
Solution to Saturn Puzzler 
In order to denote the equations, eight symbols are defined as N, NE, E, SE, SW, W, NW and W as shown below.
N represents flipping the upper half about the NS axis, E represents flipping the half on the right about the EW axis, and so.
An equation is expressed as the following format:
(a, n) x_{1}, x_{2}, ..., x_{n}
where a is the number of pieces relocated by the equation, n is the number of steps, and x_{i} are the n steps for i = 1, 2, ..., n.
For example, to move the piece M anticlockwise by 4 slots, the equation is
(6,2) S, E
A represents the piece(s) not to be affected by the equation.
M represents the piece to be relocated by the equation.
The complete solution is as follows:
(5, 3) E, SE, E 

(5, 2) SE, E 

(4, 1) E 

(6, 2) S, E 

(5, 4) SE, E, SE, E 

(4, 10) E, N, NE, N, W, SW, W, SE, S, SE 

(3, 9) N, NE, N, W, SW, W, SE, S, SE 

(2, 11) NW, W, NW, NE, E, S, SW, S, NW, W, NW 