A Rubik’s cube—named after the 20th-century Hungarian architect Ernő Rubik— is a 3-by-3-by-3 grid of cells, where any of the six nine-cell faces (top, bottom, left, right, front, back) can be rotated...

A Rubik’s cube—named after the 20th-century Hungarian architect Ernő Rubik— is a 3-by-3-by-3 grid of cells, where any of the six nine-cell faces (top, bottom, left, right, front, back) can be rotated 90◦ clockwise or counterclockwise in a single move. (See Figure 9.14.) Each face of each cell is colored with one of six colors (blue, red, green, yellow, white, and orange); initially, all nine cell-faces on each cube-face have the same color, but the cube can then be scrambled. The challenge is to use rotations to configure a scrambled cube such that each face of the cube contains nine cells of the same color.

It’s useless to rotate a face clockwise in one move, and rotate the same face counterclockwise in the next move. (You’ve just undone the previous move.) A counterclockwise move followed by a clockwise move is analogous. How many sequences are there of 26 moves that never undo the previous move