Question 151
Jasper and Rose are playing a game. Twenty-six $32$-ounce jugs are in a line, labeled Quart $A$ through Quart $Z$ from left to right. All twenty-six jugs are initially full. Jasper and Rose take turns making one of the following two moves:
- Remove a positive integer number of ounces from the leftmost nonempty jug, possibly emptying it
- Remove an equal positive integer number of ounces from the two leftmost nonempty jugs, possibly emptying one or both of them. (Attempting to remove more ounces from a jug than it currently contains is not allowed.)
Jasper plays first. A player's score is the number of ounces they take from Quart $Z$. If both players play to maximize their score, compute the maximum score that Jasper can guarantee.
- Remove a positive integer number of ounces from the leftmost nonempty jug, possibly emptying it
- Remove an equal positive integer number of ounces from the two leftmost nonempty jugs, possibly emptying one or both of them. (Attempting to remove more ounces from a jug than it currently contains is not allowed.)
Jasper plays first. A player's score is the number of ounces they take from Quart $Z$. If both players play to maximize their score, compute the maximum score that Jasper can guarantee.