Question 61
Mr. Precise wants to take his tea cup out of the microwave precisely at the front. The microwave of Mr. Precise is not precisely cooperative.
More precisely, the two of them play the following game:
Let $n$ be a positive integer. The rotating plate of the microwave takes $n$ seconds for a full turn. Each time the microwave is turned on, the plate is turned clockwise or counterclockwise for an integer number of seconds such that the tea cup can end up in $n$ possible positions. One of these positions is marked ,,front“.
At the start of the game, the microwave rotates the tea cup in one of these positions. Afterwards, for each move, Mr. Precise enters the integer number of seconds and the microwave decides whether to turn clockwise or counterclockwise.
For which $n$ can Mr. Precise ensure that after a finite number of moves, he can take out the tea cup of the microwave precisely from the front position?
(Birgit Vera Schmidt)
More precisely, the two of them play the following game:
Let $n$ be a positive integer. The rotating plate of the microwave takes $n$ seconds for a full turn. Each time the microwave is turned on, the plate is turned clockwise or counterclockwise for an integer number of seconds such that the tea cup can end up in $n$ possible positions. One of these positions is marked ,,front“.
At the start of the game, the microwave rotates the tea cup in one of these positions. Afterwards, for each move, Mr. Precise enters the integer number of seconds and the microwave decides whether to turn clockwise or counterclockwise.
For which $n$ can Mr. Precise ensure that after a finite number of moves, he can take out the tea cup of the microwave precisely from the front position?
(Birgit Vera Schmidt)