Question 110
Alice and Bob are playing a game using a blackboard. Initially, each of $2, 3, \dots, 50$ is written on the blackboard once. Also, a non-empty subset $S$ of $\{2, 3, \dots, 50\}$ is given. In the first turn, Alice erases all the elements of $S$ from the blackboard. After that, the two players, starting with Bob, take turns erasing one or more integers from the blackboard. However, they cannot erase integers that are relatively prime to all of the integers erased in the opponent's previous turn. The game ends when a player has no integers left to erase at the beginning of their turn, resulting in that player's loss and the other player's win.
How many sets $S$ are there such that Alice can always win, regardless of Bob's actions?
How many sets $S$ are there such that Alice can always win, regardless of Bob's actions?