Question 28
Let $n \gt 1$ be an integer. Alice and Bob play the following game using an $n \times 2021$ grid. First, Alice colors each square either black or white. Bob places a piece in one of the squares in the top row and designates one square in the bottom row as the goal. Then, Alice repeatedly does the following operation $n-1$ times:
> If the piece is on a white square, she moves the piece to the square one below.
> Otherwise, she moves the piece to one left or right and then to one below.
Find the minimum possible value of $n$ such that Alice can always move the piece to the goal regardless of Bob's choice.
> If the piece is on a white square, she moves the piece to the square one below.
> Otherwise, she moves the piece to one left or right and then to one below.
Find the minimum possible value of $n$ such that Alice can always move the piece to the goal regardless of Bob's choice.