Discrete Mathematics Warmups: Arrangement Puzzles
You have three colors of paint, and you want to paint all sides of a regular tetrahedron (a polyhedron with 4 equilateral triangles faces).
How many ways are there to do so if you only paint each face one of your three colors? (Rotations of the tetrahedron are considered equivalent).
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1 solution
Each of a tetrahedron's four faces is adjacent to (shares an edge with) all the other faces. Thus, if two tetrahedrons have the same number of faces colored each color, they can be rotated to look identical. In particular, this means that if a is the number of faces colored the first color and similarly for b and c , we are looking for the number of solutions in non-negative integers to
a + b + c = 4
which can be found with stars and bars as
( 2 6 ) = 1 5