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As we were trying to find the answer to the problems, we noticed some patterns. We realized that jumps of even numbers, with a factor of the number of dots would not work. This is because with jumps of two, half of the dots will be missed in the first round and if the number of jumps is a factor of the number of dots, then it will look like it only goes around once.
We also saw a pattern appear in the shape left between lines. We noticed that however many dots there were, the shape in the middle would be a shape with that number of sides. This is true if the lines are created with any number of jumps (except that number itself which creates no lines).