Distinct Patterns Problem

[bradleyharmon]

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If each side of a tetrahedron is an equilateral triangle painted white or black, five distinct patterns are possible: all sides white, all black, just one side white, just one black, and two sides white and two black. If each side of an octahedron is a white or black equilateral triangle, how many distinct patterns are possible?

My own approach to this problem was simply to add a new triangle of one set color and build up until I reached 4 black and 4 white since 5 white or 5 black would imply 3 black and 3 white, respectively, and so on for 6/2. So, there's only one way to make it all white or all black. And there seems to also be only one way to make it 1 black and 7 white and 1 way to make it 7 white and 1 black. Counting all of the way up, I have just over 20 total ways. Does anyone have a different approach to this problem or an equation that could simplify it?

 

[andrewkirk]

I got twenty-two different patterns, as follows:

• one with all black
• one with only one white
• three with two whites
• three with three whites
• six with four whites

We then double all the possibilities except the last, by swapping black for white. That gives $2\times(1+1+3+3)+6=22$.

My method of counting the number of patterns for each bullet point was to consider topological invariants. I used:

• number of edges with white on both adjacent faces
• maximum cluster size, where a cluster is a collection of white faces connected to one another by edges
• number of vertices touched by two white faces

Using topological invariants allows one to detect and discard patterns that are just rotations of a pattern already counted.