The only problem: You have to prove (or disprove) a decades-old mathematical conjecture that has stumped professional mathemeticians.
The Clay Mathematics Institute has recently created the "Millenium Prizes", a collection of the 7 most important and unsolved problems in the field of mathematics. Most of them come from the last century and a half, and cover all areas of mathematics, from game theory to geometry to algebra to number theory.
One of the problems is the P=NP? problem. This problem is very important not just to mathemeticians, but also to computer scientists, as it describes attempting to create polynomial time (read-- easily solvable) solutions for problems that have not presented such solutions so far. The term P refers to a class of problems that can be solved within a direct polynomial relationship to their number of inputs. To clean that up a bit, they are easy to solve, even as they get very large, because they grow in a regular fashion. An example of this type P problem would be finding a text string ("britney") from a large text database ("The Internet"). All you have to do is go to Google or use the search and replace function in your word processor to see that even in an enormous volume of text, it isn't that much harder to find a word.
The type NP refers to problems that have solutions which are easy to check, but difficult to arrive at. It is known that P-problems are also NP-problems, but it can't be said that NP-problems are necessarily P-problems. It's possible that these type NP problems are also type P, but that has not yet been proven. There are many problems that fit this type NP in such a way that if any one of them is of type P, then they all must be. These are referred to as NP-complete. If it were true that the NP-complete problems were in fact of type P, it would truly revolutionize computer science, especially in the field of Artificial Intelligence, as an ability to solve NP problems would be highly useful. Most mathematicians believe the opposite than P is not equal to NP, but no one has been able to prove it.
Which brings us to Minesweeper. "WinMine" is a great example of an NP-complete problem. If a solution (solving algorithm) could be discovered that functioned in polynomial time, it would prove the P=NP problem. In fact, the author of a number of recent papers has used Minesweeper to represent not only the NP complete problem, but also to replicate logical circuits. Reducing the problem to logical circuits is one of the currently proposed methods for proving that P is not = NP, so this may provide an inroads.
Of course, it's not an easy task. And being able to beat the expert level in under 2 minutes will not win you the prize. But it is an interesting way to see how mathematics can manifest itself in the most interesting (and banal) of places.
Notes:
The Clay Mathematics Institute
An article on the P=NP? connection to Minesweeper (with diagrams of minesweeper configurations that act as logic gates (well worth looking at for EE/computer nerds (i.e. me)))
Italian campaign to fight landmines by asking people to delete winmine from their computers (No I didn't make this up)
Finally, the games mathematicians play
