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Omni-MATH / Let n be a positive integer. Anna and Beatrice play a game with a deck of n cards labelled with the numbers 1,2,...,n. Initially, the deck is shuffled. The players take turns, star…

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problem

Let nn be a positive integer. Anna and Beatrice play a game with a deck of nn cards labelled with the numbers 1,2,...,n1, 2,...,n. Initially, the deck is shuffled. The players take turns, starting with Anna. At each turn, if kk denotes the number written on the topmost card, then the player first looks at all the cards and then rearranges the kk topmost cards. If, after rearranging, the topmost card shows the number k again, then the player has lost and the game ends. Otherwise, the turn of the other player begins. Determine, depending on the initial shuffle, if either player has a winning strategy, and if so, who does.
Plain-text mathematical notation (without MathML)
Let n be a positive integer. Anna and Beatrice play a game with a deck of n cards labelled with the numbers 1,2,...,n. Initially, the deck is shuffled. The players take turns, starting with Anna. At each turn, if k denotes the number written on the topmost card, then the player first looks at all the cards and then rearranges the k topmost cards. If, after rearranging, the topmost card shows the number k again, then the player has lost and the game ends. Otherwise, the turn of the other player begins. Determine, depending on the initial shuffle, if either player has a winning strategy, and if so, who does.
Original LaTeX notation
Let $n$ be a positive integer. Anna and Beatrice play a game with a deck of $n$ cards labelled with the numbers $1, 2,...,n$. Initially, the deck is shuffled. The players take turns, starting with Anna. At each turn, if $k$ denotes the number written on the topmost card, then the player first looks at all the cards and then rearranges the $k$ topmost cards. If, after rearranging, the topmost card shows the number k again, then the player has lost and the game ends. Otherwise, the turn of the other player begins. Determine, depending on the initial shuffle, if either player has a winning strategy, and if so, who does.

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task_id
c9495069-476c-5077-a5c5-98d3f206f485
task_key
test--c9495069-476c-5077-a5c5-98d3f206f485
task_revision_id
1
config / split
/ test

Official source

initial import