You want to paint as many tetrahedra as possible given that you're limited to these N colors, with one color per face, but ** not** requiring different colors on different faces, so that no two tetrahedra are identical. Two tetrahedra can count as non-identical even if they are mirror images, reversed.

It turns out that, given this value of N, you can make exactly as many tetrahedra with three colors as you can with four colors.

What's the value of N, the number of different colors available?

How many different tetrahedra do you have all together?