Alice and Bob play the following game: Alice picks a set for some natural number . Then starting with Bob, they alternatively choose one number from the set , according to the following conditions: initially Bob chooses any number he wants, afterwards the number chosen at each step should be distinct from all the already chosen numbers, and should differ by 1 from an already chosen number. The game ends when all numbers from the set are chosen. Alice wins if the sum of all of the numbers that she has chosen is composite. Otherwise Bob wins. Decide which player has a winning strategy.
Solution
To say that Alice has a winning strategy means that she can find a number to form the set A, so that she can respond appropriately to all choices of Bob and always end up with a composite number for the sum of her choices. If such does not exist, this would mean that Bob has a winning strategy instead.
Alice can try first to check the small values of . Indeed, this gives the following winning strategy for her: she initially picks and responds to all possible choices made by Bob as in the list below (in each row the choices of Bob and Alice are given alternatively, starting with Bob):
In all cases, Alice's sum is either an even number greater than 2, or else 15 or 21, thus Alice always wins.
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