Dogic

I'm not sure why this kind of puzzle is so hard to find. I don't actually own one. Nevertheless, I have written software to simulate one, and thereby been able to solve it. It's a fun puzzle. Interestingly, the solved state is not unique. You need only make each of the 20 faces of the icosahedron a solid color.

Step 1

Solve all but the last 2 faces. Trust me; it's easy and intuitive.

Step 2

Here I describe a tri-cycle with pictures and words. Ideas from this can be used through-out the entire solve from the beginning.

Consider the following puzzle state.

As shown, move the teal triangle to the other teal triangle's position as indicated. We then come to...

Here you can see that I've drawn a red-bordered triangle over the front-facing face. Simply trade this face with one in the top layer, then bring it back down into the middle layer where it was, but in a different, desired orientation. Specifically, one where the cyan triangle is ready to move into position. Labeling the vertices of the off-white face A, B, C, counter-clock-wise, starting in the lower-left corner, the sequence is simply (B,C,Bi),2Ci,(Ai,C,A). We now have...

If at this point you don't know what to do, then I can't help you.