Question 55
Let $n$ be a positive integer. Alice and Bob play the following game. First, Alice picks $n+1$ subsets $A_{1}, \ldots, A_{n+1}$ of $\{1, \ldots, 2^{n}\}$ each of size $2^{n-1}$. Second, Bob picks $n+1$ arbitrary integers $a_{1}, \ldots, a_{n+1}$. Finally, Alice picks an integer $t$. Bob wins if there exists an integer $1 \leq i \leq n+1$ and $s \in A_{i}$ such that $s+a_{i} \equiv t\pmod{2^{n}}$. Otherwise, Alice wins.
Find all values of $n$ where Alice has a winning strategy.
Find all values of $n$ where Alice has a winning strategy.