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 is known that, from any configuration, 26 moves suffice to solve the cube. (Note that we’re counting every 90◦ rotation as a move; if you rotate the same face 180◦ by using two consecutive 90◦ moves, it counts as two moves.) How many sequences of 26 moves are possible?