Question 139
Julian and Johan play a game with an even number, say $2n$, of cards ($n \in \mathbb{Z}_{\gt 0}$). Each card carries a positive integer. The cards are shuffled and laid out in a row on the table with the numbers visible. A player whose turn it is may take either the leftmost card or the rightmost card. The players move alternately.
Johan starts, so Julian eventually takes the last card. Johan's score is the sum of the numbers on the $n$ cards he has taken, and likewise for Julian. Prove that Johan can always achieve a score at least as high as Julian's.
Johan starts, so Julian eventually takes the last card. Johan's score is the sum of the numbers on the $n$ cards he has taken, and likewise for Julian. Prove that Johan can always achieve a score at least as high as Julian's.