MathDB

Alice and Bob play a game in which they take turns choosing integers from 1 to

n

. Before any integers are chosen, Bob selects a goal of "odd" or "even". On the first turn, Alice chooses one of the

n

integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the integers that has not yet been chosen, until the

n

th turn, which is forced and ends the game. Bob wins if the parity of

\{k

: the number

k

was chosen on the

k

th turn

\}

matches his goal. For which values of

n

does Bob have a winning strategy?

Problem Statement