A Characterization of Achievable Patterns of the MN-Puzzle Problem

One important MN-puzzle problem is to find a sequence of up-down and right-left movements of the empty-cell so that a goal-pattern can be achieved from a given initial-pattern.The second problem is to characterize the totality of the goal-pattern from a given initial-pattern.The third problem is finding the fewest movements to achieve it.In this paper,these problems will be completely solved.