1098 - Robots on Ice

Inspired by the ice sculptures in Harbin, the members of the programming team from Arctic University of Robotics and Automata have decided to hold their own ice festival when they return home from the contest. They plan to harvest blocks of ice from a nearby lake when it freezes during the winter. To make it easier to monitor the thickness of the ice, they will lay out a rectangular grid over the surface of the lake and have a lightweight robot travel from square to square to measure ice thickness at each square in the grid. Three locations in the grid are specified as ``check-in" points and the robot is supposed to radio a progress report from these points when it is one-fourth, one-half, and three-fourths of the way through its tour of inspection. To avoid unnecessary wear and tear on the surface of the ice, the robot must begin its tour of the grid from the lower left corner, designated in (row,column) coordinates as (0,0), visiting every other grid location exactly once and ending its tour in row 0, column 1. Moreover, if there are multiple tours that the robot can follow, then a different one is to be used each day. The robot is able to move only one square per time step in one of the four compass directions: north, south, east, or west.

You are to design a program that determines how many different tours are possible for a given grid size and a sequence of three check-in points. For example, suppose the lake surface is marked off in a 3 x 6 grid and that the check-in points, in order of visitation, are (2,1), (2,4), and (0,4). Then the robot must start at (0,0) and end at (0,1) after visitin