What's your favourite theorem?
Well, it would have to be Fermat's Last, wouldn't it.......
Ummm, is it normal to know a lot of these theorems offhand?
Either Gauss-Bonnet or the Theorema Egregium. Both are classical theorems related to Gaussian curvature, which is like a measure of how twisty a surface is at a point (sphere has positive curvature because it's bent in the same direction, hyperbola has negative curvature because it's bent in opposite directions, plane and cylander have 0 curvature because they're flat in a direction)
Gauss-Bonnet: if you start with a surface, and you stretch or bend it somehow, the integral of the curvature stays the same. This shows that, for instance, no matter how weirdly you stretch and bend a sphere, it stays a sphere (in some sense)
Theorema Egregium: if you keep distances between points fixed, then you also keep curvature fixed. This is why it is not possible to have a perfectly accurate map, because the paper is flat everywhere but the sphere is not flat anywhere. So any time you try to map any portion of a sphere, there will be some distortion of distances and area.
You guys are awesome. Unfortunately I don't know much. My brain is stained with dirty physics.