A mysterious slot machine
Ann and Drew have purchased a mysterious slot machine; each time it is spun, it chooses a random positive integer such that k is chosen with probability 2^-k for every positive integer k, and then it outputs k tokens. Let N be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann going first. Ann wins if she receives at least N total tokens from the slot machine before Drew receives at least M = 2^2018 total tokens, and Drew wins if he receives M tokens before Ann receives N tokens. If each person has the same probability of winning, compute the remainder when N is divided by 2018.
The answer is 5.
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1 solution
We claim that in general the game is fair if N = M + 1. This implies that the answer is 5 by Fermat. Indeed, consider the following game: Ann and Drew flip a coin repeatedly, and Ann wants to get to N heads, while Drew wants to get to M tails. By replacing "runs" (i.e. maximal strings of only H or T) with a spin of a slot machine, we see that the two games are the same, so long as the first flip is heads, since the first run is heads. So, we need N = M + 1.