BrainTwister #38: Rotating dice

#38 Rotating dice

set by Christopher Dearlove

Put a cube on a surface and trace a square around its base. How many ways can you place the cube flat on the table and perfectly within the square so that its faces are oriented differently?

Imagine an axis going through the middle of two opposite faces of the cube. You can rotate it around this axis by 90 degrees, and the cube would take up the same space (but in a different orientation). How many different axes are there going through the cube where you can do this with some rotation…