Consider the cat and mouse problem. Assume this five rooms maze is just the first level of a tower composed by n identical levels. A controllable bidirectional passageway connects room j of level 5*i+j to room j of 5*i+j+1 (for i = 0, 1, 2 ..., and j = 1, 2, 3, 4, 5). The first level is only connected with the second, the last level is only connected with the last-but-one. There are initially k cats in room 1 of the first level and k mice in room 5 of the last level.