 Let us look now at working backwards when using this strategy. What happens is that you know the end result you are given the end result and You are either given the process by which you reached that end result or Can somehow? Figure out how you get to that end result So what we do is we start from the end result and then work backwards Towards the beginning. Let us look now at the following problem Two players take turns at removing one to four coins from an original pile of 16 coins Is there a winning strategy for either player? You may want to pause and play this game for a few minutes to see if you can find a winning strategy Notice that for a player to win the other player must have five coins to choose from in the next to last move so that he leaves one to four coins and You can win Let's look at it this way That last move for you to win Let's call you the winner for you to win the other person must leave you one to four Coins so it must have had five So you want to give the one who will become the loser You want to leave your opponent with five Coins so that he can at most remove four and you can still remove the last one But what happens prior to that were prior to that you wish To be able to have nine to six coins to choose from So that you can choose one to four and leave Your opponent with five coins to choose from and hence so that you can win So that means that the loser must face 10 coins and so on in the previous step you want to have 14 to 11 coins so that you can force the your opponent to have to choose from 10 coins and So once here you want to force your opponent to choose from 15 and since you have 16 coins if you are playing first You will take one coin so that he Chooses from 15 and then the winning strategy is if you can Or is your opponent to choose from a pile of multiples of five you're guaranteed a win So the first player always wins provided that he knows this strategy and follows through