% \iffalse meta-comment
%
%% rubikcube.dtx
%% Copyright 2014  RWD Nickalls (dick@nickalls.org) and A Syropoulos (asyropoulos@yahoo.com)
%
% This work may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.3
% of this license or (at your option) any later version.
% The latest version of this license is in
%   http://www.latex-project.org/lppl.txt
% and version 1.3 or later is part of all distributions of LaTeX
% version 2005/12/01 or later.
%
% This work has the LPPL maintenance status `maintained'.
% 
% The Current Maintainer of this work are  RWD Nickalls and A Syropoulos
%
%
% This work consists of the files rubikcube.dtx and rubikcube.ins
% and the derived file rubikcube.sty.
%
%<*readme>
%
% The rubikcube package provides a collection of LaTeX commands and macros 
% for the typesetting of Rubik cube configurations and rotation 
% sequences using the TikZ graphic language.
%
% Please report errors or suggestions for improvement to
%
%         Dick Nickalls or Apostolos Syropoulos
%
% This package requires the basic TikZ package to be loaded already
%</readme>
%
%<*driver>
\listfiles
\documentclass{ltxdoc}
\IfFileExists{rubikcube.sty}{\usepackage{rubikcube}}{%
    \GenericWarning{rubikcube.dtx}{Package file rubikcube.sty not found.
    Documentation will be messed up!^^J
    (Generate rubikcube.sty by (La)TeXing rubikcube.ins, and then^^J
    process rubikcube.dtx again)^^J}\stop
}%
\usepackage{ifpdf}
\usepackage{url}  %% for references
\usepackage{supertabular} %% for Notation table
%%\OnlyDescription
\EnableCrossrefs
\CodelineIndex
\CodelineNumbered
\RecordChanges
\setcounter{StandardModuleDepth}{1}
\begin{document}
  \DocInput{rubikcube.dtx}
  \PrintChanges
  \PrintIndex
\end{document}
%</driver>
% \fi
%
%
% 
% \CheckSum{3912}
%
%%% \CharacterTable
%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
%%   Digits        \0\1\2\3\4\5\6\7\8\9
%%   Exclamation   \!     Double quote  \"     Hash (number) \#
%%   Dollar        \$     Percent       \%     Ampersand     \&
%%   Acute accent  \'     Left paren    \(     Right paren   \)
%%   Asterisk      \*     Plus          \+     Comma         \,
%%   Minus         \-     Point         \.     Solidus       \/
%%   Colon         \:     Semicolon     \;     Less than     \<
%%   Equals        \=     Greater than  \>     Question mark \?
%%   Commercial at \@     Left bracket  \[     Backslash     \\
%%   Right bracket \]     Circumflex    \^     Underscore    \_
%%   Grave accent  \`     Left brace    \{     Vertical bar  \|
%%   Right brace   \}     Tilde         \~}
%
%
% \title{The \textsf{rubikcube} package}
%
% \author{
%      RWD Nickalls (dick@nickalls.org) \\
%     A Syropoulos (asyropoulos@yahoo.com)
%       }
%  \date{This file describes version \RCfileversion, last revised \RCfiledate}
%  \maketitle
%
%  \begin{abstract}
%  The \rubikcube\ package provides LaTeX commands and macros 
%  for  typesetting  Rubik cube (3x3x3) notation,  configurations, and 
%  rotation sequences  using the  TikZ graphic language.
%  \end{abstract}
%
% \bigskip
% \hfil
% \begin{minipage}{2cm}
% \centering
% \begin{tikzpicture}[scale=0.4]
% \RubikFaceUpAll{W}
% \RubikFaceFrontAll{O}
% \RubikFaceRightAll{G}
% \DrawRubikCubeRU
% \end{tikzpicture}
% \end{minipage}
% \begin{minipage}{2cm}
% \centering
% \begin{tikzpicture}[scale=0.4]
% \DrawRubikLayerFaceAll{W}
% \DrawRubikLayerSideT {R}{R}{R}
% \DrawRubikLayerSideLR{B}   {G}
%                      {B}   {G}
%                      {B}   {G}
% \DrawRubikLayerSideB {O}{O}{O}
% \draw[thick,->,color=magenta] (1.5,0.5) -- (2.4, 1.4);
% \draw[thick,->] (2.5,1.5) -- (1.6, 2.4);
% \draw[thick,->,color=magenta] (1.3, 2.3) -- (1.3, 0.5);
% \draw[thick,<->,  color=blue] (0.5,2.6) -- (2.5, 2.6);
% \draw[thick,<->,  color=blue] (0.5,0.3) -- (2.5, 0.3);
% \end{tikzpicture}
% \end{minipage}
% \RubikF\RubikR\RubikU\RubikRp\RubikUp\RubikFp\
% \ \ $\longrightarrow$
% \begin{minipage}{2cm}
% \centering
% \begin{tikzpicture}[scale=0.4]
% \DrawRubikLayerFace{W}{W}{B}
%                    {W}{W}{O}
%                    {W}{R}{B}
% \DrawRubikLayerSideT {G}{G}{R}
% \DrawRubikLayerSideLR{R}   {W}
%                      {B}   {W}
%                      {O}   {W}
% \DrawRubikLayerSideB {G}{W}{O}
% \end{tikzpicture}
% \end{minipage}
% \hfil
%
% \bigskip
%
%  \tableofcontents
%
% \pagebreak
%
% \section{Introduction}
%
% The  \rubikcube\  package   provides  a collection of \LaTeX\ commands 
% and macros for  typesetting  Rubik cube  configurations  and layer-rotation 
% instructions using the  PGF/TikZ graphic languages---see examples below.
% We have extended slightly the  layer-rotation  hieroglyphic notation,  originally  
% developed  by Garfath-Cox (1981), and improved by Duvoid (2010, 2011).
%
%  The \textsf{rubikcube} package has been road-tested  on a Microsoft 
%  platform (with MikTeX and Strawberry Perl)
%  and on a Linux platform ({\TeX}Live and Mandriva).
%
% 
%  Useful web sites for solvers are the Speedsolving website, and  those 
% maintained  by Duvoid,  by Fridrich, by Jelinek,  by Reid, and by Vandenburgh (see References).
% For the mathematics and group theory associated with   Rubik's cube  see Chen (2004), 
% Joyner (2008), Hutchings (2011),  Rokicki \textit{et al.}\ (2013). 
% 
%
%  \subsection{Requirements}
%
%     The \rubikcube\ package requires the TikZ package.
%
%
%  \subsection{Supporting tool---the \textsf{rubikrotation} package}
%  \label{sec:addons}
%
%  The \textsf{rubikrotation} package, is a dynamic extension to
% the \rubikcube\ package. It consists of  the  Perl program  
% \texttt{rubikrotation.pl}  and style option \texttt{rubikrotation.sty}. 
%  The \textsf{rubikrotation} package  implements rotation sequences and 
%  random  scrambling of the 3x3x3 Rubik cube on-the-fly using a
%   \cmd{\RubikRotation}\marg{rotation-sequence} command. It returns the 
% new state in a form which is then used by the \rubikcube\ package.
%
%  Since the  \cmd{\RubikRotation} command works by \textsc{call}ing the  
%  \texttt{rubikrotation.pl} program, it follows that the \textsf{rubikrotation}
%   package requires  (a)~Perl to be installed, 
%  and (b)~\LaTeX\ needs to be run using the  \texttt{--shell-escape} commandline option.
%  See the \textsf{rubikrotation}  documentation for details.
%  The \textsf{rubikrotation} package has been road-tested  on a Microsoft 
%  platform (with MikTeX and Strawberry Perl)
%  and on a Linux platform ({\TeX}Live and Mandriva).
%
%
%  \subsection{Copyright}
%  Copyright 2014 RWD Nickalls and A Syropoulos.
%
% \medskip
% {\noindent}This work may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either
% version 1.3c of this license or any
% later version. The latest version of this licence is in 
%  http://www.latex-project.org/lppl.txt
%
%
% \section{Installation}
%
%
%  \subsection{Generating the files}
%
%  Place the file \texttt{rubikcube.zip} into a temporary directory, and unzip it. 
% This will generate the following files:
% \begin{verbatim}
% rubikcube.ins
% rubikcube.dtx
% rubikcube.sty
% rubikcube.pdf
% example-cube.tex
% example-cube.pdf
% Rubik_doc_figA.pdf
% Rubik_doc_figB.pdf
% Rubik_doc_figC.pdf
% Rubik_doc_figD.pdf
% Rubik_doc_figE.pdf
%\end{verbatim}
%  The  style option \texttt{rubikcube.sty} is generated by  running (pdf)\LaTeX\ on  
% the file \texttt{rubikcube.ins}  as follows:
%\begin{verbatim}
%   pdflatex  rubikcube.ins 
%\end{verbatim}
% The documentation file (\texttt{rubikcube.pdf}) was generated using the following
% steps:
% \begin{verbatim}
%  pdflatex   rubikcube.dtx
%  pdflatex   rubikcube.dtx
%  makeindex -s gind.ist  rubikcube
%  makeindex -s gglo.ist -o rubikcube.gls  rubikcube.glo
%  pdflatex   rubikcube.dtx
%\end{verbatim}
%
%
%  \subsection{Placing the files}
%
% Place the files either in the local working directory, or where your system 
% will find them. For a Linux system with a standard \TeX\ Directory Structure (TDS), then:
%
%\medskip
%{\noindent}*.sty  $\rightarrow$ 
%   \texttt{/usr/local/texlive/texmf-local/tex/latex/local/rubikcube/}
%{\newline}*.pdf  $\rightarrow$  \texttt{/usr/local/texlive/texmf-local/doc/rubikcube/}
%
% \medskip
% {\noindent}Finally, (depending on your system) update the  \TeX\ file database using the 
% \texttt{texhash} command.
%
%
% \subsection{Usage}
%
% Load the package by using the command \cmd{\usepackage\{rubikcube\}}.
% Note that the \rubikcube\ package requires the TikZ package, and so always load TikZ 
% before \rubikcube\  as follows:
% \begin{quote}
%\begin{verbatim}
% \usepackage{tikz}
% \usepackage{rubikcube}
%\end{verbatim}
% \end{quote}
% However, the \rubikcube\ package does check for the presence of TikZ, and will 
% load it  if TikZ is not already loaded.
%
%
%    \section{TikZ picture environment}
%
%  For a basic introduction to the use of TikZ see the following manuals
%  (from CTAN or from \texttt{http://altermundus.com/}).
%  \begin{itemize}
%  \item \texttt{pgfmanual.pdf}, version 2.10 (2012) (900 pages)
%  \item tkz-base-screen.pdf
%  \end{itemize}
% An example of the TikZ picture environment for use with \rubikcube\ is as follows 
%  If no scale is used (default scale = 1), then each of the small cubie sides 
%  will have a length of 1~cm.
% \begin{quote}
%\begin{verbatim}
% \begin{tikzpicture}[scale=0.5]
% ....
% \end{tikzpicture}
%\end{verbatim}
% \end{quote}
% A very useful feature of  TikZ  is that it automatically 
% minimises the surrounding  white-space, and consequently this is 
% mostly quite sufficient.
%
% However,  when making complicated  side-by-side figures it is 
% sometimes helpful to place each sub-figure in a surrounding minipage.
% In this case, a useful approach is to  first adjust 
% the `scale' parameter  to obtain the appropriate size, and 
% then adjust the minipage width as necessary.
% 
% Note that sometimes a temporary \cmd{\fbox\{\}} can be a useful aid  when 
% trying to  visualise the  full extent of the minipage 
% (and its whitespace). In this case remember to include a 
% trailing \% after the first brace \{ of the fbox command in order not to 
% collect any extra `space', as follows:
% \begin{quote}
%\begin{verbatim}
% \fbox{%
%    \begin{minipage}{2cm}
%    \centering
%    \begin{tikzpicture}[scale=0.5]
%    ....
%    \end{tikzpicture}
%    \end{minipage}
% } 
%\end{verbatim}
% \end{quote}
% Unexpected spacing between two adjacent images, or between an 
% image and adjacent text, is usually related to inappropriate 
% (and hidden) white-space associated with the image or its 
% associated \cmd{\minipage}.
%
% 
% \subsection{\rubikcube\ commands}
%
% Although the \rubikcube\ package has been designed with TikZ in mind, 
% it is important to appreciate that of all the  various \rubikcube\  
% commands only the `Draw...' commands actually have to be  used within
% a  TikZ picture environment. 
%
% Indeed, using \rubikcube\ 
% commands outside the TikZ picture environment can make for useful 
% flexibility when a documant is generating  more than one figure
% or image. This is because the scope of a command when used inside
% a TikZ environment is constrained to be `local' to that particular
%  environment, and hence its influence is  not accessable outside 
% the environment.  
%
% For example,  
%
% \subsection{\cmd{\draw} error message}
% \label{drawerrormessage}
%
% If a  \cmd{\Draw...} command is used \textit{outside} a TikZ picture
%  environment, then \LaTeX\  issues a  \cmd{\draw} ``Undefined control
% sequence'' error message, indicating that it is trying to draw 
% something using an undefined TikZ \cmd{\draw} command. 
%
% For example, if we  try to run the \rubikcube\ command 
%  \cmd{\DrawRubikFlat}  without a surrounding TikZ picture environment 
%  you will see something like  the following error message.
%\begin{verbatim} 
%! Undefined control sequence.
%\FlatUp ... }{#1}\pgfmathsetmacro {\uy }{#2}\draw
%                                                  [line join=round,...
%l.56 \DrawRubikFlat
%\end{verbatim}
%
%
%     \section{Command conventions}
%     \label{sec:conventions}
%
%
% \DescribeMacro{\rubikcube}
% This command generates the logo \rubikcube.
%
%
%   \subsection{Capital letters}
%
%  Each `word' in a command (except the word `text')  starts with a capital letter. 
%  For example,  \cmd{\DrawRubikCubeRU}, \cmd{\DrawCubieRU}. 
%  However, in line with \LaTeX,  `text' commands start with a lowercase `t'; 
%  for example \cmd{\textCubieRU}.  Letters  for  colours (R, O, Y, G, B, W, X)  are 
%  always written in uppercase. How the six  face letters  are  written depends 
% on  the command; sometimes in uppercase (eg \cmd{\rrD}, \cmd{\rrDs}, \cmd{\rrDa} ),
%  and  sometimes in lowercase  (eg~\cmd{\rrSu}, \cmd{\rrd}, \cmd{\rrhf}).
