How Many Distinct Regular Polygons Can Be Formed Using 36 Equally Spaced Points?

This problem involves a detailed exploration of how to form distinct regular polygons by connecting 36 equally spaced points on a circle. Let's dive into the steps required to solve this fascinating mathematical puzzle.

Step 1: Determining the Number of Vertices

A regular polygon can be formed by choosing any number of vertices from 3 to 36. This means we can have a polygon with as few as 3 vertices (a triangle) up to 36 vertices (a 36-gon). For the number of vertices, denoted as (n), we have:

(n 3, 4, 5, ldots, 36)

Step 2: Determining the Step Size

To form a regular polygon with (n) vertices, we need to select vertices by skipping (k) points in each step. The vertices can be represented as: ( P_1, P_{1 k}, P_{1 2k}, ldots, P_{1 (n-1)k} ).

To stay within the 36 points on the circle, the indices are taken modulo 36. Thus, we require:

[ 1 (n-1)k equiv 1 pmod{36} ]

This simplifies to:

[ (n-1)k equiv 0 pmod{36} ]

Which means (k) must be a multiple of (frac{36}{gcd(n, 36)}).

Step 3: Counting Distinct Polygons

For each (n), we can choose (k) values such that (k) ranges from 1 to (frac{36}{n}). However, (k) must satisfy the condition that (1 (n-1)k) is a multiple of 36. The number of distinct values of (k) is given by the number of integers (k) such that:

[ 1 leq k leq frac{36}{n} quad text{and} quad k frac{36}{d} text{ for } d text{ dividing } n. ]

Step 4: Finding Valid Pairs (n, k)

To find how many distinct regular polygons can be formed, we count valid pairs ((n, k)):

For each (n) from 3 to 36, we find (d gcd(n, 36)). The number of valid (k) values is given by (frac{36}{d}). However, each polygon defined by ((n, k)) is equivalent to that defined by ((n, 36 - k)), so we only count half of them, plus one if (k frac{36}{2d}).

Final Calculation

Summarizing the counts for all (n) from 3 to 36, we get:

(n 3): 12 polygons, steps: 1, 2, 3, 4, 6, 12. (n 4): 9 polygons, steps: 1, 2, 3, 4, 6, 12. (n 5): 7 polygons, steps: 1, 2, 3, 4, 6, 12. (n 6): 6 polygons, steps: 1, 2, 3, 4, 6, 12. Continuing this for all (n) up to 36, the total number of distinct regular polygons is: boxed{36}

This represents the distinct regular polygons formed by selecting the vertices from the available points on the circle.