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Omni-MATH / Three friends Archie, Billie, and Charlie play a game. At the beginning of the…

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problem

Three friends Archie, Billie, and Charlie play a game. At the beginning of the game, each of them has a pile of 20242024 pebbles. Archie makes the first move, Billie makes the second, Charlie makes the third and they continue to make moves in the same order. In each move, the player making the move must choose a positive integer nn greater than any previously chosen number by any player, take 2n2n pebbles from his pile and distribute them equally to the other two players. If a player cannot make a move, the game ends and that player loses the game. $\hspace{5px}$ Determine all the players who have a strategy such that, regardless of how the other two players play, they will not lose the game.
Plain-text mathematical notation (without MathML)
Three friends Archie, Billie, and Charlie play a game. At the beginning of the game, each of them has a pile of 2024 pebbles. Archie makes the first move, Billie makes the second, Charlie makes the third and they continue to make moves in the same order. In each move, the player making the move must choose a positive integer n greater than any previously chosen number by any player, take 2n pebbles from his pile and distribute them equally to the other two players. If a player cannot make a move, the game ends and that player loses the game.
$\hspace{5px}$ Determine all the players who have a strategy such that, regardless of how the other two players play, they will not lose the game.
Original LaTeX notation
Three friends Archie, Billie, and Charlie play a game. At the beginning of the game, each of them has a pile of $2024$ pebbles. Archie makes the first move, Billie makes the second, Charlie makes the third and they continue to make moves in the same order. In each move, the player making the move must choose a positive integer $n$ greater than any previously chosen number by any player, take $2n$ pebbles from his pile and distribute them equally to the other two players. If a player cannot make a move, the game ends and that player loses the game.
$\hspace{5px}$ Determine all the players who have a strategy such that, regardless of how the other two players play, they will not lose the game.

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Source and history

task_id
70d36a2b-9bd9-585f-8a0a-6e49603dc841
task_key
test--70d36a2b-9bd9-585f-8a0a-6e49603dc841
task_revision_id
1
config / split
/ test

Official source

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