MathDB

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

.
Proposed by Brandon Wang

Problem Statement