Problem 4: Alice and Bob play a game with a pile of n 1 rocks. Alice moves first. On each turn, the players have the choice to either take half of the stones (rounded up) or one stone. The winner is the person who takes the last stone. Prove that Alice has a winning strategy whenever n is odd. (Hint: Induction and a strategy- stealing argument work great here!) Bonus: For which n does Bob have the winning strategy? What is that strategy? An example game follows. We start with 15 stones. Alice takes one, leaving 14. Bob takes one, leaving 13. Alice takes half (rounding up!), leaving 6. Bob takes half, leaving 3. Alice takes one, leaving 2. Bob takes one, leaving 1. Alice takes the last stone, winning.