November Problem of the Month!

Here’s the new problem of the month, which can also be found on the 2nd floor display case of PHSC, with rules for submission.

Forester’s Floating Ferrises

Forester found that the fish in his pond like to ride ferris wheels.  So he spends his Friday afternoons making floating ferris wheels.  He makes 3 sizes of ferris wheels: 6 seaters, 10 seaters and 12 seaters. Each wheel is perfectly symmetric, so that the seats are equally spaced around a circle, and equally weighted so that the center of mass is at the center of the wheel when empty.  (The weight of the base of the ferris wheel will not be important here.)

However, since the ferris wheels have a floating base, k fish can only ride an n-seat ferris wheel without it tipping over if they can sit in some arrangement of seats so that the center of mass remains at the center of the wheel (each fish weighs the exact same, and only 1 fish per seat).

Question. For each size of ferris wheel, determine the integers k such that k fish can ride the ferris wheel without tipping it over.