Question 40
Let $n \ (\ge 2)$ be a positive integer. Alice distributes $2n$ candies into $4n$ boxes $B_1, B_2, \dots, B_{4n}$. After checking the number of candies Alice puts in each box, Bob chooses $2n$ boxes $B_{k_1}, B_{k_2}, \dots, B_{k_{2n}}$ out of the $4n$ boxes satisfying the following, and then takes all candies in the chosen boxes.
$$
k_i - k_{i-1} \in \{1, 3\} \text{ for each } i = 1, 2, \dots, 2n, \text{ and } k_{2n} = 4n.\ (k_0 = 0)
$$
Alice gets all candies in the $2n$ boxes Bob did not choose. If Alice and Bob use their best strategies to take as many candies as possible, how many candies can Alice take?
$$
k_i - k_{i-1} \in \{1, 3\} \text{ for each } i = 1, 2, \dots, 2n, \text{ and } k_{2n} = 4n.\ (k_0 = 0)
$$
Alice gets all candies in the $2n$ boxes Bob did not choose. If Alice and Bob use their best strategies to take as many candies as possible, how many candies can Alice take?