%
%     \subsection{XYZ argument ordering}
%
% Many commands have an appended two (XY) or three (XYZ) ordered letter code 
% which is used to for  specifying some feature of the command; perhaps 
%  either  face or cubie colours or a viewpoint direction. 
%
% The convention is that the letter codes are ordered in the XYZ order; 
% ie the first code relates to  an X-related parameter; 
% for example  L (Left) or R (Right); the second relates to  a Y-related 
% parameter;  for example  U (Up) or D (Down); the third (if required) 
% relates to  a Z-related  parameter;  for example  F (Front) or B (Back).
%
% \textsc{example}: |\DrawCubieRU{G}{Y}{O}|  draws a cubie from 
% the RightUp viewpoint. The sequence of  colour codes for the three visible 
% faces are XYZ ordered, and hence result in the cube having a Green Right face, 
% Yellow Up face and Orange Front face.
%
%
%  \subsection{Trailing \% on end of \rubikcube\ commands}
%
% It is important to include a trailing \% on the end of \rubikcube\ 
% commands when used \textit{outside}  a TikZ picture environment.
% This is to prevent  unwanted `space' characters appearing in 
% the graphics. Note that a useful feature  of the TikZ environment
% is that it  automatically minimises any such space.
%
%
%  \section{Rubik rotation commands and notation}
%  \label{sec:RubikCommands} 
%
% We  use the standard Rubik cube notation  of WCA (2012)---see article~12---and
% also the `s' (Slice) and `a' (anti-Slice) notation described in   the 
% `Notation and terminology' section
% in the `Pretty patterns' page on the website of  
% Fridrich (\url{http://www.ws.binghamton.edu/fridrich/ptrns.html}).
% By the term \rubikcube\ we  denote only  the familiar
% 3x3x3 puzzle cube. We  denote similar puzzle cubes  of a different size by 
% the term  NCubes (NxNxN). 
% 
% It is recommended  that  commas  are used to separate sequestial 
% Rubik moves or commands to avoid  ambiguity, especially when using just
%  lettercodes on their own. 
% For example, in the following sequence the commas usefully remove any ambiguity:
%  \rrU,\rrLw2,\rrUsp,\rrDa \ \ (|\rrU,\rrLw2,\rrUsp,\rrDa|).
%
% 
%
% \subsection{Overview}
%
% The \rubikcube\  notation  comprises  a range of 
% commands for moves or rotations (e.g.,~\rrR, \rry, \rrBw) and their equivalent  hieroglyphs 
% (e.g.,~\rrhR, \rrhy, \rrhBw), as well as commands for  drawing 3x3x3 cubes and single cubies.
%
% Note that there are a few rotation commands which  do not have arrow 
% hieroglyphs---their their rotation  is not visible from the \textsc{front} face 
% and hence cannot easily be rendered as  an arrow hieroglyph. Consequently these 
% rotations  have a simple `letter' hieroglyph 
% in the form of the letter-code in a square; for example \rrhBw, \rrhSb. 
%
% \DescribeMacro{\rr..}
%  The letter-code of a rotation is typeset  (as in text) using the rubik-rotation 
%  \cmd{\rr..} stem: ie \rrR\ is typeset using the command \cmd{\rrR}. 
% The hieroglyph of a rotation command is generated  (in text) by
% \DescribeMacro{\rrh..}
% using instead the stem \cmd{\rrh..}. \ Thus the command \cmd{\rrhR} generates 
% \rrhR\  which is the hieroglyph associated with \rrR.  
%
% \DescribeMacro{\Rubik..}
% A vertically combined lettercode  and its hieroglyph  is generated using the command 
% stem \cmd{\Rubik..}. For example, \RubikR\ is generated by the command \cmd{\RubikR}, 
% with  the square hieroglyph sitting on the baseline.
% For some hieroglyphs,  eg~the \rrhx, \rrhy, \rrhz\ denoting 90~degree axis rotations, 
% the only  difference between the \cmd{\rrh..} and \cmd{\Rubik..} forms is that 
% the \cmd{\Rubik..}  form is eleveted  to sit on the baseline just like the other
%  \cmd{\Rubik..} hieroglyphs. 
% For example \cmd{\rrhyp} generates \rrhyp, while
% \cmd{\Rubikyp} generates \Rubikyp.
% 
% \DescribeMacro{\textRubik..}
% A horizontally combined  lettercode and its hieroglyph (in sequence as in text) 
% is generated using  the command stem  \cmd{\textRubik..}. 
% For example,  \textRubikR\ is typeset using the command \cmd{\textRubikR}.
% A list of all commands and their associated  hieroglyphs is given in
%  Section~\ref{sec:listofcommands}. 
%
% 
%
% \subsection{Face rotations}
%
% \DescribeMacro{\rrU}
% \DescribeMacro{\rrD}
% \DescribeMacro{\rrL}
% \DescribeMacro{\rrR}
% \DescribeMacro{\rrF}
% \DescribeMacro{\rrB}
% The six main faces of the cube are denoted as \textsc{front} (towards the observer), 
% \textsc{back},  \textsc{left}, \textsc{right},  \textsc{up}, \textsc{down}.
% The uppercase initial letter  of each face-name (\rrF, \rrB, \rrL, \rrR, \rrU, \rrD) 
% denotes a clockwise 90-degree rotation of the  face as shown in  
% Figure~\ref{fig:notation}. 
% For example, \rrD\ is generated by the `rubik rotation' command \cmd{\rrD}.
%
% \DescribeMacro{\rrUp}
% \DescribeMacro{\rrDp}
% \DescribeMacro{\rrLp}
% \DescribeMacro{\rrRp}
% \DescribeMacro{\rrFp}
% \DescribeMacro{\rrBp}
%  An appended  prime~$^\prime$ indicates an anticlockwise rotation; eg \rrFp. 
%  This is sometimes written as  \rrF$\boldmath{^{-1}}$. The `prime' notation is 
%  achieved by appending a lowercase `p' to the face rotation command. For example, \rrRp\ 
%  is generated by \cmd{\rrRp}. 
%
% \begin{figure}[hbt]
%  \centering
%  \ifpdf
%    \includegraphics[height=6cm]{Rubik_doc_figA.pdf}
%  \fi
% \caption{\label{fig:notation}} 
% \end{figure}
%
%
%  \subsection{Slice rotations}
%
% \DescribeMacro{\rrSu}
% \DescribeMacro{\rrSd}
% \DescribeMacro{\rrSl}
% \DescribeMacro{\rrSr}
% \DescribeMacro{\rrSf}
% \DescribeMacro{\rrSb}
% The Rubik cube (3x3x3) has three orthogonal so-called `inner' slices, whose +ve 
% rotation  direction follows that of a named face. For example the inner slice 
% rotation  between the \textsc{right} and \textsc{left} faces  whose rotation 
% direction follows the rotation \rrR\ (rotation is isomorphic to \rrR) is denoted 
% as \rrSr, which is typeset using the command \cmd{\rrSr}. Note that in these cases
%  the trailing  \texttt{r}  in the command  is lowercase.
%
% \DescribeMacro{\rrSup}
% \DescribeMacro{\rrSdp}
% \DescribeMacro{\rrSlp}
% \DescribeMacro{\rrSrp}
% \DescribeMacro{\rrSfp}
% \DescribeMacro{\rrSbp}
% Each of these slice rotations (S rotations) has a reversed (primed)  p-form,
% the command for which is generated by appending the suffix `p'.
% For example  the inner slice rotations  \rrSlp\ (\cmd{\rrSlp}) and 
% \rrSr\ (\cmd{\rrSr}) are identical.
% The equivalence is more obvious when we see their respective hieroglyphs.
% For example, in this case \linebreak \rrSlp\ (\cmd{\rrSlp}) 
%  $\equiv$ \rrhSlp\ (\cmd{\rrhSlP}), 
%  and  \rrSr\ (\cmd{\rrSr}) $\equiv$ \rrhSr\ (\cmd{\rrhSr}). 
%
% \pagebreak
%
% \subsubsection*{MES slice notation}
%
% \DescribeMacro{\rrM}
% \DescribeMacro{\rrE}
% \DescribeMacro{\rrS}
% \DescribeMacro{\rrMp}
% \DescribeMacro{\rrEp}
% \DescribeMacro{\rrSp}
% An alternative and somewhat confusing (and hence is non-standard) slice notation
%  which is sometimes used  is the following 
% so-called MES notation,  as used in the Waterman algorithm
%  (Treep and Waterman 1987).
% \begin{itemize}
% \item[\rrM] \ (\textsc{middle} slice, between the \textsc{left} 
%      and \textsc{right} faces;  direction  follows  \rrL),
% \item[\rrE] \  (\textsc{equator} slice, between the \textsc{up} and \textsc{down} 
%      faces;  direction follows \rrD),
%  \item[\rrS] \  (\textsc{standing} slice, between the \textsc{front} and 
%      \textsc{back} faces; 
%  direction follows \rrF).
% \end{itemize}
% Each of these also has a reversed (prime) version.
% The equivalent S notations are therefore as follows:  \rrE\ $\equiv$ \rrSd, 
% \rrEp\ $\equiv$ \rrSu, 
% \rrM\ $\equiv$ \rrSl, \rrMp\ $\equiv$ \rrSr, \rrS\ $\equiv$ \rrSf, \rrSp\ $\equiv$ \rrSb. 
%
% 
%
% \subsubsection*{Singmaster slice notation}
%
% \DescribeMacro{\rrUs}
% \DescribeMacro{\rrDs}
% \DescribeMacro{\rrLs}
% \DescribeMacro{\rrRs}
% \DescribeMacro{\rrFs}
% \DescribeMacro{\rrBs}
% These are an alternative (but somewhat less intuitive) form of  
% slice notation  which can be thought of as complementing the 
% inner slice  rotations. These were originally described by 
% Singmaster (Frey and Singmaster, 1982).
% See the link to `notation' on the `Pretty patterns' page of the 
% Fridrich website.
%
% Each of these commands denotes a  rotation of  two opposite faces 
% in the same  direction. For example, \textRubikUs\ $\equiv$ \textRubikU\ + \textRubikDp, 
% which is typeset as: 
% \newline |\textRubikUs\ $\equiv$ \textRubikU\ + \textRubikDp|, 
% i.e., for  \rrUs\ both face-rotations are in the \textit{same} direction 
% as \rrU. 
%
%
% \subsubsection*{Anti-slice notation}
%
% \DescribeMacro{\rrUa}
% \DescribeMacro{\rrDa}
% \DescribeMacro{\rrLa}
% \DescribeMacro{\rrRa}
% \DescribeMacro{\rrFa}
% \DescribeMacro{\rrBa}
% Each of these commands denotes a  rotation of  two opposite faces 
% in \textit{opposite}  directions. 
% For example, \textRubikUa\ $\equiv$ \textRubikU\ + \textRubikD, 
% which is typeset as: 
% \newline |\textRubikUa\ $\equiv$ \textRubikU\ + \textRubikD|.
% See the link to `notation' on the `Pretty patterns' page of the 
% Fridrich website.
%
%
%      \subsection{Wide rotations}
%
% \DescribeMacro{\rrUw}
% \DescribeMacro{\rrDw}
% \DescribeMacro{\rrLw}
% \DescribeMacro{\rrRw}
% \DescribeMacro{\rrFw}
% \DescribeMacro{\rrBw}
% The  clockwise \textit{combined} rotation of  an outer face AND its inner slice 
% (officially known as a `double outer slice' rotation) is denoted by appending a 
% lowercase  \textbf{\textsf{\footnotesize{w}}} (denoting `wide') to a face rotation command.
% For example,  a \textsc{right} double outer slice rotation is  denoted as \rrRw. 
% Similarly, the prime $^\prime$ version \rrLwp\  is generated by \cmd{\rrLwp}.
%
%  The superscript~$^2$, or sometimes just an ordinary 2, indicates that the rotation
%  is applied twice. For example \rrR\textbf{$^2$} or \rrR\textbf{2} 
%  denote  \textit{two} successive 90~degree clockwise rotations of the \textsc{right} face.
%
%
%
%     \subsection{Axis rotations}
%
% \DescribeMacro{\rrx}
% \DescribeMacro{\rry}
% \DescribeMacro{\rrz}
% \DescribeMacro{\rrhx}
% \DescribeMacro{\rrhy}
% \DescribeMacro{\rrhz}
% Whole-cube clockwise rotations  of 90-degrees about about the orthogonal  axes centred 
% on the  \textsc{right}, \textsc{up}, \textsc{front}  faces are denoted 
% as \rrx, \rry, \rrz\ (the \cmd{\rr..} forms) respectively, 
% with their hieroglyphs (the \cmd{\rrh..} forms) being denoted 
% as \rrhx, \rrhy, \rrhz\ in order to distinguish them from  square layer-rotation 
% hieroglyphs.
% Note that \rrx, \rry, \rrz\ rotations are always expressed in lowercase, 
% and hence this practice is extended also to the commands. 
%
% An \rrx\textbf{2} rotation (two \rrx\ rotations one after 
% the other, ie~\rrhx\ \rrhx) denotes rotating 
% the cube 180~degrees about its x axis so as to bring the \textsc{down} face  
% into the \textsc{up} position.
%
%  An appended  prime~$^\prime$ indicates an anticlockwise rotation; 
% for example, \rrxp\ (which is generated by appending a `p' to the end of 
% the command, ie~\cmd{\rrxp}). 
%
% \DescribeMacro{\Rubikx}
% \DescribeMacro{\Rubiky}
% \DescribeMacro{\Rubikz}
% These  \cmd{\Rubik..}  forms (and their prime `p' versions) generate the same
%  hieroglyphs  as their \cmd{\rrh..} versions, except that their spacing is 
%  similar to that associated with the `square box'  \cmd{\Rubik..}  hieroglyphs.  
% Consequently  when typesetting  an axis command in a sequence of  `square-box' 
% \cmd{\Rubik..} commands, it is better to use the \cmd{\Rubik..} form rather than
% the equivalent \cmd{\rrh..} form (see the examples in Section~\ref{sec:examples}).
% There are no \cmd{\textRubik..} forms for the axis commands (since they are 
% not necessary).
