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You have a plenty of segments of the same length at your disposal.

Put a segment and put another to meet at their ends. Set the counterclockwise angle between two segments at 180/5 degrees. If you repeat this process, it ends after five steps (see the picture).

pentagram

Then, you can see a regular pentagon inside your figure.

If you put the segments with an angle of 180/n degrees, which polygon will you see inside the resulting figure?

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    $\begingroup$ I'm voting to close this question as off-topic because it's a straightforward geometry problem that belongs on math.SE. $\endgroup$
    – xnor
    Feb 15, 2015 at 17:21
  • $\begingroup$ I don't think so. This is a straightforward geometry problem. Right. But, this needs no heavy calculations, no auxiliary lines, no mathematical knowledge. This problem can be solved just by drawing lines by hands, but it might give a somewhat unexpected result. So I think this problem should belong on puzzling.SE. $\endgroup$
    – P.-S. Park
    Feb 16, 2015 at 5:55

1 Answer 1

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Adding together equal length segments with a constant counterclockwise angle of $180°/n$ results in

  • for $n$ odd: a closed loop of $n$ segments, and
  • for $n$ even: a closed loop of $2n$ segments.

So for $n$ odd you get an $n$-gon, and for $n$ even a $2n$-gon.

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