Rotations and Mirroring



Clearly,  if you take a solution and spin it about the vertical axis (the bottom remains on the bottom),  then the four positions are really the same solution.  Now any of the six faces can be rotated to be on the bottom and these are also the same solution.  So there are 24 ways to rotate a solution that should always be regarded as the same solution.  But if the solution contains a letter A piece,  then only one of the 24 solutions puts the A piece in its canonical position as discussed in the Standard Rotation section.  So the counting of solutions in the Zobrist Code Book only counts positions where the first lettered piece is in its canonical position.   The count for pieces KLMNO are correspondingly divided by two,  because these pieces can rotate two ways to the canonical position.


Mirrored A and H pieces

Mirroring is handled differently than rotations.  Mirror positions are separately counted since they offer the game player multiple chances to solve the puzzle.  This would apply to mirror positions within a code from the code book.  But only a few codes allow the possibility of mirror positions.  The code must consist entirely of pieces that self mirror (DEMTUVWZ and 2)  or piece pairs that are mirrors of each other (A-H B-J C-F G-I K-L N-O and R-S).  Here is an example of mirror positions within a code.


n  n  n    n  o  o    t  t  v
w  m  n    w  w  o    t  v  v
m  m  m    w  m  o    t  v  o



o  o  o    n  n  o    v  t  t
o  m  w    n  w  w    v  v  t
m  m  m    n  m  w    n  v  t

There are also mirror codes.  For example,  “abdtuv” and “dhjtuv” exactly mirror each other.  Mirror codes always appear in pairs and the letters that are different in the two mirror codes must be  letter-mirror-pairs (excuse the double hyphen).  Letters that are common to the two mirror codes can also be instances of self-mirroring pieces.  The solution count will be exactly the same for the mirror codes and each solution for one will be mirrored in the other.

Game for the Brain!