# Problem 411: Uphill paths
Let n be a positive integer. Suppose there are stations at the
coordinates (x, y) = (2i mod n, 3i mod n) for 0 ≤ i ≤ 2n. We will
consider stations with the same coordinates as the same station. We wish
to form a path from (0, 0) to (n, n) such that the x and y coordinates
never decrease. Let S(n) be the maximum number of stations such a path
can pass through. For example, if n = 22, there are 11 distinct
stations, and a valid path can pass through at most 5 stations.
Therefore, S(22) = 5. The case is illustrated below, with an example of
an optimal path: It can also be verified that S(123) = 14 and S(10000) =
48. Find ∑ S(k5) for 1 ≤ k ≤ 30.