%
%
% \subsubsection*{The u, d, l, r, f, b notation} 
%
% \DescribeMacro{\rru}
% \DescribeMacro{\rrd}
% \DescribeMacro{\rrl}
% \DescribeMacro{\rrr}
% \DescribeMacro{\rrf}
% \DescribeMacro{\rrb}
% A commonly used  alternative for the \rrx, \rry, \rrz\ notation 
% (and endorsed  by the WCA)  uses the 
% lowercase face letter to denote a 90~degree whole-cube rotation  in the same 
% sense as that of the   standard face rotations. 
%
% {\noindent}Thus  
% \rru\ $\equiv$ \rry, \    \rrd\ $\equiv$ \rryp, \ 
% \rrl\ $\equiv$ \rrxp, \   \rrr\ $\equiv$ \rrx, \ 
% \rrf\ $\equiv$ \rrz, \    \rrb\ $\equiv$ \rrzp,
%
% {\noindent}For example, \rrd\ is generated by  the command \cmd{\rrd}.
%
% {\noindent}Note that these moves do not have prime $^\prime$ versions  
% since \rru\ is the opposite of \rrd, \rrl\ is the opposite of \rrr, and
% \rrf\ is the opposite of \rrb.
%
% \DescribeMacro{\rrh..}
% \DescribeMacro{\Rubik..}
% As with the \rrhx, \rrhy, \rrhz\ forms (described above) there also 
% equivalent \cmd{\rrh..} and \cmd{\Rubik..} forms. For example, 
% \rrhd\ is generated by the command \cmd{\rrhd}.
%
%
%  \subsection{Examples}
%  \label{sec:examples}
%  \begin{quote}
%    \rrR\  is generated by  the `rubik rotation' command \cmd{\rrR}
%
%    \rrFw\  is generated by  the `rubik rotation' command \cmd{\rrFw}
%
%    \rrL$^2$ is generated by \cmd{\rrL}\verb!$^2$!
%
%    \rrRp\ is generated by \cmd{\rrRp}
%
%    \rrFwp\  is generated by \cmd{\rrFwp}
%
%        \rrx\ and \rrhy\ and \Rubikzp\ are  generated by 
%   \cmd{\rrx}  and \cmd{\rrhy} and \cmd{\Rubikzp}
%
%    \rrf\ and \rrhb\ are  generated by \cmd{\rrf} and \cmd{\rrhb}
%
%  The sequence  \rrU\rrU\rrR\rrR\ is generated by  \cmd{\rrU}\cmd{\rrU}\cmd{\rrR}\cmd{\rrR}
%  \end{quote}
%  
%  \bigskip
%         
%  {\noindent}\RubikF\RubikR\RubikU\Rubiky\RubikRp\RubikLwp\
%  \ \  \verb!\RubikF\RubikR\RubikU\Rubiky\RubikRp\RubikLwp!
%  
%  \bigskip
%  
%  {\noindent}The  equivalent `text' version  of the first three 
%  rotations above (and their commands)  are as follows:
%  
%  \bigskip
%  
%  {\noindent}\textRubikF\ \textRubikR\ \textRubikU\ \ \ \  
%       \verb!\textRubikF\ \textRubikR\ \textRubikU!
%  
%  \bigskip
%  {\noindent}Writing the first sequence using  only  letter-codes or only hieroglyphs, we have:
%  
%  \bigskip
%  
%  {\noindent}\rrF\rrR\rrU\rry\rrRp \ \ \ \
%  \verb!\rrF\rrR\rrU\rry\rrRp!
%  
%  \bigskip
%  
%  {\noindent}\rrhF\rrhR\rrhU\rrhy\rrhRp \ \ \ \
%  \verb!\rrhF\rrhR\rrhU\rrhy\rrhRp\! 
%
%  \bigskip
% 
% The following shows the importance of using commas to avoid ambiguity:
%
% \bigskip
%
% {\noindent}\rrD,\rrU2,\rrF2,\rrDs2,\rrB,  \ \ \ \  \verb!\rrD,\rrU2,\rrF2,\rrDs2,\rrB,!
%
%  \bigskip
%
% {\noindent}\rrhD,\rrhU2,\rrhF2,\rrhDs2,\rrhB, \ \ \ \  \verb!\rrhD,\rrhU2,\rrhF2,\rrhDs2,\rrhB,!
% 
%
%  \subsection{Listing of all rotation commands}
%   \label{sec:listofcommands}
%
% Note that all these commands also have a \cmd{\Rubik..} equivalent form which 
% typesets both the hieroglyph and its lettercode in a vertical format, 
% as shown in the above `Examples' section. These have been ommitted  here
% owing to the difficulty of including this form easily in the following table.
%
% Note also that some \cmd{\rrh..} commands (eg~the \cmd{\rrhB..} commands)  
% show only the lettercode in a square box. This is because these  rotations, which 
%  do not have a `true' visual representation as seen from the \textsc{front} face,
% are somewhat ambiguous unless typeset with the associated lettercode (ie using the 
% \cmd{\Rtbik..}  or \cmd{\textRubik..} commands). 
%
%  \newcommand{\dnstrut}{\rule{0pt}{17pt}}
%  \newcommand{\dns}{\hspace{2mm}}
%  \newcommand{\dnsp}{\hspace{2mm}} 
%
%
%
%  \begin{supertabular}[lll]{p{3cm} p{3cm} p{4cm}}
%  \dnstrut\rrU\dns\cmd{\rrU}
%  & \rrhU\dns\cmd{\rrhU}
%  & \textRubikU\dns\cmd{\textRubikU} \nonumber\\ 
%  \dnstrut\rrUp\dns\cmd{\rrUp}
%  & \rrhUp\dns\cmd{\rrhUp}
%  & \textRubikUp\dns\cmd{\textRubikUp} \nonumber\\ 
%  \dnstrut\rrUw\dns\cmd{\rrUw}
%  & \rrhUw\dns\cmd{\rrhUw}
%  & \textRubikUw\dns\cmd{\textRubikUw} \nonumber\\ 
%  \dnstrut\rrUwp\dns\cmd{\rrUwp}
%  & \rrhUwp\dns\cmd{\rrhUwp}
%  & \textRubikUwp\dns\cmd{\textRubikUwp} \nonumber\\ 
%  \dnstrut\rrUs\dns\cmd{\rrUs}
%  & \rrhUs\dns\cmd{\rrhUs}
%  & \textRubikUs\dns\cmd{\textRubikUs} \nonumber\\ 
%  \dnstrut\rrUsp\dns\cmd{\rrUsp}
%  & \rrhUsp\dns\cmd{\rrhUsp}
%  & \textRubikUsp\dns\cmd{\textRubikUsp} \nonumber\\ 
%  \dnstrut\rrUa\dns\cmd{\rrUa}
%  & \rrhUa\dns\cmd{\rrhUa}
%  & \textRubikUa\dns\cmd{\textRubikUa} \nonumber\\ 
%  \dnstrut\rrUap\dns\cmd{\rrUap}
%  & \rrhUap\dns\cmd{\rrhUap}
%  & \textRubikUap\dns\cmd{\textRubikUap} \nonumber\\
%  \end{supertabular}
%
%
%  \begin{supertabular}[lll]{p{3cm} p{3cm} p{4cm}}
%  \dnstrut\rrD\dns\cmd{\rrD}
%  & \rrhD\dns\cmd{\rrhD}
%  & \textRubikD\dns\cmd{\textRubikD} \nonumber\\ 
%  \dnstrut\rrDp\dns\cmd{\rrDp}
%  & \rrhDp\dns\cmd{\rrhDp}
%  & \textRubikDp\dns\cmd{\textRubikDp} \nonumber\\ 
%  \dnstrut\rrDw\dns\cmd{\rrDw}
%  & \rrhDw\dns\cmd{\rrhDw}
%  & \textRubikDw\dns\cmd{\textRubikDw} \nonumber\\ 
%  \dnstrut\rrDwp\dns\cmd{\rrDwp}
%  & \rrhDwp\dns\cmd{\rrhDwp}
%  & \textRubikDwp\dns\cmd{\textRubikDwp} \nonumber\\ 
%  \dnstrut\rrDs\dns\cmd{\rrDs}
%  & \rrhDs\dns\cmd{\rrhDs}
%  & \textRubikDs\dns\cmd{\textRubikDs} \nonumber\\ 
%  \dnstrut\rrDsp\dns\cmd{\rrDsp}
%  & \rrhDsp\dns\cmd{\rrhDsp}
%  & \textRubikDsp\dns\cmd{\textRubikDsp} \nonumber\\ 
%  \dnstrut\rrDa\dns\cmd{\rrDa}
%  & \rrhDa\dns\cmd{\rrhDa}
%  & \textRubikDa\dns\cmd{\textRubikDa} \nonumber\\ 
%  \dnstrut\rrDap\dns\cmd{\rrDap}
%  & \rrhDap\dns\cmd{\rrhDap}
%  & \textRubikDap\dns\cmd{\textRubikDap} \nonumber\\
%  \end{supertabular}
%
%
%  \begin{supertabular}[lll]{p{3cm} p{3cm} p{4cm}}
%  \dnstrut\rrL\dns\cmd{\rrL}
%  & \rrhL\dns\cmd{\rrhL}
%  & \textRubikL\dns\cmd{\textRubikL} \nonumber\\ 
%  \dnstrut\rrLp\dns\cmd{\rrLp}
%  & \rrhLp\dns\cmd{\rrhLp}
%  & \textRubikLp\dns\cmd{\textRubikLp} \nonumber\\ 
%  \dnstrut\rrLw\dns\cmd{\rrLw}
%  & \rrhLw\dns\cmd{\rrhLw}
%  & \textRubikLw\dns\cmd{\textRubikLw} \nonumber\\ 
%  \dnstrut\rrLwp\dns\cmd{\rrLwp}
%  & \rrhLwp\dns\cmd{\rrhLwp}
%  & \textRubikLwp\dns\cmd{\textRubikLwp} \nonumber\\ 
%  \dnstrut\rrLs\dns\cmd{\rrLs}
%  & \rrhLs\dns\cmd{\rrhLs}
%  & \textRubikLs\dns\cmd{\textRubikLs} \nonumber\\ 
%  \dnstrut\rrLsp\dns\cmd{\rrLsp}
%  & \rrhLsp\dns\cmd{\rrhLsp}
%  & \textRubikLsp\dns\cmd{\textRubikLsp} \nonumber\\ 
%  \dnstrut\rrLa\dns\cmd{\rrLa}
%  & \rrhLa\dns\cmd{\rrhLa}
%  & \textRubikLa\dns\cmd{\textRubikLa} \nonumber\\ 
%  \dnstrut\rrLap\dns\cmd{\rrLap}
%  & \rrhLap\dns\cmd{\rrhLap}
%  & \textRubikLap\dns\cmd{\textRubikLap} \nonumber\\
%  \end{supertabular}
%
%
%  \begin{supertabular}[lll]{p{3cm} p{3cm} p{4cm}}
%  \dnstrut\rrR\dns\cmd{\rrR}
%  & \rrhR\dns\cmd{\rrhR}
%  & \textRubikR\dns\cmd{\textRubikR} \nonumber\\ 
%  \dnstrut\rrRp\dns\cmd{\rrRp}
%  & \rrhRp\dns\cmd{\rrhRp}
%  & \textRubikRp\dns\cmd{\textRubikRp} \nonumber\\ 
%  \dnstrut\rrRw\dns\cmd{\rrRw}
%  & \rrhRw\dns\cmd{\rrhRw}
%  & \textRubikRw\dns\cmd{\textRubikRw} \nonumber\\ 
%  \dnstrut\rrRwp\dns\cmd{\rrRwp}
%  & \rrhRwp\dns\cmd{\rrhRwp}
%  & \textRubikRwp\dns\cmd{\textRubikRwp} \nonumber\\ 
%  \dnstrut\rrRs\dns\cmd{\rrRs}
%  & \rrhRs\dns\cmd{\rrhRs}
%  & \textRubikRs\dns\cmd{\textRubikRs} \nonumber\\ 
%  \dnstrut\rrRsp\dns\cmd{\rrRsp}
%  & \rrhRsp\dns\cmd{\rrhRsp}
%  & \textRubikRsp\dns\cmd{\textRubikRsp} \nonumber\\ 
%  \dnstrut\rrRa\dns\cmd{\rrRa}
%  & \rrhRa\dns\cmd{\rrhRa}
%  & \textRubikRa\dns\cmd{\textRubikRa} \nonumber\\ 
%  \dnstrut\rrRap\dns\cmd{\rrRap}
%  & \rrhRap\dns\cmd{\rrhRap}
%  & \textRubikRap\dns\cmd{\textRubikRap} \nonumber\\
%  \end{supertabular}
%
%
%  \begin{supertabular}[lll]{p{3cm} p{3cm} p{4cm}}
%  \dnstrut\rrF\dns\cmd{\rrF}
%  & \rrhF\dns\cmd{\rrhF}
%  & \textRubikF\dns\cmd{\textRubikF} \nonumber\\ 
%  \dnstrut\rrFp\dns\cmd{\rrFp}
%  & \rrhFp\dns\cmd{\rrhFp}
%  & \textRubikFp\dns\cmd{\textRubikFp} \nonumber\\ 
%  \dnstrut\rrFw\dns\cmd{\rrFw}
%  & \rrhFw\dns\cmd{\rrhFw}
%  & \textRubikFw\dns\cmd{\textRubikFw} \nonumber\\ 
%  \dnstrut\rrFwp\dns\cmd{\rrFwp}
%  & \rrhFwp\dns\cmd{\rrhFwp}
%  & \textRubikFwp\dns\cmd{\textRubikFwp} \nonumber\\ 
%  \dnstrut\rrFs\dns\cmd{\rrFs}
%  & \rrhFs\dns\cmd{\rrhFs}
%  & \textRubikFs\dns\cmd{\textRubikFs} \nonumber\\ 
%  \dnstrut\rrFsp\dns\cmd{\rrFsp}
%  & \rrhFsp\dns\cmd{\rrhFsp}
%  & \textRubikFsp\dns\cmd{\textRubikFsp} \nonumber\\ 
%  \dnstrut\rrFa\dns\cmd{\rrFa}
%  & \rrhFa\dns\cmd{\rrhFa}
%  & \textRubikFa\dns\cmd{\textRubikFa} \nonumber\\ 
%  \dnstrut\rrFap\dns\cmd{\rrFap}
%  & \rrhFap\dns\cmd{\rrhFap}
%  & \textRubikFap\dns\cmd{\textRubikFap} \nonumber\\
%  \end{supertabular}
%
%
%  \begin{supertabular}[lll]{p{3cm} p{3cm} p{4cm}}
%  \dnstrut\rrB\dns\cmd{\rrB}
%  & \rrhB\dns\cmd{\rrhB}
%  & \textRubikB\dns\cmd{\textRubikB} \nonumber\\ 
%  \dnstrut\rrBp\dns\cmd{\rrBp}
%  & \rrhBp\dns\cmd{\rrhBp}
%  & \textRubikBp\dns\cmd{\textRubikBp} \nonumber\\ 
%  \dnstrut\rrBw\dns\cmd{\rrBw}
%  & \rrhBw\dns\cmd{\rrhBw}
%  & \textRubikBw\dns\cmd{\textRubikBw} \nonumber\\ 
%  \dnstrut\rrBwp\dns\cmd{\rrBwp}
%  & \rrhBwp\dns\cmd{\rrhBwp}
%  & \textRubikBwp\dns\cmd{\textRubikBwp} \nonumber\\ 
%  \dnstrut\rrBs\dns\cmd{\rrBs}
