I started with assembling ten equal, symmetric pieces:
I figured if I could keep the pieces sorta “facing” the same way, it would work itself out. And then:
Right. Duh. Five-sided holes are hard to construct from pre-assembled three-pieced things.
So, I took a few of my pre-constructs apart, but only as many as I had to. And it worked!
In other words, I had a semi-correct sorta kinda plan than didn’t really work as I thought, but did get the results I wanted.
So, I’ve answered half of my question (although I don’t know of a way to sorta “calculate” it beforehand, I’m inclined to think it’s something along the lines of the four-color problem), and it is possible to construct the model so that you only use three colors without similar-colored neighbors. But the other question still remains: is there a way to guarantee that this works? How about for other models?