When you watch your favorite crime drama, what is happening? Yes, of course, they're trying to catch the murderer, but do they just follow the evidence to the criminal, and end with their answer? No, usually they're looking not only to find the murderer, but to understand the events and evidence that led to the motive, the means and the opportunity.
Mathematics is very similar to this in many respects. Mathematics seeks to understand relationships in a precise a manner as possible, and seeks not just the answers themselves, but an understanding of the answers. Just like any good crime drama, this search has led to a few places that seem to be dead ends, but actually open up new directions of study.
Back in 1985, the PBS program NOVA focused on the history of these mathematical brick walls, in a program called Mathematical Mystery Tour. It's just under 54 minutes long, and is presented in 8 parts below (just click the right arrow to go to the next part). It's a fascinating look at the challenges that mathematicians have faced and even some that they still face.
This special doesn't have time to go into too much detail on many of its subjects, so I'm including some links below where you can go for a closer look:
* Better Explained (Two especially relevant articles are How to Develop a Mindset for Math and A Quirky Introduction To Number Systems)
* Euclid's Elements
* Fermat's Last Theorem (video on how it was finally solved in 1994)
* Imagining The Tenth Dimension (with YouTube link)
* Millenium Problems (unsolved mathematical problems)