One of the most famous examples of stuckness - both for maths as a whole and for a mathematician in particular - is Fermat's Last Theorem, which states that there is no solution to $a^n + b^n = c^n$ for whole numbers $a$, $b$, $c$ and $n$ unless $n$ is 1 or 2. Pierre de Fermat wrote this down in a book in 1637, and noted "I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."
Any proof he had of this would indeed have been marvellous: despite the best efforts of many thousands of mathematicians, the 'theorem' remained unproved until the mid-1990s.
After learning of a related breakthrough in 1986, Professor Andrew Wiles dedicated himself to solving the problem. He worked in almost total secrecy for close to seven years and finally announced an outline of his proof in a series of lectures at Cambridge in June 1993.
Unfortunately, there was a problem: Wiles had made a small, but significant error somewhere in the middle of his proof. He spent another year trying to put it right. By September 1994, he was ready to give up; he told his colleague, Richard Taylor, he could see no point in carrying on trying to fix it.
Wiles - one of the most brilliant mathematicians in history - was well and truly stuck. Properly stuck.
Taylor suggested giving keeping on until the end of the month. Wiles found the approach that worked on September 19th by cobbling together two different approaches, neither of which worked on their own.
After eight years of work for Wiles - and more than 350 years of the entire mathematical world being stuck - Fermat's Last Theorem was, finally, proved.
So, how did Wiles get himself unstuck from his big sticking point? I can see several strategies that might work for anyone, from primary school students to professors.
Tell me, dear readers, what are your best hacks for getting unstuck?