If you didn't get one of these for Christmas (this year, or in 1985), you can create and play with one in SketchUp! This project shows how to model the completed cube and how to spin the sides, but sorry - you'll have to solve it for yourself!
This is a model of a popular optical illusion. Starting with a simple checkerboard . . .
Anyone can design anything in 3D! http://www.3dvinci.net/