Answer: it is only possible if N = 2.
Step-by-step explanation:
if you have N heads up coins.
turn one, you turn N -1 coins up.
Now you have N - 1 coins tails up and one coin heads up.
Second turn, you turn N - 1 coins, (if you select the same N - 1 coins as before, you will end in the initial situation, so now we will change one coin, this time we will select the coin that is still heads up and we will leave one of the tails up unchanged)
Then we have N - 2 coins heads up and 2 coins tails up.
Now, we can select any N -1 coins to flip, and we will end with with N - 1 coins tail up and 1 coin heads up.
and this cycle will repeat.
Then the only situation is having N = 2.
Becuase when we have N -2 coins heads up and 2 coins tails up we will have: N - 2 = 0, so we have 0 coins heads up and 2 coins tails up.
This means that the statement is only true if N = 2.
