A unicursal curve in the plane is a curve that you get when you put down your pencil, and draw until you get back to the starting point. As you draw, your pencil mark can intersect itself, but you're not supposed to have any triple intersections. You could say that your pencil is allowed to pass over an point of the plane at most twice. This property of not having any triple intersections is generic: if you scribble the curve with your eyes closed (and somehow magically manage to make the curve finish exactly where it began), the curve won't have any triple intersections.

1. Draw a circle.
2. Mark a number of equally spaced points on the circle (2,3,4,5,6,7,8, up to 17. Maybe try them all or just try quite a few of them).
3. With a ruler, connect the dots so that one could draw out the whole pattern without lifting your pencil (i.e. unicursally).

You are studying the relationship between the size of the jump, the number of dots, and the shape that they produce.

Investigate the following questions:

1. For a given number of points, how many 'distinct' ways are there to make a unicursal dot connecting pattern?
2. What counts as a 'distinct' pattern in the above? How else could you define 'distinct'? What does this change in the definition do to your answer to 1 above?
3. For a given number of points, how many 'symmetrical' ways are there to make a unicursal dot connecting pattern?
4. Same as 2 but redefining 'symmetrical'
5. Repeat 1-4 with the 'unicursal' condition lifted.