The applet displays several (N) triangles, all pointing upwards initially. On any move, you can turn over any M of them. (You do that by clicking on M triangles in turn.) The question is, Is it possible to have all N triangles to point downwards?

The original puzzle [Mathematical Circles, p. 132] stated for N = 7 and M = 4 has negative solution. It is easily seen that, in this case, the number of inverted triangles is always even and, therefore, can't be 7.

The solution begs for a generalization: what can be said about other pairs of M and N?