Three regular polygons, all with unit sides, share a common vertex and are all coplanar. Each polygon has a different number of sides, and each polygon shares a side with the other two; there are no gaps or overlaps. Find the number of sides for each polygon. There are multiple answers.

If we let i, j, and k be the number of sides of the polygons, then

180*(i-2)/i + 180*(j-2)/j + 180*(k-2)/k = 360   or

k = 2*i*j/(i*j-2*(i+j))

Using the following Perl program:

for($i=3;$i<1000;$i++) {
  for($j=$i+1;$j<1000;$j++) {
    $m = $i*$j-2*($i+$j);
    if ($m>0) {
      $k = 2*$i*$j/$m;
      $m = int($k);
      if ($k==$m) {
        if ($k>$j) {
          print $i,$j,$k;
        }
      }
    }
  }
}

I get the six answers found by Charlie.

Posted by Jerry on 2004-07-17 17:08:47