%  & \rrhBs\dns\cmd{\rrhBs}
%  & \textRubikBs\dns\cmd{\textRubikBs} \nonumber\\ 
%  \dnstrut\rrBsp\dns\cmd{\rrBsp}
%  & \rrhBsp\dns\cmd{\rrhBsp}
%  & \textRubikBsp\dns\cmd{\textRubikBsp} \nonumber\\ 
%  \dnstrut\rrBa\dns\cmd{\rrBa}
%  & \rrhBa\dns\cmd{\rrhBa}
%  & \textRubikBa\dns\cmd{\textRubikBa} \nonumber\\ 
%  \dnstrut\rrBap\dns\cmd{\rrBap}
%  & \rrhBap\dns\cmd{\rrhBap}
%  & \textRubikBap\dns\cmd{\textRubikBap} \nonumber\\
%  \end{supertabular}
%
%
%  \begin{supertabular}[lll]{p{3cm} p{3cm} p{4cm}}
%  \dnstrut\rrSu\dns\cmd{\rrSu}
%  & \rrhSu\dns\cmd{\rrhSu}
%  & \textRubikSu\dns\cmd{\textRubikSu} \nonumber\\ 
%  \dnstrut\rrSup\dns\cmd{\rrSup}
%  & \rrhSup\dns\cmd{\rrhSup}
%  & \textRubikSup\dns\cmd{\textRubikSup} \nonumber\\ 
%  \dnstrut\rrSd\dns\cmd{\rrSd}
%  & \rrhSd\dns\cmd{\rrhSd}
%  & \textRubikSd\dns\cmd{\textRubikSd} \nonumber\\ 
%  \dnstrut\rrSdp\dns\cmd{\rrSdp}
%  & \rrhSdp\dns\cmd{\rrhSdp}
%  & \textRubikSdp\dns\cmd{\textRubikSdp} \nonumber\\ 
%  \dnstrut\rrSl\dns\cmd{\rrSl}
%  & \rrhSl\dns\cmd{\rrhSl}
%  & \textRubikSl\dns\cmd{\textRubikSl} \nonumber\\ 
%  \dnstrut\rrSlp\dns\cmd{\rrSlp}
%  & \rrhSlp\dns\cmd{\rrhSlp}
%  & \textRubikSlp\dns\cmd{\textRubikSlp} \nonumber\\ 
%  \dnstrut\rrSr\dns\cmd{\rrSr}
%  & \rrhSr\dns\cmd{\rrhSr}
%  & \textRubikSr\dns\cmd{\textRubikSr} \nonumber\\ 
%  \dnstrut\rrSrp\dns\cmd{\rrSrp}
%  & \rrhSrp\dns\cmd{\rrhSrp}
%  & \textRubikSrp\dns\cmd{\textRubikSrp} \nonumber\\ 
%  \dnstrut\rrSf\dns\cmd{\rrSf}
%  & \rrhSf\dns\cmd{\rrhSf}
%  & \textRubikSf\dns\cmd{\textRubikSf} \nonumber\\ 
%  \dnstrut\rrSfp\dns\cmd{\rrSfp}
%  & \rrhSfp\dns\cmd{\rrhSfp}
%  & \textRubikSfp\dns\cmd{\textRubikSfp} \nonumber\\ 
%  \dnstrut\rrSb\dns\cmd{\rrSb}
%  & \rrhSb\dns\cmd{\rrhSb}
%  & \textRubikSb\dns\cmd{\textRubikSb} \nonumber\\ 
%  \dnstrut\rrSbp\dns\cmd{\rrSbp}
%  & \rrhSbp\dns\cmd{\rrhSbp}
%  & \textRubikSbp\dns\cmd{\textRubikSbp} \nonumber\\ 
%  \end{supertabular}
%
%
%  \begin{supertabular}[lll]{p{3cm} p{3cm} p{4cm}}
%  \dnstrut\rrE\dns\cmd{\rrE}
%  & \rrhE\dns\cmd{\rrhE}
%  & \textRubikE\dns\cmd{\textRubikE} \nonumber\\ 
%  \dnstrut\rrEp\dns\cmd{\rrEp}
%  & \rrhEp\dns\cmd{\rrhEp}
%  & \textRubikEp\dns\cmd{\textRubikEp} \nonumber\\ 
%  \dnstrut\rrM\dns\cmd{\rrM}
%  & \rrhM\dns\cmd{\rrhM}
%  & \textRubikM\dns\cmd{\textRubikM} \nonumber\\ 
%  \dnstrut\rrMp\dns\cmd{\rrMp}
%  & \rrhMp\dns\cmd{\rrhMp}
%  & \textRubikMp\dns\cmd{\textRubikMp} \nonumber\\ 
%  \dnstrut\rrS\dns\cmd{\rrS}
%  & \rrhS\dns\cmd{\rrhS}
%  & \textRubikS\dns\cmd{\textRubikS} \nonumber\\ 
%  \dnstrut\rrSp\dns\cmd{\rrSp}
%  & \rrhSp\dns\cmd{\rrhSp}
%  & \textRubikSp\dns\cmd{\textRubikSp} \nonumber\\ 
%  \end{supertabular}
%
%
%  \begin{supertabular}[lll]{p{3cm} p{3cm} p{4cm}}
%  \dnstrut\rrx\dns\cmd{\rrx}
%  & \rrhx\dns\cmd{\rrhx}
%  & \Rubikx\dns\cmd{\Rubikx} \nonumber\\ 
%  \dnstrut\rrxp\dns\cmd{\rrxp}
%  & \rrhxp\dns\cmd{\rrhxp}
%  & \Rubikxp\dns\cmd{\Rubikxp} \nonumber\\ 
%  \dnstrut\rry\dns\cmd{\rry}
%  & \rrhy\dns\cmd{\rrhy}
%  & \Rubiky\dns\cmd{\Rubiky} \nonumber\\ 
%  \dnstrut\rryp\dns\cmd{\rryp}
%  & \rrhyp\dns\cmd{\rrhyp}
%  & \Rubikyp\dns\cmd{\Rubikyp} \nonumber\\ 
%  \dnstrut\rrz\dns\cmd{\rrz}
%  & \rrhz\dns\cmd{\rrhz}
%  & \Rubikz\dns\cmd{\Rubikz} \nonumber\\ 
%  \dnstrut\rrzp\dns\cmd{\rrzp}
%  & \rrhzp\dns\cmd{\rrhzp}
%  & \Rubikzp\dns\cmd{\Rubikzp} \nonumber\\ 
%  \end{supertabular}
%
%
%  \begin{supertabular}[lll]{p{3cm} p{3cm} p{4cm}}
%  \dnstrut\rru\dns\cmd{\rru}
%  & \rrhu\dns\cmd{\rrhu}
%  & \Rubiku\dns\cmd{\Rubiku} \nonumber\\ 
%  \dnstrut\rrd\dns\cmd{\rrd}
%  & \rrhd\dns\cmd{\rrhd}
%  & \Rubikd\dns\cmd{\Rubikd} \nonumber\\ 
%  \dnstrut\rrl\dns\cmd{\rrl}
%  & \rrhl\dns\cmd{\rrhl}
%  & \Rubikl\dns\cmd{\Rubikl} \nonumber\\ 
%  \dnstrut\rrr\dns\cmd{\rrr}
%  & \rrhr\dns\cmd{\rrhr}
%  & \Rubikr\dns\cmd{\Rubikr} \nonumber\\ 
%  \dnstrut\rrf\dns\cmd{\rrf}
%  & \rrhf\dns\cmd{\rrhf}
%  & \Rubikf\dns\cmd{\Rubikf} \nonumber\\ 
%  \dnstrut\rrb\dns\cmd{\rrb}
%  & \rrhb\dns\cmd{\rrhb}
%  & \Rubikb\dns\cmd{\Rubikb} \nonumber\\ 
%  \end{supertabular}
%
%
%
%    \section{Colours}
%
% The \rubikcube\ package defines seven colours which are defined as follows: 
% red~(R), orange~(O), yellow~(Y),  green~(G), blue~(B),  white~(W), 
% and   grey~(X). Now  according to the following webpage\,\footnote{I thank Peter Bartal  
% for pointing this out.}
%
% \medskip
% \noindent\texttt{http://The-Rubiks-Cube.deviantart.com/journal/Using-Official-Rubik}
% \newline\texttt{-s-Cube-Colors-268760351}
%
% \medskip
% {\noindent}the official Rubik cube colours are defined as
% \begin{quote}
%\begin{verbatim}
% .... colours which are red (PMS 200C*), green (PMS 347C*),
% blue (PMS 293C*), orange (PMS 021C*), yellow (PMS 012C*)
% and white.
% .....
% Pantone colors can not be accurately converted to RGB colors, 
% the colors the web runs on. But they can be approximated. 
% Through some research, I have found some estimations which 
% may help you which I have listed below. Remember, these are
% just approximate RGB equivalents to the official Rubik's Cube
%  colors.
% 
% Red: 200C #C41E3A  (www.perbang.dk/rgb/c41e3a/)
% Green: 347C #009E60 (www.perbang.dk/rgb/009e60/)
% Blue: 293C #0051BA (www.perbang.dk/rgb/0051ba/)
% Orange: 021C "Pantone Orange" #FF5800 (www.perbang.dk/rgb/ff5800/)
% Yellow: 012C "Pantone Yellow" #FFD500 (www.perbang.dk/rgb/ffd500/)
% White: N/A #FFFFFF
%
% Red    {HTML}{C41E3A}
% green  {HTML}{009E60}
% Blue   {HTML}{0051BA}
% Yellow {HTML}{FFD500}
% Orange {HTML}{FF5800}
% White  {HTML}{FFFFFF}
%\end{verbatim}
% \end{quote}
%  However, we have optimised these prescribed colours very slightly 
%  for screen \& print use (for example, the yellow  was made very 
%  slightly brighter),  and so the actual colours  as used by 
%  the \rubikcube\ package  are as follows:
%
% \begin{quote}
%\begin{verbatim}
% \definecolor{R}{HTML}{C41E33}
% \definecolor{G}{HTML}{00BE38}
% \definecolor{B}{HTML}{0051BA}
% \definecolor{Y}{HTML}{FFFF00}
% \colorlet{O}{orange}
% \colorlet{W}{white}
% \colorlet{X}{black!30}%
%\end{verbatim}
% \end{quote}
% Different colours  can be allocated   to  the ROYGBWX letters  (using the 
% \cmd{\colorlet} command) as required. For example, the standard `red' 
% colour could  be allocated to the letter R using the command
% \begin{quote}
%    \cmd{\colorlet\{R\}\{red\}}
% \end{quote}
% However, it is important to appreciate that  the letter codes 
% ROYGBWX are `hardwired' into many of the macros in the \rubikcube\ package,
%  so don't change these.
%
%
%  \section{Rubik face coordinates}
%  \label{sec:coordinates}
%
% The coordinate origin  of each view of the Rubik cube  is located  
% at the bottom left corner of the \textsc{front} face, as shown in 
%  Figure~\ref{fig:cubesquaregraph}. Note also that the bottom left 
%  corner of the cube  itself is at $(-1,-1)$, 
%  and hence the default height and width of the cube is 4cm. 
%
% \begin{figure}[hbt]
% \centering
% \ifpdf
%   \includegraphics[height=3cm]{Rubik_doc_figB.pdf}
% \else
% \includegraphics[height=3cm]{Rubik_doc_figB.eps}
% \fi
%
% \parbox{9cm}{\caption{\label{fig:cubesquaregraph}Origin of coordinates 
%  is at the  bottom left corner of the grey \textsc{front} face. 
%  The bottom left corner of the cube  itself is at $(-1,-1)$, 
%  and hence the default height and width of the 3D-cube is 4cm.}}
% \end{figure}
%
% This is an important feature since knowing the location of the 
% origin enables one to easily use any of the  TikZ commands
% (eg~\cmd{\draw} and \cmd{\node} commands) to  superimpose lines, 
% arrows and text etc onto the Rubik cube (see Section~\ref{sec:arrows}).
%
%
%  \subsection{Size of cube \cmd{\minipage}}
%
%  Since the the default height and width of the 3D-cube is 4cm (see above), 
%  it follows that the width of the \cmd{\minipage} required for a cube in a TikZ 
%  picture environment can be easily calculated. For example, if the TikZ 
%  scale used is $0.5$, then the appropriate width of the required minipage 
%  is $0.5 \times 4\mbox{cm} =  2\mbox{cm}$. 
%
% The default width of the semi-flat cube representation is 10cm, and that of the 
% flat cube is 12cm.
%
%  \section{Commands}
%
% All  |rubikcube| commands assume a 3x3x3 cube by default.
% There are three primary command categories:  \cmd{\Draw..} commands 
% (which must be used \textit{inside} a TikZ picture environment), 
% `parameter-allocation' commands (either inside or outside a TikZ environment), 
% and commands which can be used in ordinary text (eg~rotation commands).
%
% A \cmd{\Draw..} command typesets  
% either a Rubik cube, cubie or a layer  using parameters set or defined via 
% previous parameter-allocation commands (eg~face colours, dimensions etc).
%
%
%    \subsection{Draw commands}
%
%   \DescribeMacro{\DrawRubikCubeXY}
%   \DescribeMacro{\DrawCubieXYxyz}
% These  commands  draw Rubik cubes and  single cubies  in one of four 
% orientations as denoted by the terminal XY viewing-direction code RU (RightUp), 
% RD (RightDown), LU (LeftUp),  LD (LeftDown). For example, the command
%  \begin{quote}
%  \cmd{\DrawRubikCubeRU}
%  \end{quote}
% will draw a Rubik cube  as viewed from the RightUp direction, as 
% shown in the following figure.
