There is an interesting epistemological thought experiment at "Close Range," a philosophy weblog.
The thought experiment is phrased in needlessly difficult terms, particularly given the trickiness of the problem, so here's my (suggested) rephrasing.
Kurt, a theoretical mathematician, has developed a very difficult proof for a particular conclusion (we'll call the conclusion P). The proof is so difficult Kurt can only see the cogency of the proof when he is at his very sharpest. He then goes through a sort of vacillation:
1. Immediately on finishing the proof, he is at his sharpest, so he believes P.
2. Then, almost immediately after this, he loses his grasp on the proof, and doubts it.
3. Then, he grasps it again, and believes it.
4. Then, he loses it again, and doubts it.
And so it goes, until finally, at some point (we'll call it N) his belief becomes stable, so even when he can't grasp the proof he believes P, given that there have been times when he could grasp it.
The question asked is, Does Kurt know P at (1)?
The author (Marc Moffett) suggests three possibilities:
A) He knows it at (1), doesn't at (2), knows it at (3), doesn't at (4), knows it at (N).
B) He doesn't know it prior to (N) because he isn't justified in believing P.
C) Kurt has justified true belief at (1) and (3) but doesn't know it until (N).
There are, I think, variations other than these three (especially variations on (A)), but we can stick with these. Marc opts for (C). (B) seems blatantly false; I would say Kurt is justified in believing P at stages (1) and (3) ex hypothesi: he has a proof for it. Marc is also right that there are problems in thinking our being justified in believing something at any one moment is contingent on our mental acuity at other moments. Drowsiness, sleep, the fullness of gluttony, distraction, and so forth would destroy almost any justification if (B) were true.
Marc suggests that (A) is "an immensely unattractive position" because it allows knowledge to flicker in and out, and, moreover, he thinks it is "intuitively wrong." I don't see why, on either point. The response on "Blogosophy" (another philosophy weblog) seems about right. There is another response on "Musings from the Lehigh Valley" (yet another philosophy weblog), more complicated, but, I think, on the nose.
As to (C), I think a case can be made for it; in particular, I think there is a sense of 'knowledge' in which (C) would be exactly right. I don't think, in fact, that all the things we call 'knowing', or, indeed, all the things commonly called 'knowing' in analytic epistemology, are all the same sort of thing, so a better answer would require a deeper analysis of the different sorts of things that can be considered knowledge.
All this reminds me why I dislike philosophy-by-thought-experiments; thought experiments are, at best, beginnings of arguments. Nonetheless, this is an interesting thought experiment, which is why I put it up despite the fact that it falls outside my usual philosophical interests.