# Problem 86: Cuboid route

![graphic](img086.gif)

A spider, S, sits in one corner of a cuboid room, measuring 6 by 5 by 3,
and a fly, F, sits in the opposite corner. By travelling on the surfaces
of the room the shortest "straight line" distance from S to F is 10 and
the path is shown on the diagram. However, there are up to three
"shortest" path candidates for any given cuboid and the shortest route
doesn't always have integer length. It can be shown that there are
exactly 2060 distinct cuboids, ignoring rotations, with integer
dimensions, up to a maximum size of M by M by M, for which the shortest
route has integer length when M = 100. This is the least value of M for
which the number of solutions first exceeds two thousand; the number of
solutions when M = 99 is 1975. Find the least value of M such that the
number of solutions first exceeds one million.