Question 137
There are $2n$ cards labelled $1, 2, \ldots, 2n$ respectively, where $n$ is a positive integer. Two players $A$ and $B$ take turns choosing a card which has not been chosen by anyone, with $A$ going first, until all cards are chosen. Let $a_1, a_2, \ldots, a_n$ be the labels of the cards chosen by $A$ in this order, and let $b_1, b_2, \ldots, b_n$ be the labels of the cards chosen by $B$ in this order. The score of $A$ is equal to
$$
(a_1 + a_2 + \dots + a_n)^2 + (b_1 + b_2 + \dots + b_n)^2 - (a_1b_1 + a_2b_2 + \dots + a_nb_n).
$$
What is the maximum score (in terms of $n$) that $A$ can guarantee to obtain regardless of how $B$ plays?
$$
(a_1 + a_2 + \dots + a_n)^2 + (b_1 + b_2 + \dots + b_n)^2 - (a_1b_1 + a_2b_2 + \dots + a_nb_n).
$$
What is the maximum score (in terms of $n$) that $A$ can guarantee to obtain regardless of how $B$ plays?