2. The algorithms I was using can calculate the worse distance = 100/cos(180/n) + 200*(n-1)*tan(180/n). Coded a simple progam, it seems when n=8, distance=688.14, which is the shortest distance.
3. This algorithm has a big problem, not effective enough.
take n=4 as an example. The step 1, 41 out of 141 is really useful. In step 2, the first 100 miles were just testing what was already verified by the first step. Similarly, the first 100 miles of step 3 and first 100 miles of step 4 were testing what was already verified too.
4. A better way of thinking is needed.