What’s the minimum length of road needed to connect n cities?
This combinatorics problem is NP-Hard (if you can find a polynomial time algorithm to solve it, then P=NP), and is therefore considered difficult to solve for a computer.
An awesome demonstration of the intersection of mathematics and physics – the tendency of soap bubbles to minimize their surface area ends up minimizing the appropriate distance and solving the problem.
Worth a watch:
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really beautiful blog entry!