0000, 0001, 0011, 0101, 0111, and 1111. Colorings of a cube The number of rotationally distinct colourings of the faces of a cube using three colours can be determined from this formula as follows. Let X be the set of 36 possible face colour combinations that can be applied to a cube in one particular orientation, and let the rotation group G of the cube act on X in the natural manner. Then two elements of X belong to the same orbit precisely when one is simply a rotation of the other. The number of rotationally distinct colourings is thus the same as the number of orbits and can be found by counting the sizes of the fixed sets for the 24 elements of G. one identity element which leaves all 36 elements of X unchanged six 90-degree face rotations, each of which leaves 33 of the elements of X unchanged three 180-degree face rotations, each of which leaves 34 of the elements of X unchanged eight 120-degree vertex rotations, each of which leaves 32 of the elements of X unchanged six 180-degree edge rotations, each of which leaves 33 of the elements of X unchanged A detailed examination of these automorphisms may be found here. The average fix size is thus Hence there are 57 rotationally distinct colourings of the faces of a cube in three colours. In general, the number of rotationally distinct colorings of the faces of a cube in n colors is given by 8 Queens Puzzle In the eight queens puzzle there are 92 solutions, of which 12 fundamental solutions are distinct up to rotation and reflection of the board. There are 8 combinations of rotations and reflections, including the null action. The null action leaves all 92 solutions unchanged. Four of the 92 solutions are symmetrical, unchanged by 180° rotation. That gives: . Notation The Lemma uses notation from group theory and set theory, and is subject to misinterpretation without that background, but is useful nonetheless. In particular, Xg does not mean exponentiation, X/G does not mean division, and |G| does not mean absolute value. Proof The first step in the proof of the lemma is to re-express the sum over the group elements g ∈ G as an equivalent sum over the set of elements x ∈ X: (Here Xg = {x ∈ X | g.x = x} is the subset of all