% 
% \bigskip
%
% \begin{minipage}{2.8cm}
% \begin{tikzpicture}[scale=0.7]
% \DrawRubikCubeRU 
% \end{tikzpicture}
% \end{minipage}
%   \hspace{5mm}
% \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
% \begin{tikzpicture}[scale=0.7]
%    \DrawRubikCubeRU 
% \end{tikzpicture}
%\end{verbatim}
% \end{minipage}
%
% \bigskip 
%  
% {\noindent}Note that in the above example all the cubies are displayed 
% in the default grey colour, since no Face colours have been specified. 
% 
% Since \LaTeX\ commands have a maximum limit of only 9 parameters, 
% it is necessary to use separate commands 
% (`Face' and `Slice'  commands---see below) in order to accommodate 
% all 27 visible colours of a 3D~Rubik cube.
%
% Since a single cubie has only three visible faces we can include 
% colour parameters in \cmd{\DrawCubie} commands. Consequently \cmd{\DrawCubie}
%  commands have the format 
% \begin{quote}
%  \cmd{\DrawCubieXY\{x\}\{y\}\{z\}}  
% \end{quote}
% where the  XY pair denote the viewing direction as before, and 
% the xyz  denote the face colours associated with each of the three axes. 
%
% For example, the  command    \cmd{\DrawCubieRU\{O\}\{Y\}\{G\}}   draws a 
%  single cubie as viewed  from the RightUp  direction, 
% with face colours Orange (x-axis), Yellow (y-axis), Green (z-axis), as follows.
% 
% \bigskip
%
% \begin{minipage}{1.35cm}
% \begin{tikzpicture}[scale=1]
% \DrawCubieRU{O}{Y}{G} 
% \end{tikzpicture}
% \end{minipage}
%    \hspace{1cm}
% \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
% \begin{tikzpicture}[scale=1]
%    \DrawCubieRU{O}{Y}{G} 
% \end{tikzpicture}
%\end{verbatim}
% \end{minipage}
% 
% \bigskip 
%
%
%
% \begin{figure}[hbt]
%   \centering
% \ifpdf
%    \includegraphics[height=4cm]{Rubik_doc_figC.pdf}
%  \else
% \includegraphics[height=4cm]{Rubik_doc_figC.eps}
% \fi
% \vspace{-5mm}\caption{\label{fig:cubiedydx}Cubie dy dx parameters} 
% \end{figure}
%
%  \DescribeMacro{\Cubiedy}
%  \DescribeMacro{\Cubiedx}
%  Minor  cubie configuration changes  can be effected 
% (within a TikZ picture environment)  by adjusting the dy and dx 
% values ($> 0$; no units)  shown in Figure~\ref{fig:cubiedydx} 
% via the two commands
% \begin{quote}
% \cmd{\Cubiedy\{\}} \\
% \cmd{\Cubiedx\{\}} 
% \end{quote}
% as shown in the folowing example.
%
% \bigskip
%
% \begin{minipage}{1.7cm}
% \begin{tikzpicture}[scale=1]
% \Cubiedy{0.4}
% \Cubiedx{0.8}
% \DrawCubieRU{O}{Y}{G} 
% \end{tikzpicture}
% \end{minipage}
% \hspace{2cm}
% \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
% \begin{tikzpicture}[scale=1]
%   \Cubiedy{0.4}
%   \Cubiedx{0.8}
%   \DrawCubieRU{O}{Y}{G} 
% \end{tikzpicture}
%\end{verbatim}
% \end{minipage}
% 
% \bigskip
%
% {\noindent}Note that the  \textsc{front} of the cubie is a unit square, 
% and the graphic origin of the cubie image is at the bottom left corner of the 
% \textsc{front}  (see also the section on Arrows).
% The default  values of dy and dx are 0.4. 
%
% \medskip
%
% \DescribeMacro{\textCubieRU}
% \DescribeMacro{\textCubieRD}
% \DescribeMacro{\textCubieLU}
% \DescribeMacro{\textCubieLD}
% For convenience, there are also four (smaller) `text' versions \textCubieRU{O}{Y}{G}   
%  of the  four \cmd{\DrawCubie} commands for 
% use in ordinary text, as follows:
% \begin{quote}
%  \textCubieRU{O}{Y}{G} \ \ |\textCubieRU{O}{Y}{G}|
%
% \medskip
%
% \textCubieRD{O}{Y}{G} \ \  |\textCubieRD{O}{Y}{G}|     
%
% \medskip
%
% \textCubieLU{O}{Y}{G}  \ \ |\textCubieLU{O}{Y}{G}|
%
% \medskip
%
%  \textCubieLD{O}{Y}{G}  \ \  |\textCubieLD{O}{Y}{G}|  
% \end{quote}
% Note that the \cmd{\textCubieXY} commands are not influenced by the 
% \cmd{\Cubiedy}, \cmd{\Cubiedx} commands as their size is preset for text use. 
%
% Other \cmd{\Draw..} commands which we shall meet later are
%  \texttt{DrawRubikLayerFace..},  \texttt{DrawRubikLayerSide..}, 
% \texttt{DrawNCube..}, \texttt{DrawRubikFlat} and \texttt{DrawRubikCubeFlat}.
%
%
% \subsubsection{\cmd{\draw} error message}
%
% See also section~\ref{drawerrormessage} regarding the error message associated with 
% using a \cmd{\Draw...} command  \textit{outside}  a TikZ picture environment.
%
%
%  \subsection{Face commands}
%
%  \DescribeMacro{\RubikFaceUp}
%  \DescribeMacro{\RubikFaceDown}
%  \DescribeMacro{\RubikFaceLeft}
%  \DescribeMacro{\RubikFaceRight}
%  \DescribeMacro{\RubikFaceFront}
%  \DescribeMacro{\RubikFaceBack}
% These  commands take nine colour arguments and  allocate   colours to the 
% individual cubies  of a  Rubik cube face.
% The ordering is isomorphic to the sequence 1--9, i.e.,~numbering the small 
% squares 1-3~(top row, left to right), 4-6~(middle row, left to right), 
% 7-9~(bottom row, left to right), as follows:
% \begin{quote}
% \fbox{
% \begin{minipage}{1.6cm}
% \#1 \#2 \#3 
%
% \#4 \#5 \#6
%
% \#7 \#8 \#9
% \end{minipage}
% }
% \end{quote}
% Conveniently, \LaTeX\ allows the colour arguments to be separated by spaces 
% (e.g.,~separated in groups of three), or even spread across several 
% lines (e.g.,~in a square block to resemble a 9-face) in order to 
% make the command more visually intuitive, as in the following examples.
% \begin{quote}
%\begin{verbatim}
% \RubikFaceUp{G}{B}{G}   {G}{W}{O}   {G}{O}{G}
%
% \RubikFaceFront{O}{W}{R}
%                {W}{W}{W}
%                {G}{W}{G}
% 
%\end{verbatim}
% \end{quote}
%
%  \DescribeMacro{\RubikFaceUpAll}
%  \DescribeMacro{\RubikFaceDownAll}
%  \DescribeMacro{\RubikFaceLeftAll}
%  \DescribeMacro{\RubikFaceRightAll}
%  \DescribeMacro{\RubikFaceFrontAll}
%  \DescribeMacro{\RubikFaceBackAll}
% For convenience, each of these commands has an associated 
% `\textsc{all}' command which allocates  the same colour to all the cubies 
% on a 9-face, as follows (only a single colour argument is required):
%
% If you want a particular face to be \textsc{all} grey, then one can 
% either omit the particular `Face' command (since the 
% default colour is grey), or  use  a `Face' command specifying the 
% colour-code~X;  for example,  \cmd{\RubikFaceUpAll\{X\}}.
% However, if you do use a Face command, then all of the command's 
% colour arguments must be used, as otherwise you will generate a 
% `missing parameter' error, and no colour will be 
% allocated (i.e.,~you will see  a black-hole).
% Use of these commands is shown in the following figure.
%
% \bigskip
% 
% \begin{minipage}{2.8cm}
% \begin{tikzpicture}[scale=0.7]
% \RubikFaceUpAll{X}
% \RubikFaceRightAll{R}
% \RubikFaceFront{W}{Y}{G}
%                {W}{Y}{G}
%                {W}{Y}{G}
% \DrawRubikCubeRU
% \end{tikzpicture}
% \end{minipage}
%    \hspace{5mm}
% \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
% \begin{tikzpicture}[scale=0.7]
%    \RubikFaceUpAll{X}
%    \RubikFaceRightAll{R}
%    \RubikFaceFront{W}{Y}{G}
%                   {W}{Y}{G}
%                   {W}{Y}{G}
%    \DrawRubikCubeRU
% \end{tikzpicture}
%\end{verbatim}
% \end{minipage}
% 
%
% \subsection{RubikCubeSolved command}
%
%  \DescribeMacro{\RubikCubeSolved}
% This command  sets all the face colours to that of a standard `solved' cube
% as shown in the following semi-flat image.
% This command is equivalent to the following set of face commands:
%\begin{verbatim}
%  \RubikFaceUpAll{W}%
%  \RubikFaceDownAll{Y}%
%  \RubikFaceLeftAll{B}%
%  \RubikFaceRightAll{G}%
%  \RubikFaceFrontAll{O}%
%  \RubikFaceBackAll{R}%
%\end{verbatim}
%
% \begin{tikzpicture}[scale=0.6]
% \RubikCubeSolved
% \DrawRubikCubeFlat
% \end{tikzpicture}
%
%  Users can easily set their own alternative `solved'  face/colour configuration by `renewing' 
% this command as follows---remember to include the trailing \%  (note that the colours 
% ROYGBWX are currently hardwired  so don't  use a different set of uppercase letters).
%\begin{verbatim}
% \renewcommand{\RubikCubeSolved}{%
%    \RubikFaceUpAll{..}%
%    \RubikFaceDownAll{..}%
%    \RubikFaceLeftAll{..}%
%    \RubikFaceRightAll{..}%
%    \RubikFaceFrontAll{..}%
%    \RubikFaceBackAll{..}%
% }
%\end{verbatim} 
%
%
%    \subsection{Slice commands}
%
%  \DescribeMacro{\RubikSliceTopX}
%  \DescribeMacro{\RubikSliceEquatorX}
%  \DescribeMacro{\RubikSliceBottomX}
%  These three commands  allocate the six visible cubie colours associated with  a
%   \textit{horizontal}  slice of a Rubik cube. 
% There are three pairs of  Slice commands; one pair 
% (Left view \& Right view) for each of the horizontal  slices Top, Equator, Bottom.
% The six colour arguments  associated with a given slice run in sequence 
% from left to right irrespective of the viewpoint, eg \#1 \#2 \#3   \#4 \#5 \#6.
%
% Since the viewpoint of the Rubik cube (from the Right or from the Left) 
% influences which face the colours are associated with, it is necessary 
% to have the view (R or L) represented in the command name.
%
% The format of the `slice' command is shown in the following example.
% The Rubik  cube is shown from  the LeftDown (LD) view 
% and consequently  each of the `slice' commands in this
% particular example  ends in L, consistent with 
% the final \cmd{\DrawRubikCubeLD} command.
%
% \bigskip
%
% \begin{minipage}{2.8cm}
% \centering
% \begin{tikzpicture}[scale=0.7]
% \RubikFaceDownAll{Y}
% \RubikSliceTopL    {G}{G}{G} {R}{R}{R}
% \RubikSliceEquatorL{R}{R}{R} {B}{B}{B}
% \RubikSliceBottomL {O}{O}{O} {G}{G}{G}
% \DrawRubikCubeLD
% \end{tikzpicture}
% \end{minipage}
%    \hspace{5mm}
% \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
% \begin{tikzpicture}[scale=0.7]
%    \RubikFaceDownAll{Y}
%    \RubikSliceTopL     {G}{G}{G} {R}{R}{R}
%    \RubikSliceEquatorL {R}{R}{R} {B}{B}{B}
%    \RubikSliceBottomL  {O}{O}{O} {G}{G}{G}
%    \DrawRubikCubeLD
% \end{tikzpicture}
%\end{verbatim}
% \end{minipage}
%
%
%  \subsection{LayerFace \& LayerSide commands}
%  \label{sec:layerface}
%
%  These commands are mainly  intended  for drawing  final layer 
%  configurations (ie typically using the yellow face in the  \textsc{up} position)
%  with or without the associated side faces of the cubies of the top layer.
%
%  \subsubsection{LayerFace}
%
%  \DescribeMacro{\DrawRubikLayerFace}
%  This LayerFace command  draws a  simple  Rubik cube 3x3 face and allocates colours to the  9 cubies.
%  The command takes nine ordered colour arguments.
%  Their use is illustrated in the following example, which shows  a yellow cross configuration.
%  
%  \bigskip
%
%  \begin{minipage}{2.5cm}
%  \centering
%  \begin{tikzpicture}[scale=0.7]
%  \DrawRubikLayerFace{X}{Y}{X}
%                     {Y}{Y}{Y}
%                     {X}{Y}{X}
%  \end{tikzpicture}
%  \end{minipage}
%  \hspace{1cm}
%  \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
%  \begin{tikzpicture}[scale=0.7]
%     \DrawRubikLayerFace{X}{Y}{X}
%                        {Y}{Y}{Y}
%                        {X}{Y}{X}
%  \end{tikzpicture}
%\end{verbatim}
%  \end{minipage}
%
% \medskip
%
% \subsubsection{\cmd{\draw} error message}
%
% See also section~\ref{drawerrormessage} regarding the error message associated with 
% using a \cmd{\Draw...} command  \textit{outside}  a TikZ picture environment.
%
%  \medskip
%
%  \subsubsection{LayerSide}
% 
%  \DescribeMacro{\DrawLayerSideXYp}
%  LayerSide commands  draw the associated side colours  of  
%  the top layer   as  small rectangular sidebars.
%  The LayerSide commands  adopt a three-letter XYp \textbf{position} notation 
%   where the  XY pair define the location (X:[Left \verb!|! Middle \verb!|! Right]; 
%  Y:[Top \verb!|! Equator \verb!|! Bottom]) of a particular cubie in the layer face.
