (here's one example on Mathoverflow). I can't quite describe how happy this makes me.
EDIT: I just discovered that the first half became Problem 5 in the 2004 Miklós Schweitzer Competition!