PL-OM 1987 Problem 6

A plane is tiled with regular hexagons of side $1$ . $A$ is a fixed hexagon vertex. Find the number of paths $P$ such that: (1) one endpoint of $P$ is $A$ , (2) the other endpoint of $P$ is a hexagon vertex, (3) $P$ lies along hexagon edges, (4) $P$ has length $60$ , and (5) there is no shorter path along hexagon edges from $A$ to the other endpoint of $P$ .