What is the trick to constructing a heptagon and a nonagon which have all their sides equal? The length of the side has to be a natural number. Your answer should include a drawing of the two polygons.


The trick is

to not make the angles equal.

The shapes below are a heptagon and a nonagon with equal sides, and both are very easy to construct (since they're just made up of a square grid, plus some equilateral triangles built off of it).

enter image description here

  • $\begingroup$ I had in mind convex polygons. Can you make them convex? $\endgroup$ – Vassilis Parassidis Jan 23 at 1:27
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    $\begingroup$ This answer is a valid method of answering the problem as stated; that it fits perfectly but isn't what you were thinking of is a fault of the puzzle and not the answer. $\endgroup$ – bobble Jan 23 at 1:41
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    $\begingroup$ @VassilisParassidis Sure - at every concave angle ABC, draw circles of radius 1 around A and C. They intersect at B, as well as some other point, which we'll call D. Then you can "pop out" that angle by replacing A-B-C with A-D-C. Repeat until all convex angles are popped out. $\endgroup$ – Deusovi Jan 23 at 1:52

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