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Problem of the Week 971

An Odd Even Game

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Consider the following expression, where * stands for either addition or multiplication.

(((((((((? * ?) * ?) * ?) * ?) * ?) * ?) * ?) * ?) * ?)

Alice moves first and replaces the leftmost ? with a digit from 0 to 9. Then Bob replaces the first operation by either PLUS or TIMES and also replaces the ? following it by one of the unused digits. Then Alice replaces the next operation by PLUS or TIMES, and the ? following it by an unused digit. At the end of the game Bob makes his two choices and all 10 digits appear, and the result evaluates to an integer. Alice wins if the integer is even.

Who has a winning strategy, Alice or Bob?

Note: this is a game with no draws, so there must be a winning strategy for either Alice or Bob.

Source: The Inquisitive Problem Solver by Paul Vaderlind, Richard Guy, and Loren Larson, MAA.