Flipping Items Simultaneously

The applet displays a number (N) of small circles in a cyclic arrangement. All the circles are red, but one, which is blue. When you click on a circle then M successive circles starting with this one clockwise change their color: from red to blue and vice versa. The question is, Is it possible after a number of such moves to arrive at the situation where all the circles are red, except the one pointed to by a small arrow? This one must be blue.

The original puzzle [Mathematical Circles, pp. 127-128] stated for N = 12 and M = 3, 4, 6 has negative solution. The solution is outlined below.

But the problem begs for a generalization: what can be said about other pairs of M and N?

References

1. D. Fomin, S. Genkin, I. Itenberg, Mathematical Circles (Russian Experience), AMS, 1996

Copyright © 1996-2008 Alexander Bogomolny