Painting a tetrahedron
You have different colors, and you wish to paint the faces of a regular tetrahedron, each in a different color. How many different ways can you do it?
Note : Different implies that for two coloring combinations and , you can't pick up and reorient it to make it look exactly like .
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1 solution
Suppose the 4 colors are red, green, yellow and blue. Without loss of generality, you can paint the bottom green. Then you really only have two unique choices for the remaining three colors: Going clockwise looking from the top... "red -> yellow -> blue" or "red -> blue -> yellow".
Therefore there are 2 solutions.
Or another way of looking at it, is that you can pick any two sides (without loss of generality) and paint them yellow and blue. This leaves you only 2 choices for the remaining two colors (red and green), which are both "different" as defined in the problem.