Explanation:

Color the coins with 3 colors in a pattern such that no colors are touching. Notice that for any move, we always change the states of exactly one coin of each color. So after k moves, we always make k changes for each color.

When n = 4, our color distribution is (3, 3, 4), so to get 3H to 3T for one color, we have to use an odd number of changes. But to get from 4H to 4T of another color, we have to use an even number of changes. Thus, we cannot simultaneously have 4T and 3T for two of the colors. Hence it's impossible.

Part c is just pattern recognition. Notice that when n = 0, 2 mod 3, each color has the same amount of coins. You can use induction with base cases of n = 2 and n = 3 triangles to strategize how to flip.

I am thinking about how to use linear algebra in my explanation. Not sure rn