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Omni-MATH / Let n be a positive integer. Two players A and B play a game in which they take turns choosing positive integers k≤n. The rules of the game are: (i) A player cannot choose a number…

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problem

Let nn be a positive integer. Two players AA and BB play a game in which they take turns choosing positive integers k≤nk \le n. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player AA takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]
Plain-text mathematical notation (without MathML)
Let n be a positive integer. Two players A and B play a game in which they take turns choosing positive integers k≤n. The rules of the game are:

(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.

The player A takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.

[i]
Original LaTeX notation
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:

(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.

The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.

[i]

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Source and history

task_id
4d353750-b381-5b95-a368-c8027b81c887
task_key
test--4d353750-b381-5b95-a368-c8027b81c887
task_revision_id
1
config / split
/ test

Official source

initial import