Let $$P$$ be a regular polygon with 8 vertices. By a labelling of $$P$$ we mean an assignment of integers $$1, 2, \ldots, 8$$ to the vertices in some order. A labelling is good if the path consisting of line segments from 1 to 2, 2 to 3, and so on up to 7 to 8 does not self-intersect. If $$N$$ is the number of good labellings, what is the remainder when $$N$$ is divided by 100?

The vertex receiving label 1 can be chosen in 8 ways, and at each later step the next label must sit at one of the two ends of the remaining boundary chain, since an interior choice would strand unvisited vertices and force a crossing. That leaves 2 choices for each of the labels $$2, 3, 4, 5, 6, 7$$, and the label 8 is forced, so $$N = 8 \times 2^6 = 512$$. The remainder on division by 100 is 12.