Discrete Mathematics Level 2

Alice and Bob encountered a stall selling chocolate milk--their favorite! However, they only have enough money to buy one drink and share. In front of them are 10 cups of milk arranged in a row, and each of them is assigned a value--the higher the value, the higher the temperature:

\[1, \, 3, \, 3, \, 7,\, 9, \, 1,\, 8, \, 4, \, 5,\, 2.\]

The problem arises when Alice wants the milk to be as warm as possible, but Bob wants it to be as cold as possible. So, they will play a game to decide which cup of milk they will buy. They will play alternately, and Alice will go first. The process for each player is as follows:

  1. Remove 3 consecutive cups from the table.
  2. Merge the rest of the cups without changing the order.

The game ends when there is only 1 cup left, and they will buy it. Assuming that both Alice and Bob play optimally, what is the temperature of the milk they will buy?


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