Backwards induction in the centipede game

Imagine the following game, which is commonly called a ‘centipede game’. There is a pile of pound coins on the table. X and Y take it in turns to take either one or two coins from the pile, and they keep the coins they take. However, as soon as either of them takes two coins, the game stops, and the rest of the coins are cleared away. So long as they each take only one coin when their turn comes, the game continues till the pile is exhausted. Suppose the number of coins is even, and X has the first turn. Assume both X and Y aim to maximise their own gain only, and they are rational throughout the game. We mean ‘rational’ to imply only that they believe the logical consequences of their beliefs, and that they do not choose an option if there is some other available option that they believe would give them more money. For the moment, assume they have a common belief in rationality throughout the game. That is to say, throughout the game, they each believe that each of them will be rational throughout the game, that each of them will believe throughout the game that each of them will be rational throughout the game, and so on.