The 3x3 grid is given as below;
How many pentagons could be drawn by connecting the dots by lines in the grid?
Rules:
- Pentagons could be convex or concave.
- The lines cannot intersect each other.
- If reflecting or rotating a pentagon forms the same pentagon you have already counted before, it should not be added.
Example:
Note: The other question referred as the duplicate question's answers are not right. and the OP is missing. That's why I believe this question should be solved here and accepted.
Answer
I get 23 pentagons. There are 126 ways of choosing 5 out of the 9 dots. After reducing for symmetry, only 23 are left, which you can check using Burnside's lemma. I drew these in the picture below. For any of these you can put a convex hull around the points. If that is a pentagon, then that is the unique one you can make from those points. If all the points are on the hull but it is a quadrilateral or triangle, then no pentagons are possible. If it has an interior point, then you have a choice of which side of the convex hull to replace by two sides connecting to the interior point.
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