Key idea. Count tails and examine every kind of move.

1. Initially there are no tails, so the tail count is 0, an even number.

2. If we flip two heads, the tail count increases by 2. If we flip two tails, it decreases by 2. If we flip one head and one tail, the count does not change.

3. In all three cases its parity stays the same. Thus the number of tails remains even after every move.

4. Having all nine coins tails up would give nine tails, which is odd. That is impossible.

Remember. Checking all types of move explains why an invariant really is preserved.