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2 player game with 2 bowls and 2019 balls

Source: 2019 Swedish Mathematical Competition p3

May 1, 2021
combinatoricsgamegame strategywinning strategy

Problem Statement

There are two bowls on a table, one white and one black. In the white bowl there 20192019 balls. Players AA and BB play a game where they make every other move (AA begins). One move consists is \bullet to move one or your balls from one bowl to the other, or \bullet to remove a ball from the white bowl, with the condition that the resulting position (that is, the number of bullets in the two bowls) have not occurred before. The player who has no valid move to make loses. Can any of the players be sure to win? If so, which one?