   Fastest Path on a Hexagon Map (with varying cost for different hexes)
  How do I quickly calculate and use the fastest route between A and B on a hexagon map where the hexes on map have different cost to cross?   
  You could use an A* algorithm.  Each tile would have six children instead of the nine children typical to the square-based-tile version.  This is part of a more general problem known as pathfinding as is more AI-oriented than math oriented.   