Question 112
A pile of $2^n - 1$ coins is placed on the table. Alice is allowed to make the following moves: If a pile has an even number of coins, she can throw half of those coins away. If a pile has an odd number of coins $k$ such that $k + 1$ is properly divisible by $2^l$ (i.e. $2^l|k + 1$ and $2^l \lt k + 1$) for some positive integer $l$, Alice can take away $2^l - 1$ coins from this pile and form a new pile with those coins. Alice continues making moves until no more moves can be made. What are the possible values for the number of coins that will remain on the table?