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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?

<hr> <h3> This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet. </h3> <hr>

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?


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

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