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There are

n

hoops on a circle. Rik numbers all hoops with a natural number so that all numbers from

1

n

occur exactly once. Then he makes one walk from hoop to hoop. He starts in hoop

1

and then follows the following rule: if he gets to hoop

k

, then he walks to the hoop that places

k

clockwise without getting into the intermediate hoops. The walk ends when Rik has to walk to a hoop he has already been to. The length of the walk is the number of hoops he passed on the way. For example, for

n = 6

Rik can take a walk of length

5

as the hoops are numbered as shown in the figure. https://cdn.artofproblemsolving.com/attachments/2/a/3d4b7edbba4d145c7e00368f9b794f39572dc5.png (a) Determine for every even

n

how Rik can number the hoops so that he has one walk of length

n

.
(b) Determine for every odd

n

how Rik can number the hoops so that he has one walk of length

n - 1

.
(c) Show that for an odd

n

there is no such numbering of the hoops that Rik can make a walk of length

n

4

Problems(1)