%
%  \begin{figure}[hbt]
%     \centering
%  \ifpdf
%     \includegraphics[height=3cm]{Rubik_doc_figD.pdf}
%  \else
%  \includegraphics[height=3cm]{Rubik_doc_figD.eps}
%  \fi
%  \vspace{-5mm}\caption{\label{fig:facenotation}LayerSide letter-codes } 
%  \end{figure}
% 
%  Since corner cubies have two side faces, the `p' parameter 
%  (p:[x\verb!|!y])  is required  to denote the directional `position' of 
%  the cubie side-face relative to the cubicle XY position
%  (x indicates adjacent along the x-axis, and  y indicates adjacent along the y-axis).
%  Since edge cubies have only one side face,  the `p' parameter is optional 
%  (for consistency), since it is not strictly  necessary.
%
%  \medskip\noindent\textbf{Commands}: \
%  Various different LayerSide commands are available: those for  
%  drawing a single colour side bar, and   others for facilitating  
%  drawing all 12 side bars.
%     For example, \textit{either} of the following commands 
%
%  \begin{quote}
%\begin{verbatim}
%  \DrawRubikLayerSideLE{G}
%  \DrawRubikLayerSideLEx{G}
%\end{verbatim}
%  \end{quote}
%  draws  a  single small vertical  green rectangle
%  \begin{tikzpicture}[scale=0.5]
%  \DrawRubikLayerSideLEx{G}
%  \end{tikzpicture}
%   just to the left of the Left Equator (LE) square of the 9-face.
%  The following commands draw all three side bars of a given side (Top, Bottom, Left, Right)
%  \begin{quote}
%\begin{verbatim}
%  \DrawRubikLayerSideT{}{}{}
%  \DrawRubikLayerSideB{}{}{}
%  \DrawRubikLayerSideL{}{}{}
%  \DrawRubikLayerSideR{}{}{}
%\end{verbatim}
%  \end{quote}
%  where the T and B forms require the colour parameters to be in left-right 
%  horizontal order (eg, left, middle, right),
%  while the L and R forms require the colour parameters to be in top-down 
%  vertical order (eg, top, equator, bottom).
%  Note that this means that the colour parameters of the  L and R commands 
%  can (if required) then be positioned intuitively (vertically) as follows:
%  \begin{quote}
%\begin{verbatim}
%  \DrawRubikLayerSideL{}
%                      {}
%                      {}
%\end{verbatim}
%  \end{quote}
%  Extending this idea, the L and R forms are combined in the LR command, 
%  which takes six colour arguments ordered in left-right pairs,
%  \begin{quote}
%\begin{verbatim}
%  \DrawRubikLayerSideLR{}{} {}{} {}{}
%\end{verbatim}
%  \end{quote}
%  so that they can also be written vertically as left-right pairs.
%  Use of these commands is shown in the following two examples.
%  
%  \bigskip
% 
%  \begin{minipage}{2.5cm}
%  \centering
%  \begin{tikzpicture}[scale=0.7]
%  
%  
%  \DrawRubikLayerFace{X}{Y}{X}
%                     {Y}{Y}{Y}
%                     {X}{Y}{X}
%  
%  \DrawRubikLayerSideLTy{B}
%  \DrawRubikLayerSideLTx{O}
%  \DrawRubikLayerSideLE{G}
%  \DrawRubikLayerSideR{O}
%                      {B}
%                      {R}
%  \DrawRubikLayerSideB{R}{G}{O}
%  \node (LT) at (0.5, 2.5)  [red]{\small\textsf{LT}};
%  \node (LE) at (0.5, 1.5)  [red]{\small\textsf{LE}};
%  \end{tikzpicture}
%  \end{minipage}
%      \hspace{1cm}
%  \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
%  \begin{tikzpicture}[scale=0.7]
%     \DrawRubikLayerFace{X}{Y}{X}
%                        {Y}{Y}{Y}
%                        {X}{Y}{X}
%     
%     \DrawRubikLayerSideLTy{B}
%     \DrawRubikLayerSideLTx{O}
%     \DrawRubikLayerSideLE{G}
%  
%     \DrawRubikLayerSideR{O}{B}{R}
%     \DrawRubikLayerSideB{R}{G}{O}
%  
%     \node (LT) at (0.5, 2.5) 
%                [red]{\small\textsf{LT}};
%     \node (LE) at (0.5, 1.5)  
%                [red]{\small\textsf{LE}};
%  \end{tikzpicture}
%\end{verbatim}
%  \end{minipage}
%  
%  \bigskip
%  
%  \begin{minipage}{2.5cm}
%  \centering
%  \begin{tikzpicture}[scale=0.7]
%  
%  \DrawRubikLayerFace{G}{Y}{R}
%                     {Y}{Y}{Y}
%                     {B}{Y}{Y}
%  
%  \DrawRubikLayerSideT {Y}{B}{B}
%  \DrawRubikLayerSideLR{R}   {Y}
%                       {R}   {O}
%                       {Y}   {O}
%  \DrawRubikLayerSideB {O}{G}{G}
%  
%  \end{tikzpicture}
%  \end{minipage}
%      \hspace{1cm}
%  \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
%  \begin{tikzpicture}[scale=0.7]
%     \DrawRubikLayerFace{G}{Y}{R}
%                        {Y}{Y}{Y}
%                        {B}{Y}{Y}
%     
%     \DrawRubikLayerSideT {Y}{B}{B}
%     \DrawRubikLayerSideLR{R}   {Y}
%                          {R}   {O}
%                          {Y}   {O}
%     \DrawRubikLayerSideB {O}{G}{G}  
%  \end{tikzpicture}
%\end{verbatim}
%  \end{minipage}
%  
%  \bigskip
%  
%  The default values (size) of the  sidebars are as follows:
%  width (0.3), length(1) and separation  from the square face (0.3). 
%  Note that the default value of the length of a cubie side is 1. 
%  These sidebar values (decimal values $\geq 0$; no units) can be 
%  changed from their default values using the three commands.
%  \begin{quote}
%  \cmd{\RubikSideBarWidth\{\}}\\
%  \cmd{\RubikSideBarLength\{\}}\\
%  \cmd{\RubikSideBarSep\{\}}
%  \end{quote}
%  Values set in the document preamble will apply globally, while values 
%  set within a TikZ picture environment will apply only locally to that 
%  particular environment, as shown in the following example where we have 
%  set both the sidebar width and length to 0.7.
%  
%  \bigskip
%
%  \begin{minipage}{2.5cm}
%  \centering
%  \begin{tikzpicture}[scale=0.7]
%  
%  
%  \DrawRubikLayerFace{X}{Y}{X}
%                     {Y}{Y}{Y}
%                     {X}{Y}{X}
%  
%  \RubikSideBarWidth{0.7}
%  \RubikSideBarLength{0.7}
%  \DrawRubikLayerSideMTy{G}
%  \end{tikzpicture}
%  \end{minipage}
%      \hspace{1cm}
%  \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
%  \begin{tikzpicture}[scale=0.7]
%     \DrawRubikLayerFace{X}{Y}{X}
%                        {Y}{Y}{Y}
%                        {X}{Y}{X}
%     \RubikSideBarWidth{0.7}
%     \RubikSideBarLength{0.7}
%     \DrawRubikLayerSideMTy{G}
%  \end{tikzpicture}
%\end{verbatim}
%  \end{minipage}
%  
%  \bigskip
%  
%  Note also that changing the  sidebar-width or sidebar-separation 
%  values may well also change the surrounding white-space (use \cmd{\fbox} 
%  to visualise this) and may therefore require some fine-tuning of the 
%  minipage width setting in order to optimise  appearance. 
% 
%  Since the \textsc{front} face  drawn using  the \cmd{\DrawRubikCube} command is 
%  identical with that drawn using the \cmd{\DrawLayerFace} command it 
%  follows that LayerSide commands can also be used in conjunction with 
%  the \textsc{front} face drawn using  \cmd{\DrawRubikCube} command, as 
%  shown in the following example.
%  
%  \bigskip
%  
%  \begin{minipage}{2.5cm}
%  \centering
%  \begin{tikzpicture}[scale=0.7]
%  \RubikFaceUp   {X}{W}{X}%
%                 {W}{W}{W}%
%                 {X}{W}{X}%
%  
%  \RubikFaceFront{O}{O}{X}%
%                 {O}{O}{X}%
%                 {X}{X}{W}%
%  
%  \RubikFaceRight{X}{G}{G}%
%                 {X}{G}{G}%
%                 {G}{X}{X}%
%  \DrawRubikCubeRU
%  
%  \DrawRubikLayerSideL{O}{B}{R}
%  \DrawRubikLayerSideB{R}{G}{O}
%  \end{tikzpicture}
%  \end{minipage}
%        \hspace{1cm}
%  \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
%  \begin{tikzpicture}[scale=0.7]
%     \RubikFaceUp   {X}{W}{X}%
%                    {W}{W}{W}%
%                    {X}{W}{X}%
%  
%     \RubikFaceFront{O}{O}{X}%
%                    {O}{O}{X}%
%                    {X}{X}{W}%
%  
%     \RubikFaceRight{X}{G}{G}%
%                    {X}{G}{G}%
%                    {G}{X}{X}%
%     \DrawRubikCubeRU
%  
%     \DrawRubikLayerSideL{O}{B}{R}
%     \DrawRubikLayerSideB{R}{G}{O}
%  \end{tikzpicture}
%\end{verbatim}
%  \end{minipage}
%  
%  \bigskip
%  
%  {\noindent}Note that since using a left or a right \cmd{\DrawRubikLayerSide}  
%  command in conjunction with a \cmd{\DrawRubikCube} command will necessarily 
%  increase the width of the image, one may also have to adjust the width of 
%  the associated minipage. 
%  
% 
%  \subsection{Flat commands}
%  \label{sec:flatcommands}
%  
%  \DescribeMacro{\DrawRubikFlat}
%  \DescribeMacro{\DrawRubikCubeFlat}
%  Flat commands are used to draw `flat' representations of the cube. 
%  The most  standard format, which is generated by the command \cmd{\DrawRubikFlat},
%  is  shown in the following example.
%  
%  \bigskip
%  
%  \begin{minipage}{0.4\textwidth}
%  \centering
%  \begin{tikzpicture}[scale=0.4]
%  \RubikCubeSolved
%
%  \DrawRubikFlat
%  \node (U) at (1.5, 4.5)   [black]{\small\textsf{U}};
%  \node (D) at (1.5, -1.5)  [black]{\small\textsf{D}};
%  \node (L) at (-1.5, 1.5)  [black]{\small\textsf{L}};
%  \node (R) at (4.5, 1.5)   [black]{\small\textsf{R}};
%  \node (F) at (1.5, 1.5)   [black]{\small\textsf{F}};
%  \node (B) at (7.5, 1.5)   [black]{\small\textsf{B}};
%  \end{tikzpicture}
%  \noindent\strut\hspace{5mm}\texttt{$\backslash$DrawRubikFlat}
%  \end{minipage}
%  \begin{minipage}{5cm}
%\begin{verbatim}
%  \begin{tikzpicture}[scale=0.4]
%     \RubikCubeSolved
%     \DrawRubikFlat
%  \end{tikzpicture}
%\end{verbatim}
%  \end{minipage}
%  
%  \bigskip
%  
%  The addition of  text (numbers or letters) in the faces is 
%  straightforward---the origin of the 1-unit grid is located at the 
%  bottom left corner of the front face (green here). 
%  The letters were placed using the following TikZ code:
%  
%\begin{verbatim}
%  \node (U) at (1.5, 4.5)   [black]{\small\textsf{U}};
%  \node (D) at (1.5, -1.5)  [black]{\small\textsf{D}};
%  \node (L) at (-1.5, 1.5)  [black]{\small\textsf{L}};
%  \node (R) at (4.5, 1.5)   [black]{\small\textsf{R}};
%  \node (F) at (1.5, 1.5)   [black]{\small\textsf{F}};
%  \node (B) at (7.5, 1.5)   [black]{\small\textsf{B}};
%\end{verbatim}
%  
%  {\noindent}An interesting `semi-flat' alternative format  uses the standard 
%  view of the cube  and  appends the three hidden sides 
%  (see Rokicki \textit{etal.}, 2013),  which is generated by the 
%  command \cmd{\DrawRubikCubeFlat} as follows.
%  
%  \bigskip
%
%  \begin{minipage}{5cm}
%  \begin{tikzpicture}[scale=0.5]
%  \RubikCubeSolved
%  \DrawRubikCubeFlat
%  \end{tikzpicture}
%  \end{minipage}
%  \begin{minipage}{5cm}
%\begin{verbatim}
%  \begin{tikzpicture}[scale=0.5]
%     \RubikCubeSolved
%     \DrawRubikCubeFlat
%  \end{tikzpicture}
%\end{verbatim}
%  \end{minipage}
%  
% \medskip
%
% \subsubsection{\cmd{\draw} error message}
%
% See also section~\ref{drawerrormessage} regarding the error message associated with 
% using a \cmd{\Draw...} command  \textit{outside}  a TikZ picture environment.
%
%  
%  \section[NCube]{NCube (NxNxN)}
%  \label{sec:NCube}
% 
%  \DescribeMacro{\DrawNCubeAll} 
%  An `NCube' is  a solved NxNxN cube drawn from the RU direction; 
%  (ie.~only shows faces \textsc{up}, \textsc{front}, \textsc{right}). 
%  The cubie colours of each face are All the same.
%  \begin{quote}
%    \cmd{\DrawNCubeAll\{N\}\{Xcolour\}\{Ycolour\}\{Zcolour\}}.
%  \end{quote}
%  This command takes four ordered parameters (N, X, Y, Z)---the number 
%  (integer; $N>0$) of cubies along an edge, followed by three face 
%  colours in XYZ order.
%  Since the viewpoint is only from the RU direction, the  three colour 
%  parameters are: X(Right), Y(Up), Z(Front).
%  
%  \bigskip
%  
%  \begin{minipage}{0.4\textwidth}
%  \begin{tikzpicture}[scale=0.5]
%     \DrawNCubeAll{5}{O}{Y}{G}
%  \end{tikzpicture}
%  \end{minipage}
%  \begin{minipage}{0.5\textwidth}
%\begin{verbatim}
%  \begin{tikzpicture}[scale=0.5]
%     \DrawNCubeAll{5}{O}{Y}{G}
%  \end{tikzpicture}
%\end{verbatim}
%  \end{minipage}
%
% \medskip
%
% \subsubsection{\cmd{\draw} error message}
%
% See also section~\ref{drawerrormessage} regarding the error message associated with 
% using a \cmd{\Draw...} command  \textit{outside}  a TikZ picture environment.
