Question 4
Two players alternately choose the sign for one of the numbers $1, 2, \ldots, 20$. Once a sign has been chosen it cannot be changed. The first player tries to minimize the final absolute value of the total and the second player to maximize it. What is the outcome (assuming both players play perfectly)?
Example: the players might play successively: $1$, $20$, $-19$, $18$, $-17$, $16$, $-15$, $14$, $-13$, $12$, $-11$, $10$, $-9$, $8$, $-7$, $6$, $-5$, $4$, $-3$, $2$. Then the outcome is $12$. However, in this example the second player played badly!
Example: the players might play successively: $1$, $20$, $-19$, $18$, $-17$, $16$, $-15$, $14$, $-13$, $12$, $-11$, $10$, $-9$, $8$, $-7$, $6$, $-5$, $4$, $-3$, $2$. Then the outcome is $12$. However, in this example the second player played badly!