Question 23
$n$ people have been invited to a lunch, who will sit around a round table, whose seats have been marked from 1 to $n$ using appropriate placeholder cards, distributed by a master of ceremonies.
The waiter has decided to serve the courses following an original procedure: he chooses a guest, serves him, then moves counterclockwise a number of places equal to the placeholder number of the guest just served, serves the guest where he is now, and so on, always moving counterclockwise based on the placeholder number of the last guest served.
Determine for which $n$ the master of ceremonies can arrange the place cards so that the waiter can, starting from an appropriate guest and following the procedure described, serve all the guests.
The waiter has decided to serve the courses following an original procedure: he chooses a guest, serves him, then moves counterclockwise a number of places equal to the placeholder number of the guest just served, serves the guest where he is now, and so on, always moving counterclockwise based on the placeholder number of the last guest served.
Determine for which $n$ the master of ceremonies can arrange the place cards so that the waiter can, starting from an appropriate guest and following the procedure described, serve all the guests.