%  
%
%     \section{Arrows}
%     \label{sec:arrows}
%  
%  The \rubikcube\ package does not offer any special commands for drawing 
%  arrows since it is straightforward just to include the appropriate TikZ
%  `draw' commands in the \texttt{tikzpicture} environment. 
%  
%  In order to facilitate using the standard TikZ `draw' commands both 
%  the  RubikCubeFaceFront  and LayerFace commands have the coordinate origin 
%  at the bottom left corner, and draw `faces' consisting of 9~unit-squares 
%  in a  3x3 grid, as shown in  Figure~\ref{fig:facegraph}.
%  Consequently the start and finish coordinates for any arrow or line are 
%  easy to determine.
%  
%  \begin{figure}[hbt]
%     \centering
%  \ifpdf
%     \includegraphics[height=3cm]{Rubik_doc_figE.pdf}
%  \else
%  \includegraphics[height=3cm]{Rubik_doc_figE.eps}
%  \fi
%  \vspace{-5mm}\caption{\label{fig:facegraph}RubikCubeFaceFront and RubikLayerFace coordinates} 
%  \end{figure}
%  
%  Thus, to draw an `ultra thick' green  arrow  from the centre of cubie LB 
%  (say point A) to the centre of cubie ME 
%  (say point B)---see Figure~\ref{fig:facegraph}---we first determine 
%  the coordinates of $A$ \& $B$, namely  $A (0.5, 0.5)$,  $B (1.5, 1.5)$.  
%  Now, to draw the arrow from $A$ to $B$ we  just include  the following TikZ 
%  command in the \texttt{tikzpicture} environment.
%  \begin{quote}
%\begin{verbatim}
%  \draw[->,ultra thick,color=green] (0.5,0.5) -- (1.5, 1.5);
%\end{verbatim}
%  \end{quote}
%  The following example shows the  cubie changes in the \textsc{up} face 
%  generated by the rotation sequence \rrF\rrR\rrU\rrRp\rrUp\rrFp.
%  The magenta arrows indicate movement \textit{with} cubie rotations, 
%  while the black arrow indicates movement \textit{without} rotation.
%  
%  This example also highlights the fact that when there are several arrows, 
%  the start and end positions often need to be offset slightly away from cubie centres.
%  
%  \bigskip
%  \noindent
%  \begin{minipage}{2.5cm}
%  \centering
%  \begin{tikzpicture}[scale=0.7]
%  
%  \DrawRubikLayerFaceAll{W}
%  
%  \draw[->,thick,color=magenta] (1.5,0.5) -- (2.4, 1.4);
%  \draw[->,thick] (2.5,1.5) -- (1.6, 2.4);
%  \draw[->,thick,color=magenta] (1.3, 2.3) -- (1.3, 0.5);
%  \draw[<->,thick,color=magenta] (0.5,2.6) -- (2.5, 2.6);
%  \draw[<->,thick,color=magenta] (0.5,0.3) -- (2.5, 0.3);
%  \end{tikzpicture}
%  \end{minipage}
%  \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
%  \begin{tikzpicture}[scale=0.7]
%  \DrawRubikLayerFaceAll{W}
%  \draw[->,thick,color=magenta] (1.5,0.5) -- (2.4, 1.4);
%  \draw[->,thick] (2.5,1.5) -- (1.6, 2.4);
%  \draw[->,thick,color=magenta] (1.3, 2.3) -- (1.3, 0.5);
%  \draw[<->,thick,color=magenta] (0.5,2.6) -- (2.5, 2.6);
%  \draw[<->,thick,color=magenta] (0.5,0.3) -- (2.5, 0.3);
%  \end{tikzpicture}
%\end{verbatim}
%  \end{minipage}
%  
%  \bigskip
%  Since the coordinates shown in Figure~\ref{fig:facegraph} extend 
%  outwards in all directions, they can also be used as a guide for drawing 
%  arrows (or other structures) outside this 3x3 `face' square. This approach
%  is shown in the following example, where we have changed the TikZ scale 
%  to~0.4 in order to generate a small figure, and also  reduced the minipage
%  width to minimise  surrounding white-space, in order to facilitate placing the 
%  figure and the code side-by-side.
%  
%  \bigskip
%  
%  \noindent
%  \begin{minipage}{1.8cm}
%  \centering
%  \begin{tikzpicture}[scale=0.4]
%  
%  \DrawRubikLayerFace{G}{Y}{R}%
%                     {Y}{Y}{Y}%
%                     {B}{Y}{Y}%
%  
%  \DrawRubikLayerSideT {Y}{B}{B}
%  \DrawRubikLayerSideLR{R}   {Y}
%                       {R}   {O}
%                       {Y}   {O}
%  \DrawRubikLayerSideB {O}{G}{G}
%  
%  \draw[->,ultra thick,color=green] (0.5,5) -- (0.5, 4);
%  \end{tikzpicture}
%  \end{minipage}
%  \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
%  \begin{minipage}{1.8cm}
%  \centering
%  \begin{tikzpicture}[scale=0.4]
%     \DrawRubikLayerFace{G}{Y}{R}
%                        {Y}{Y}{Y}
%                        {B}{Y}{Y}
%     
%     \DrawRubikLayerSideT {Y}{B}{B}
%     \DrawRubikLayerSideLR{R}   {Y}
%                          {R}   {O}
%                          {Y}   {O}
%     \DrawRubikLayerSideB {O}{G}{G}  
%     \draw[->,ultra thick,color=green] (0.5,5) -- (0.5, 4);
%  \end{tikzpicture}
%  \end{minipage}
%\end{verbatim}
%  \end{minipage}
%  
%  \bigskip
%  
%  {\noindent}The following example shows an arrow on the Rubik cube. 
%  The origin of coordinates is at the bottom left corner of the 
%  \textsc{front} face (see Section~\ref{sec:coordinates}).
%  
%  \bigskip
%  \noindent
%  \begin{minipage}{2.8cm}
%  \centering
%  \begin{tikzpicture}[scale=0.7]
%  
%  \RubikFaceFront{O}{O}{O}
%                 {O}{O}{X}
%                 {X}{O}{X}
%  
%  \RubikFaceRight{G}{G}{G}
%                 {X}{G}{G}
%                 {X}{X}{X}
%  
%  \RubikFaceDown {X}{G}{X}
%                 {X}{Y}{X}
%                 {X}{X}{X}
%  
%  \DrawRubikCubeRD
%  \draw[ultra thick,->,color=blue] 
%            (1.5,0.5) -- (2.5, 1.5);
%  \end{tikzpicture}
%  \end{minipage}
%     \hspace{1cm}
%  \begin{minipage}{0.6\textwidth}
%\begin{verbatim}
%    \begin{tikzpicture}[scale=0.7]
%  
%    \RubikFaceFront{O}{O}{O}
%                   {O}{O}{X}
%                   {X}{O}{X}
%  
%    \RubikFaceRight{G}{G}{G}
%                   {X}{G}{G}
%                   {X}{X}{X}
%  
%    \RubikFaceDown {X}{G}{X}
%                   {X}{Y}{X}
%                   {X}{X}{X}
%  
%    \DrawRubikCubeRD
%    \draw[ultra thick,->,color=blue] 
%                (1.5,0.5) -- (2.5, 1.5);  
%  \end{tikzpicture}
%\end{verbatim}
%  \end{minipage}
%  
%
%
% \section{Final example}
%
% We now present, as a final example, the code used to draw the  front page figure:
%
% \bigskip
% \hfil
% \begin{minipage}{2cm}
% \centering
% \begin{tikzpicture}[scale=0.4]
% \RubikCubeSolved
% \DrawRubikCubeRU
% \end{tikzpicture}
% \end{minipage}
% \begin{minipage}{2cm}
% \centering
% \begin{tikzpicture}[scale=0.4]
% \DrawRubikLayerFaceAll{W}
% \DrawRubikLayerSideT {R}{R}{R}
% \DrawRubikLayerSideLR{B}   {G}
%                      {B}   {G}
%                      {B}   {G}
% \DrawRubikLayerSideB {O}{O}{O}
% \draw[thick,->,color=magenta] (1.5,0.5) -- (2.4, 1.4);
% \draw[thick,->] (2.5,1.5) -- (1.6, 2.4);
% \draw[thick,->,color=magenta] (1.3, 2.3) -- (1.3, 0.5);
% \draw[thick,<->,  color=blue] (0.5,2.6) -- (2.5, 2.6);
% \draw[thick,<->,  color=blue] (0.5,0.3) -- (2.5, 0.3);
% \end{tikzpicture}
% \end{minipage}
% \RubikF\RubikR\RubikU\RubikRp\RubikUp\RubikFp\
% \ \ $\longrightarrow$
% \begin{minipage}{2cm}
% \centering
% \begin{tikzpicture}[scale=0.4]
% \DrawRubikLayerFace{W}{W}{B}
%                    {W}{W}{O}
%                    {W}{R}{B}
% \DrawRubikLayerSideT {G}{G}{R}
% \DrawRubikLayerSideLR{R}   {W}
%                      {B}   {W}
%                      {O}   {W}
% \DrawRubikLayerSideB {G}{W}{O}
% \end{tikzpicture}
% \end{minipage}
% \hfil
%
% \bigskip
%
%\begin{verbatim}
% \hfil
% \begin{minipage}{2cm}
%   \centering
%   \begin{tikzpicture}[scale=0.4]
%     \RubikCubeSolved
%     \DrawRubikCubeRU
%   \end{tikzpicture}
% \end{minipage}
% \begin{minipage}{2cm}
%   \centering
%   \begin{tikzpicture}[scale=0.4]
%     \DrawRubikLayerFaceAll{W}
%     \DrawRubikLayerSideT {R}{R}{R}
%     \DrawRubikLayerSideLR{B}   {G}
%                          {B}   {G}
%                          {B}   {G}
%     \DrawRubikLayerSideB {O}{O}{O}
%     \draw[thick,->,color=magenta] (1.5,0.5) -- (2.4, 1.4);
%     \draw[thick,->] (2.5,1.5) -- (1.6, 2.4);
%     \draw[thick,->,color=magenta] (1.3, 2.3) -- (1.3, 0.5);
%     \draw[thick,<->,  color=blue] (0.5,2.6) -- (2.5, 2.6);
%     \draw[thick,<->,  color=blue] (0.5,0.3) -- (2.5, 0.3);
%   \end{tikzpicture}
% \end{minipage}
% \RubikF\RubikR\RubikU\RubikRp\RubikUp\RubikFp\
% \ \ $\longrightarrow$
% \begin{minipage}{2cm}
%   \centering
%   \begin{tikzpicture}[scale=0.4]
%     \DrawRubikLayerFace{W}{W}{B}
%                        {W}{W}{O}
%                        {W}{R}{B}
%     \DrawRubikLayerSideT {G}{G}{R}
%     \DrawRubikLayerSideLR{R}   {W}
%                          {B}   {W}
%                          {O}   {W}
%     \DrawRubikLayerSideB {G}{W}{O}
%   \end{tikzpicture}
% \end{minipage}
% \hfil
%\end{verbatim}
%
%
%  \section{Known issues \& shortcomings}
%
% Please contact the authors regarding any errors, shortcomings and ideas etc. 
%
% \begin{itemize} 
%  
%  \item  The rotation hieroglyphs are optimised for a 10pt font, and  do 
%    not scale  with document  font size. This may be implemented in the future, 
%   and will require careful mapping of the  TikZ scale factor  to font point 
%   size, or just replacing all length parameters by a multiple of \cmd{\em} or \cmd{\en} etc.
%   At the moment it seems to work reasonably well with both 11pt and 12pt fonts.
%
%  \item The sidebars can only be drawn in relation to  either a Front face of a 
%     cube, or a  LayerFace. 
%
%  \item The \textsc{front} face  as drawn using the  \cmd{\DrawNCubeAll} macro  does 
%    not have its origin at the bottom left corner. Not regarded as a significant 
%   problem since this macros is really just a cosmetic fix/tool for drawing a 
%   neat NxNxN cube. 
% 
% \end{itemize}
%
%
%  \section{Acknowledgements}
%  
%  We would like to thank Peter Bartal and  Peter Grill for useful ideas and  
% suggestions.   We have built on some of the initial 
%  ideas of Bartal (2011), and have cited these instances in the code documentation. 
%  
%
%
%   \section{Future supporting packages}
%  
%  Since  Rubik-type cubes currently exist in  a variety of sizes (from 2x2x2 to 11x11x11)
%  it is possible that similar packages dealing with cubes of other sizes may be made in 
%  the future.  Supporting tools  may also be  made
%  using various programming languages, eg Lua etc. Consequently  the naming of such 
%  packages and tools, and even the associated CTAN directories, needs to be given 
%  some thought in order to prevent possible confusion.
%  We suggest, therefore, the following directory structure  for CTAN might be  appropriate.
%  
%  $$
%  \textsc{rubik}  \left\{
%    \begin{array}{l} 
%    \mbox{rubiktools}\\
%    \mbox{rubik2x}\\
%    \mbox{rubik3x}\\
%    \ldots\\
%    \mbox{rubik11x}\\
%    \end{array}
%         \right.\\
%  $$
%
%    \section{References}
%  
%  
%  \begin{itemize}
%  
%  
%  \item  Bartal P (2011)
%  
%  \url{http://tex.stackexchange.com/questions/34482/}
%  
%  \item Chen JJ (2004). Group theory and the Rubik's cube.
%  \url{http://www.math.harvard.edu/~jjchen/docs/Group_Theory_and_the_Rubik's_Cube.pdf}
%  
%  \item Demaine ED, Demaine ML, Eisenstat S, Lubiw A and Winslow A (2011).
%  Algorithms for solving Rubik's cubes.
%  \url{http://www.arxiv.org/abs/1106.5736/}
%  
%  \item Garfath-Cox, A (1981). \textit{The cube},  (Bolden Publishing Co., 
%       East Molesey, Surrey) pp.32.  [copy in British Library]
%  
%  
%  \item Duvoid T (2010).
%  M\'{e}thode simple pour remonter le Rubik's cube. 
