2. Ana and Bojan agree to play such a game: First, Ana says a number from 1 to 7 (including 1 and 7), then Bojan adds one of the numbers between 1 and 7 (including 1 and 7) to the number said and states the sum, then Ana does the same, and so on. The winner is the one who first says the number 100. Can Ana be sure that she will be the first to say the number 100 with the right choice of the first number, or if she makes a mistake once, can Bojan win? How should Ana play to win? Can Bojan win in the game, and if so, explain how?
Solution. Ana will be sure of victory if she manages to say 92 first, as Bojan can say at most 99, and then she will say 100. This means Ana needs to say 92, 84, 76, ..., so she should start with the number 4, and then always say a number that is 8 greater (which is possible, since Bojan can add at most 7, and Ana can add at least 1).

If Ana makes a mistake at any point, i.e., says a number that is not of the form $4+8k$ $(k=0,1, \ldots, 11)$, Bojan can take advantage of this by saying the nearest number of the form $4+8k$ and continuing to play as Ana should have if she hadn't made a mistake.