2019 BMT Team 11
Source:
January 7, 2022
combinatorics
Problem Statement
A baseball league has people, each with a different -digit binary number whose base- value ranges from to . When any player bats, they do the following: for each pitch, they swing if their corresponding bit number is a , otherwise, they decide to wait and let the ball pass. For example, the player with the number has binary number . For the first and second pitch, they wait; for the third, they swing, and so on. Pitchers follow a similar rule to decide whether to throw a splitter or a fastball, if the bit is , they will throw a splitter, and if the bit is , they will throw a fastball.
If a batter swings at a fastball, then they will score a hit; if they swing on a splitter, they will miss and get a “strike.” If a batter waits on a fastball, then they will also get a strike. If a batter waits on a splitter, then they get a “ball.” If a batter gets strikes, then they are out; if a batter gets balls, then they automatically get a hit. For example, if player pitched against player (binary is ), the batter would get a ball for the first pitch, a ball for the second pitch, a strike for the third pitch, a strike for the fourth pitch, and a hit for the fifth pitch; as a result, they will count that as a “hit.” If player pitched against player (binary is ), however, then the fifth pitch would be the batter’s third strike, so the batter would be “out.”
Each player in the league plays against every other player exactly twice; once as batter, and once as pitcher. They are then given a score equal to the number of outs they throw as a pitcher plus the number of hits they get as a batter. What is the highest score received?