Number of regular polygons possibleThirty six equally spaced points are plotted on a circle, and some of these points are joined successively to form a polygon. How many distinct such regular polygons are possible.What I thought that the answer should be 36 C 3 + 36 C 4 + 36 C 5 + 36 C 6 + 36 C 7 + ⋯ + 36 C 36 but this is not the correct answer that has been provided. What am I doing wrong?

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Number of regular polygons possibleThirty six equally spaced points are plotted on a circle, and some of these points are joined successively to form a polygon. How many distinct such regular polygons are possible.What I thought that the answer should be                 36            C    3        +          36            C    4        +          36            C    5        +          36            C    6        +          36            C    7    +  ⋯      +          36            C          36       but this is not the correct answer that has been provided. What am I doing wrong?

penzionekta · Accepted Answer

Explanation:Since the k vertices of a regular polygon inscribed in a circle must be equally spaced, to form a regular polygon using the 36 equally spaced points on the circle, the number of sides must be a divisor of 36. Hence, the regular polygon must have either 3,4,6,9,12,18, or 36 sides. There are 36/k distinguishable regular polygons with k sides since there are 36 possible points at which you could start drawing the regular polygon, but doing so counts each such regular polygon k times, once for each of the k vertices of the polygon where you could start your drawing. Hence, the number of distinct regular polygons which could be drawn using 36 equally spaced points on the circle is       36    3    +      36    4    +      36    6    +      36    9    +      36    12    +      36    18    +      36    36    =  12  +  9  +  6  +  4  +  3  +  2  +  1  =  37

Thirty six equally spaced points are plotted on a circle, and some of these points are joined successively to form a polygon. How many distinct such regular polygons are possible.

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