Next, some labeling. For an n-pointed star, label the vertices of an n-gon 0, 1, 2...n-1. If we start at 0, we can then describe any star by the pair (n, s), where s is the label that corresponds to the first vertex we connect to. For example, in this system a five-pointed star is represented as (5, 2), and traverses the vertices in the order 0, 2, 4, 1, 3, (0).

From the above, our stars are really just the sequences 0, s, 2s...(n-1)s, where the values are taken modulo n. In order to be a valid star, however, none of the terms in this sequence can be equal to each other, since this would represent a repeated vertex in our drawing. This is equivalent to placing the condition that n and s be relatively prime. Lastly, since reflections are to be ignored, our final total must be divided by 2 since each star (n, s) will have an equivalent star (n, (n-s)).

Since 1000 = 2^3 * 5^3, we must exclude all multiples of 2 (500) and all multiples of 5 (200). This double counts all multiples of 10 (100), however, so the final count of pairs (n, s) that will not result in repeated vertices is (1000-500-200+100)/2 = 200. Finally, we must also exclude the pair (1000, 1), since this corresponds to drawing a 1000-gon, not a star. So the total number of possible 1000-point stars is 199.