Problem: 10% of the surface of a sphere is colored green, and the rest is colored blue. Show that no matter how the colors are arranged, it is possible to inscribe a cube in the sphere so that all of its vertices are blue.
Hint: Use probabilistic analysis. Consider a random cube and calculate the expected number of vertices that are blue.
Solution: Posted by connect2ppl - Giridhar Addepalli (CSE, IITK alumnus and Yahoo! Sr. Software Engineer) in comments!