Brouwer fixed-point theorem
The Brouwer fixed-point theorem is an amazing result in topology and one of the most useful theorems in mathematics. Suppose there are two sheets of paper, one lying directly on top of the other. Take the top sheet, crumple it up, and put it back on top of the other sheet. Brouwer's theorem says that there must be at least one point on the top sheet that is in exactly the same position relative the bottom sheet as it was originally. The same idea works in three dimensions. Take a cup of coffee and stir it as much as you like. Brouwer's theorem insists that there must be some point in the coffee that is in exactly the same spot as it was before you started stirring (though it might have moved around in between). Moreover, if you stir again to move that point out of its original position, you can't help but move another point back into its original position!
Not surprisingly, the formal definition of Brouwer's theorem makes no mention of sheets of paper or cups of coffee. It states that a continuous function from an n-ball into an n-ball (that is, any way of mapping points in one object that is topologically the same as the filling of an n-dimensional sphere to another such object) must have a fixed point. Continuity of the function is essential: for example, if you rip the paper in the example above then there may not be a fixed point.