Here's a puzzle I thought of… a farmer digs 60 pits in the ground in a big circle.
If he chooses an empty pit, drops an apple seed in it, and drops an apple seed in a pit immediately adjacent to it, and then chooses another empty pit, drops an apple seed in it, and drops an apple seed in a pit immediately adjacent, and repeats this process, he is able easily to place apple seeds in all of the pits without placing more than one seed in any one pit.
If he chooses an empty pit, drops an apple seed in it, and drops an apple seed in a pit two spaces away, chooses another empty pit, &c., and repeats this process, he is also able to place apple seeds in all of the pits without placing more than one seed in any pit.
He finds, similarly, that he is able to do this by following the same proceedure, only spacing the pairs three spaces apart. (Boy, this farmer sure seems to have a lot of free time on his hands!)
However, by spacing the the pairs four spaces apart, he is unable to place apple seeds in all the pits without placing more than one seed in any pit, no matter which way he does it.
For how many other spacings, under 60 (a full rotation), is this feat impossible. Why?