String art is an arrangement of thread strung between nails to form geometric patterns or representational designs such as a ship's sails, sometimes with other artist material comprising the remainder of the work.
Let say we have $4$ nails, and we would like to make as many different shapes as we can where nails are put in the vertices of a square, then we could have at most two different shapes as shown below:
The rules while forming the shapes are simple:
- the string cannot pass from the same nail again,
- the string needs to pass from every nail.
- and the string needs to go back where it starts at the end to attach the string again.
- while counting, do not count reflected, symmetrical or rotated shapes again.
1- If we have a regular hexagon shape where nails are located on its vertices, how many different shapes could you form?
2- Is it possible to generalize how many shapes we can have at most for any kind of regular polygon?