I had the opportunity to have a professor (in functional analysis) who had a lot of personal experiences of John F Nash Jr. Basically, the movie is almost entirely fiction. The book, however, is supposedly fairly close to real events.
Regarding that problem on the board, if you pay attention and freeze frames, you'll notice the "problem" on the board changes between cuts. So not only is it a bogus problem, it's many bogus problems. (As far as I recall from doing this a few years back when the movie was new - at the time I did not really possess the mathematical knowledge needed to interpret it, but I asked some who did).