Notation for the basic moves: Let L be one of the longitudinal lines going from the north pole to the south pole. Let f_{1} denote the longitudinal rotation by 180 degrees along L. (The f stands for "flip".) Going lefttoright (i.e., counterclockwise from above), let the other "flips" be denoted f_{2}, ..., f_{8}, resp.. Let r_{1} denote the rotation of the north polar cap by 45 degrees righttoleft (i.e., clockwise from above). Positive exponents will be used to apply this more than once: for example, let r_{1}^{3} denote the rotation of the north polar cap by 135(=3x45) degrees righttoleft. Let r_{2} denote the rotation of the northoftheequatoral belt by 45 degrees righttoleft, r_{3} the rotation of the southoftheequatoral belt by 45 degrees righttoleft, and r_{4} the rotation of the south polar cap by 45 degrees righttoleft. Each of these moves has an "inverse" move going in the reverse direction which we denote by putting a superscript of 1 on it. For example, r_{1}^{1} denotes the rotation of the north polar cap by 45 degrees lefttoright (i.e., counterclockwise from above). Notice that each f_{1},...,f_{8} is equal to its inverse move. If you want to make several moves in sequence we simply multiplify these symbols together lefttoright: to move f_{1} then r_{3} twice then the inverse of r_{4} you could simply write f_{1}*r_{3}^{2}*r_{4}^{1}. (Note that the order is in general important but in this case r_{3}*r_{4}=r_{4}*r_{3}.)


We call the length of a move the smallest number of these generators or their inverses (r_{1},...,r_{4}, f_{1},...,f8,r_{1}^{1},...,f_{8}^{1}) required to make the move. For example, f_{1}*r_{3}^{2}*r_{4}^{1} is length 4 but r_{4}*r_{3}^{2}*r_{4}^{1} is length 2.
Question: What is the longest move of the masterball?
A 2cycle will swap exactly 2 facets : 2cycle position (~27K) The shortest move I know for an honest 2cycle is very long (discovered by using GAP). If you know of a short one, please let me know.
Close to this is Andrew Southern's product of two 2cycles (from the beachball pattern):
f_{1}*r_{3}*r_{4}*f_{2}*f_{4}*r_{1}* r_{4}^{1}*f_{4}*r_{4}^{4}*f_{4}* r_{4}*r_{1}^{1}* f_{4}*r_{4}^{4}* f_{2}*r_{3}^{1}*r_{4}^{1}*f_{1}
Here are two pictures of this (from different orientations):
Let's call such a product of two 2cycles (where one of the transpositions only affects facets along the same column) a column double swap . In other words, a column double swap will swap two facets in different rows *and* swap two facets in the same column (which would not be noticed if that column had already been "solved"). Such moves are very useful to know. Along with "fishing" (see the documentation above), one only needs to know some column double swaps to solve the puzzle.
Andrew Southern's pictures of a column double swap . Here's another column double swap:
f_{1}*r_{1}*f_{4}* r_{1}^{1}*r_{4}*f_{4}* r_{4}^{1}*f_{1}
Both of these moves were discovered by Andrew Southern.