If one analyzed carefully the proof presented in Chapter 3 that a sphere may be duplicated using rotations, one would find that that proof used ten pieces. More precisely, S2 = A ∪ B where A ∩ B = ∅ and A ∼4S2 and B ∼6S2. (By 3.6, 2.6, and 1.10, S2\D splits into A′ and B′ with A′ ∼2S2\D ∼3B′; 3.9 shows that S2 ∼2S2\D, whence A′ and B′ yield A, B ⊆ S2 with the properties claimed.) It is easy to see that at least four pieces are necessary whenever a set X, acted upon by a group G, is G-paradoxical. For if X contains disjoint A, B with A ∼mX ∼nB and m + n < 4, then one of m or n equals 1. If, say, m = 1, then X = g(A) for some g ∈ G whence A = g-1(X) = X and B = ∅, a contradiction. It turns out that an interesting feature of the rotation group's action on the sphere allows the minimal number of pieces to be realized: there are disjoint sets A, B ⊆ S2 such that A ∼2S2 ∼2B. Moreover, the techniques used to cut the number of pieces to a minimum lead to significant new ideas on how to deal with the fixed points of an action of a free group, adding to our ability to recognize when a group's action is paradoxical.