# Rainbow masterball

1.  Notation for the basic moves: Let L be one of the longitudinal lines going from the north pole to the south pole. Let f1 denote the longitudinal rotation by 180 degrees along L. (The f stands for "flip".) Going left-to-right (i.e., counterclockwise from above), let the other "flips" be denoted f2, ..., f8, resp.. Let r1 denote the rotation of the north polar cap by 45 degrees right-to-left (i.e., clockwise from above). Positive exponents will be used to apply this more than once: for example, let r13 denote the rotation of the north polar cap by 135(=3x45) degrees right-to-left. Let r2 denote the rotation of the north-of-the-equatoral belt by 45 degrees right-to-left, r3 the rotation of the south-of-the-equatoral belt by 45 degrees right-to-left, and r4 the rotation of the south polar cap by 45 degrees right-to-left. 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, r1-1 denotes the rotation of the north polar cap by 45 degrees left-to-right (i.e., counterclockwise from above). Notice that each f1,...,f8 is equal to its inverse move. If you want to make several moves in sequence we simply multiplify these symbols together left-to-right: to move f1 then r3 twice then the inverse of r4 you could simply write f1*r32*r4-1. (Note that the order is in general important but in this case r3*r4=r4*r3.)

We call the length of a move the smallest number of these generators or their inverses (r1,...,r4, f1,...,f8,r1-1,...,f8-1) required to make the move. For example, f1*r32*r4-1 is length 4 but r4*r32*r4-1 is length 2.

Question: What is the longest move of the masterball?

 Here is a small catalog of moves for the masterball: some masterball moves .

A 2-cycle will swap exactly 2 facets : 2-cycle position (~27K) The shortest move I know for an honest 2-cycle 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 2-cycles (from the beachball pattern):

f1*r3*r4*f2*f4*r1* r4-1*f4*r44*f4* r4*r1-1* f4*r44* f2*r3-1*r4-1*f1
Here are two pictures of this (from different orientations):

Let's call such a product of two 2-cycles (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:

f1*r1*f4* r1-1*r4*f4* r4-1*f1
Both of these moves were discovered by Andrew Southern.

2. ## Some pictures

Here are some pictures of some MAPLE 3d plots created by this package:

 Some pictures of these basic moves: r1 (~12K) r2 (~12K) r3 (~12K) r4 (~18K) f1 (~21K) f2 (~21K) f3 (~21K) f4 (~21K)

### Some pretty patterns

These are all based on ideas of Andrew Southern.
1. quadrantized pattern (side view) (~39K), quadrantized pattern (front and back) . In the above notation, this pattern can be reached from the beachball pattern by the moves
(r3*r4)4*f1* (r3*r4)2*f2* (r3*r4)2*f3
2. banded pattern
r3-4*r1-4*f1*r32 *r12*f2*r32*r12*f3
3. checkered pattern (top view), checkered pattern (side), checkered pattern (front and back)
move1=r1*r3*f1*r3-1* r1-1*f7*f2,
move2=r1*r3*f1*r3-1* r1-1*f1,
move3=r1*r3*f3* r1-1*r3-1*f4*f5,
move4=r1*r3*f2* r1-1*r3-1*f2,
move5=f3*f4*f5*f6* f3*f4*f5* f6*r1*r2*r3*r4 (swaps columns 1,2),
move6=f4*f5*f6*f7*f4*f5* f6*f7*r1*r2* r3*r4 (swaps columns 2,3)
checkered=move1*move2*move3*move4*move5*move6

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Created 1997. Last modified 8-26-2007.