I've encountered a mathematical puzzle with an enlightening solution - sadly I recall only the outline of the solution without the specifics. The puzzle went somewhat as follows:
A king with many wives and 1000 daughters offered to spare a young man who had offended him, with a condition. He could either marry his most beautiful daughter, or be executed. The catch is, he may see each daughter only once, and must decide on the spot if she is the most beautiful. Accepting that the young man has never seen any of the daughters before, beauty is well ordered (admittedly a stretch), and the order the daughters are presented is perfectly random (no hidden tricks by the king), what strategy will maximize the young man's probability of survival? What are his odds?
I recall the solution was: wait for the first n daughters to pass by, remembering the most beautiful seen thus far. After that, pick the first daughter seen who is more beautiful than the best found in the first n. What would be the formula for finding n?
As lulu said, this is the secretary problem. The accepted solution is to apply the odds algorithim. You reject the first n / e daughters (e being the base of the natural logarithm) and then accept the next applicant who is better than all the other rejected applicants. In this case, that is after 1000 / e daughters (or 368 rounded to whole figures), you then choose the next daughter who is better than all the previous daughters.
This has an approximately 37% success rate; sadly there is no way to guarantee a success.