This question is similar to a previous one about "urban legends", but not the same. It is established that Milnor proved the Fáry-Milnor theorem as an undergraduate at Princeton. For the record, Fáry was a professor in France and proved the result independently. Milnor has a solely authored paper in the Annals that acknowledges Fox. The problem had been posed by Borsuk in 1947. In Milnor's version of the theorem, the infimal total curvature of any smooth knot equals its bridge number. There are many stories to that Milnor thought it was a homework problem or a test question, that he came to class late and solved it on the spot, etc. My question is: Is there any good evidence that when Milnor solved this problem, that he thought it was anything other than an open problem? I am not so interested in answers with "names withheld to spare embarrassment", nor otherwise in digressions or mischievous answers. I am more interested in a convincing citation, either to print or to a named person. (Like Milnor himself, although he has the right not to discuss the matter.) Frankly I think that some versions of this Milnor story are a little bit tasteless. At least in my own mind, I'd like to have it in a more dignified form, if it's true at all.
The answer by "none" is very good and at first I accepted it. But when I checked Nasar's book, to my surprise it does not completely put the issue to rest. Nasar says that Milnor, as a freshman, showed the proof to his differential geometry professor Albert Tucker with the request, "Would you be good enough to point out the flaw in this attempt. I'm sure there is one, but can't find it." (Tucker then passed around the proof to Fox and Chern.) It would make no sense for Milnor to say this if he truly thought that it was homework. But Nasar also contradicts this inference with the statement, "The story goes that Milnor mistook the conjecture for a homework assignment." For this statement she cryptically cites "Princeton University Archives".
Besides the conflicting evidence, the mistaken-for-homework story as attributed to Milnor is also suspiciously similar to the confirmed story about Dantzig.
Update: I sent e-mail to Milnor and I got a short reply that begins, "It seems a pity to contradict such a pleasant tall tale; but for my version see…" As I see it, the story that Milnor mistook Borsuk's curvature conjecture for homework has now died three deaths: (1) Milnor says that it's not true. (2) It isn't consistent with either Milnor's or Tucker's account of what happened. (3) It is similar to a true story about Dantzig that has mutated and that has also been pasted onto other mathematicians, such as for instance Fefferman.
By 1991, Tucker said that "as a bad joke", he called Borsuk's conjecture "an assignment". But this is 40 years after the fact, when history was already muddied by the mutating story that started with Dantzig. Actually it's not necessarily so bad, if it happened, because it should have been clear from context that it was only a joke.
Milnor referred me to a short autobiographical account, "Growing up in the Old Fine Hall". This version of the story says that Tucker first discussed Fenchel's theorem that total curvature of any topological circle is at least $2\pi$, and then stated Borsuk's conjecture; then a few days later Milnor had a draft of a proof. This account emphasizes the mathematics more than the human drama. Even so, it's just not consistent with the "thought it was homework" story.
In various sources (many provided in the second answer by "none" below), Tucker gives a consistent account that Milnor first thought that his proof was wrong and asked Tucker to find the mistake. This detail is decidedly absent from Milnor's published accounts. Obviously he asked Tucker to check his proof; that could easily be confused with being sure that there is a mistake. Maybe only two people were there; I would leave it at saying that one of them says so and the other one doesn't say.
I re-accepted the first answer given by "none" since it is most of the whole story.
Finally the true case of Dantzig. One reason that this story is so well-known, and a reason that it has mutated so much and gotten a little tacky, began with a coincidence on a plane trip to or from California. Dantzig happened to be seated next to mega-Reverend Robert Schuller and told him his account. Schuller relayed it to his flock with some silly exaggerations, and the story made its way into many other churches. Eventually Dantzig's name disappeared from this story spreading mostly among non-mathematicians. It would then be easy to fill in the name of some other mathematician. This history is told in the book "Curses! Broiled Again!" (which apparently got it from the College Mathematics Journal) For the record, in the true version of the story, Dantzig was in graduate school in statistics, it was 1939, he came in late to class, he saw two open problems on the board and thought they were homework. The professor was Neyman, who must have been a bit impenetrable to his students, because he didn't tell Dantzig for six weeks what had really happened.
Best Answer
The story is told in some detail in Sylvia Nasar's "A Beautiful Mind", a biography of John Nash, who was a fellow student of Milnor at one point. In that version, Milnor knew that Borsuk's conjecture was an open problem; he wrote up his apparent answer not believing it to be correct, and asked Fox to look it over since he (Milnor) hadn't been able to find the error himself. Fox told him to write up the result for publication; the final result was generalized considerably over the original version. It's pretty likely that Nasar interviewed Milnor (because of his biographical connection with Nash) while writing the book, so her version is probably as good as you'll find.
The "came to class late and thought it was a homework problem" story is about George Dantzig and is easy to find on the internet (e.g. Wikipedia or Snopes). It was about some problem in statistics. I think there may have actually been two open problems involved.
Sometimes the Dantzig story gets told with SIX open problems. That might be confabulated with Grothendieck's PhD thesis. Dieudonné and Schwartz had written a paper on functional analysis ending with six apparently difficult open problems. They turned these over to Grothendieck saying something like "see if you can make some progress on any of these, and that can be your thesis." Within a few months Grothendieck developed the theory of nuclear spaces, that turned all six problems into trivial calculations, and basically killed off the research direction originally proposed (by completely solving it). That story is from Allyn Jackson's biographical profile of Grothendieck in Notices of the AMS, I think.