there are 144 movable centers, consisting of six sets of 24 pieces each. within each set there are four centers of each color. centers from one set cannot be exchanged with those from another set. each set can be arranged in 24! different ways. assuming that the four centers of each color in each set are indistinguishable, the number of permutations of each set is reduced to 24!/(24) arrangements, all of which are possible. the reducing factor comes about because there are 24 (4!) ways to arrange the four pieces of a given color. this is raised to the sixth power because there are six colors. the total number of permutations of all movable centers is the permutations of a single set raised to the sixth power, 24!/(24).