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ProbIEm 1 Consider the following gamer played among two players, Alice and Bob. There are two piles of n matches. each. Let us call them File 0 and File 1. Whenever it is a player's turn. the player chooses one of the two piles, and removes one or two matches from that pile. As long as there are some matches left. a player has to remove at least one match from one of the two piles. The player who removes the last match wins. Initially, there are :1 matches on each pile for some positive integer n. Alice has the first turn. The goal of this problem is to prove that there is a winning strategy for Bob, for any positive integer n. (a) Suppose n = 2, so each pile has initially 2 matches. Assume further that in her first turn Alice removes exactly one match from Pile D. IIv'vhat should Bob do in his first turn. so that he is guaranteed to win the game? Briefly explain you answer. {b} lnformallyr describe a winning strategy for Bob. That is. explain how many matches Bot: removes from which pile after each of Afice's turns. A justification is not required. (c) Prove by induction that Bob has a winning strategy for any positive integer n. Clearly indicate the predicate that you are proving. the base case. and the inductive step