Suppose you have 9 evenly spaced dots in a circle on a piece of paper. You want to draw a 9-pointed star by connecting dots around the circle without lifting your pencil, skipping the same number of dots each time.

Determine the number of different stars that can be drawn, if the regular nonagon does not count as a star.

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![](https://cdn.mathpix.com/cropped/2024_04_17_09d1ad279a9140cbcb62g-1.jpg?height=238&width=736&top_left_y=908&top_left_x=730)

There are two different stars that can be drawn, by skipping 1 dot each time (as in the first star shown above) or skipping 3 dots each time (as in the third star shown above). Skipping 2 dots each time does not work because we end up drawing three separate equilateral triangles and having to lift our pencil, and skipping any other number of dots gives the regular nonagon or one of the three stars above.