%  \newline\url{http://duvoid.fr/rubik/rubik-debutant-couleurs.pdf}
%  \newline\url{http://duvoid.fr/rubik/sources/notation_en.eps}
%  \newline\url{http://duvoid.fr/rubik/sources/rubik-debutant-couleurs.tex}
%  
%  \item Duvoid T (2011).
%  M\'{e}thode avanc\'{e}e  pour remonter le Rubik's cube. 
%  \newline\url{http://duvoid.fr/rubik/rubik-friddrich-couleurs.pdf}
%  \newline\url{http://duvoid.fr/rubik/sources/rubik-friddrich-couleurs.tex}
%  
%  \item Fridrich J. \ \  \url{http://www.ws.binghamton.edu/fridrich/}.  
%  See the useful `notation' section  on the `Pretty patterns' webpage at 
%  \url{http://www.ws.binghamton.edu/fridrich/ptrns.html}.
%
% \item Golomb SW (1982). Rubik's cube and quarks. \textit{American Scientist}; 
%    \underline{70}, pp.~257--259.

%  
%  \item Gymrek M (2009). The mathematics of toys and games.
%  \newline\url{http://wen.mit.edu/sp.268/www/rubik.pdf}
%  
%  \item Hofstadter D (1981). Rubik cube. \textit{Scientific American}; March issue.
%  
%  \item Hutchings M (2011). The mathematics of Rubik's cube (slide presentation). 
%        \url{http://math.berkeley.edu/~hutching/}
%
%
% \item Jelinek website (Jelinek J). Rubik's cube solution method. \url{http://rubikscube.info/}    
%  
%  \item Joyner D (2008). \textit{Adventures in group theory: 
%       Rubik's cube, Merlin's machine and other mathematical toys}.
%  \url{http://mike.verdone.ca/media/rubiks.pdf}
%  
%  
%  \item Kriz J and Siegel P (2008). Rubik's cube-inspired puzzles demonstrate math's simple groups.
%  \textit{Scientific American}; [? July 2008] 
%  
%  
%  \item Learntofish (2010). 
%   \url{http://learntofish.wordpress.com/2010/10/08/20-moves-suffice-to-solve-the-rubiks-cube/}
%
%  \item Randelhofer website (Randelhofer W).  Pretty patterns. 
%  \url{http://www.randelhofer.ch/rubik/patterns/U080.01.html}
%
%   \item Reid M. \ \  \url{http://www.math.ucf.edu/~reid/Rubik/}
%
%  
%  \item Rokicki T, Kociemba H, Davidson M and Dethridge J (2013). The diameter of the 
%       Rubik's cube is twenty.  \textit{SIAM.\ J.\ Discrete Math.}, \underline{27}, 1082--1105.
%  (\url{http://tomas.rokicki.com/rubik20.pdf})
%
%  \item Rubik's cube. See Section on notation. 
%  \newline\url{http://en.wikipedia.org/wiki/Rubik's_Cube}
%  
%  \item Speedsolving website. \url{www.speedsolving.com/}
%  
%  \item Tran R (2005). A mathematical approach to solving Rubik's cube.
%  \url{http://www.math.ubc.ca/~cass/courses/m308/projects/rtran/rtran.pdf}
%  
%
%  \item Treep A and  Waterman M (1987). Marc Waterman's Algorithm, Part 2. 
%    \textit{Cubism For Fun 15}, p.~10 (Nederlandse Kubus Club)
%    [cited from \textit{Wikipedia} (Rubik's cube)]

%  
%  \item Vandenbergh L. \ \ \textsc{cubezone} (\url{http://www.cubezone.be}).
%  
%  \item WCA (2012). World Cube Association Regulations. See Section~12 for notation.
%   \url{http://www.worldcubeassociation.org/regulations.htm}
%  
%  
%  \end{itemize}
%  
%
%
% ^^A ==================================================
% \StopEventually{\PrintIndex}
%
%
%
% \section{The code} 
%  
%  The conventions  we adopt regarding  capital letters and the 
%  XYZ argument ordering are detailed in  Section~\ref{sec:conventions}.
%
%
%   \subsection{    Package heading}
%
%    \begin{macrocode}
%<*rubikcube>
\def\RCfileversion{2.0}%
\def\RCfiledate{2014/01/20}%
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{rubikcube}[\RCfiledate\space (v\RCfileversion)]
%    \end{macrocode}
%  The package requires TikZ
%    \begin{macrocode}
\@ifpackageloaded{tikz}{}{%
  \typeout{---rubikcube requires the TikZ package.}%
  \RequirePackage{tikz}}%
%    \end{macrocode}
%
%    \begin{macro}{\rubikcube}
%  First we create a suitable logo
%    \begin{macrocode}
\newcommand{\rubikcube}{\textsf{rubikcube}}%
%    \end{macrocode}
%    \end{macro}
%
%
%   \subsection{    Some useful internal commands}
%
%  \begin{macro}{\@rr} 
%  \begin{macro}{\@rrp}
%  \begin{macro}{\@rrw} 
%  \begin{macro}{\@rrwp} 
%  \begin{macro}{\@rrs} 
%  \begin{macro}{\@rrsp}
%  \begin{macro}{\@rra} 
%  \begin{macro}{\@rrap} 
%  \begin{macro}{\@xyzh} 
%  \begin{macro}{\@xyzhp} 
%  \begin{macro}{\@xyzRubik} 
%  \begin{macro}{\@xyzRubikp} 
%  \begin{macro}{\@SquareLetter} 
%  \begin{macro}{\@hRubik} 
% Internal commands. These  are used to generate the
% prime, w, w-prime, s, s-prime, a, a-prime   
% rotation commands. The \cmd{\@xyz..} commands are used to generate the 
% x,y,z,u,d,l,r,f,b, and their prime rotation commands.
% The \cmd{\@SquareLetter} command is used to form the separate square hieroglyph  form
% used for rotations with no \textsc{front} representation 
% (eg B.., Fs, Fsp, Fa, Fap, S, Sp, Sf, Sfp, Sb, Sbp).
% The \cmd{\@hRubik} is the vertical shift  used  to raise the box carying the rotation 
% letter-code in \cmd{\Rubik..} commands not visible from the front.
%
%  The idea is that by using these internal
% tools we will be able to more easily standardise the  size and position 
% of all the various glyphs. However, these internal tools are currently 
% only partially implemented (= work in progress).
% \begin{macrocode}
\newcommand{\@rr}[1]{\textbf{\textsf{#1}}}
\newcommand{\@rrp}[1]{\textbf{\textsf{#1}$^\prime$}}
\newcommand{\@rrw}[1]{\textbf{\textsf{#1\footnotesize{w}}}}
\newcommand{\@rrwp}[1]{\textbf{\textsf{#1\footnotesize{w}}$^\prime$}}
\newcommand{\@rrs}[1]{\textbf{\textsf{#1\footnotesize{s}}}}
\newcommand{\@rrsp}[1]{\textbf{\textsf{#1\footnotesize{s}}$^\prime$}}
\newcommand{\@rra}[1]{\textbf{\textsf{#1\footnotesize{a}}}}
\newcommand{\@rrap}[1]{\textbf{\textsf{#1\footnotesize{a}}$^\prime$}}
\newcommand{\@xyzh}[1]{\textbf{[\textsf{#1}]}\,}
\newcommand{\@xyzhp}[1]{\textbf{[\textsf{#1}$^\prime$]}\,}
\newcommand{\@xyzRubik}[1]{\raisebox{3.45pt}{\textbf{[\textsf{#1}]}}}
\newcommand{\@xyzRubikp}[1]{\raisebox{3.45pt}{\textbf{[\textsf{#1}$^\prime$]}}}
\newcommand{\@SquareLetter}[1]{\setlength\fboxsep{2.5pt}\fboxrule=0.8pt%
   \fbox{\rule[-1pt]{0pt}{8.5pt}\raisebox{-0.5pt}{#1}}}
\newlength\@hRubik%
\setlength{\@hRubik}{0.185cm}%
%    \end{macrocode}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%
%
%   \subsection{    Colours}
%
%  These colour allocations were initially defined by Peter Bartal (2011).
%  Now modified so that grey is defined as {black!30}.
%
%    \begin{macrocode}
\definecolor{R}{HTML}{C41E33}%
\definecolor{G}{HTML}{00BE38}%
\definecolor{B}{HTML}{0051BA}%
\definecolor{Y}{HTML}{FFFF00}%
\colorlet{X}{black!30}% grey
\colorlet{O}{orange}%
\colorlet{W}{white}%
%    \end{macrocode}
%
%  \subsection{    Face commands}
%
% Cubeface notation = U,D,L,R,F,B (Singmaster)
% {\newline}Cubiesquare notation = t,b,l,r = top,bottom,left,right.
% Need to use t,b for cubiesquares to avoid confusion with cubeFace notation.
% Number the cubiesquares on a face 1-9, starting topleft, ending bottom right, as follows: 
% {\newline}\strut\hspace{1cm}Up row \ \ \ \ \ (1,2,3) = tl, tm, tr
% {\newline}\strut\hspace{1cm}middle row (4,5,6) = ml, mm, mr
% {\newline}\strut\hspace{1cm}bottom row (7,8,9) = bl, bm, br
%
%  \begin{macro}{\RubikFaceUp} 
%  \begin{macro}{\RubikFaceDown} 
%  \begin{macro}{\RubikFaceLeft} 
%  \begin{macro}{\RubikFaceRight} 
%  \begin{macro}{\RubikFaceFront}  
%  \begin{macro}{\RubikFaceBack}  
% These 5 commands allocate a colour to each of the 9 cubiesquares in the 
% specified face (Up, Down, Left, Right, Front, Back).  Each command takes 9 arguments 
% (colour codes) in the order 1--9 as specified above. 
%
% {\noindent}For example, \cmd{\RubikFaceUp\{R\}\{O\}\{Y\} \{G\}\{B\}\{W\} \{X\}\{R\}\{G\}}
%
%
% {\noindent}Each of the 9 \cmd{\def\{\}} commands below allocates one colour 
% to a specific cubiesquare, using a simple three-letter encoding. 
% Each letter is an initial letter of the words Up, Down, Left, Right, Front, Back, 
% left, middle, right, top, middle, bottom. 
% For example in the command \cmd{\Urt\{\#1\}} 
% the U stands for the Up face of the cube, while the \texttt{rt} stands 
% for the ``right-top'' cubiesquare on this face. Note that the order 
% of the two lowercase letters (in this case \texttt{rt}) are written 
% in the $x,y$ order; ie the first of the two lowercase letters relates to the $x$ 
% direction (either left, middle, or right), while the second  lowercase letter 
% relates to the $y$  direction (either top, middle, or bottom)
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro} 
%  \end{macro}
%  \end{macro}
%    \begin{macrocode}
\newcommand{\RubikFaceUp}[9]{%
\def\Ult{#1}\def\Umt{#2}\def\Urt{#3}% 
\def\Ulm{#4}\def\Umm{#5}\def\Urm{#6}%
\def\Ulb{#7}\def\Umb{#8}\def\Urb{#9}%
}   
\newcommand{\RubikFaceFront}[9]{%
\def\Flt{#1}\def\Fmt{#2}\def\Frt{#3}% 
\def\Flm{#4}\def\Fmm{#5}\def\Frm{#6}%
\def\Flb{#7}\def\Fmb{#8}\def\Frb{#9}%
}
\newcommand{\RubikFaceRight}[9]{%
\def\Rlt{#1}\def\Rmt{#2}\def\Rrt{#3}% 
\def\Rlm{#4}\def\Rmm{#5}\def\Rrm{#6}%
\def\Rlb{#7}\def\Rmb{#8}\def\Rrb{#9}%
}
\newcommand{\RubikFaceDown}[9]{%
\def\Dlt{#1}\def\Dmt{#2}\def\Drt{#3}% 
\def\Dlm{#4}\def\Dmm{#5}\def\Drm{#6}%
\def\Dlb{#7}\def\Dmb{#8}\def\Drb{#9}%
}
\newcommand{\RubikFaceLeft}[9]{%
\def\Llt{#1}\def\Lmt{#2}\def\Lrt{#3}% 
\def\Llm{#4}\def\Lmm{#5}\def\Lrm{#6}%
\def\Llb{#7}\def\Lmb{#8}\def\Lrb{#9}%
}
\newcommand{\RubikFaceBack}[9]{%
\def\Blt{#1}\def\Bmt{#2}\def\Brt{#3}% 
\def\Blm{#4}\def\Bmm{#5}\def\Brm{#6}%
\def\Blb{#7}\def\Bmb{#8}\def\Brb{#9}%
}
%    \end{macrocode}
%
%  \begin{macro}{\RubikFaceUpAll} 
%  \begin{macro}{\RubikFaceDownAll} 
%  \begin{macro}{\RubikFaceLeftAll} 
%  \begin{macro}{\RubikFaceRightAll} 
%  \begin{macro}{\RubikFaceFrontAll}  
%  \begin{macro}{\RubikFaceBackAll}  
% These 5 commands allocate the same colour to all 9 cubiesquares in the specified face (Up, Down, Left, Right, Front).  Each command therefore  takes only  1 argument (one of the colour codes).
%
% {\noindent}For example, \cmd{\RubikFaceUpAll\{R\}}
%  \end{macro}
%  \end{macro}
%  \end{macro}
%  \end{macro} 
%  \end{macro}
%  \end{macro}
%    \begin{macrocode}
\newcommand{\RubikFaceUpAll}[1]{%
\def\Ult{#1}\def\Umt{#1}\def\Urt{#1}% 
\def\Ulm{#1}\def\Umm{#1}\def\Urm{#1}%
\def\Ulb{#1}\def\Umb{#1}\def\Urb{#1}%
}
\newcommand{\RubikFaceFrontAll}[1]{%
\def\Flt{#1}\def\Fmt{#1}\def\Frt{#1}% 
\def\Flm{#1}\def\Fmm{#1}\def\Frm{#1}%
\def\Flb{#1}\def\Fmb{#1}\def\Frb{#1}%
}
\newcommand{\RubikFaceRightAll}[1]{%
\def\Rlt{#1}\def\Rmt{#1}\def\Rrt{#1}% 
\def\Rlm{#1}\def\Rmm{#1}\def\Rrm{#1}%
\def\Rlb{#1}\def\Rmb{#1}\def\Rrb{#1}%
}
\newcommand{\RubikFaceLeftAll}[1]{%
\def\Llt{#1}\def\Lmt{#1}\def\Lrt{#1}% 
\def\Llm{#1}\def\Lmm{#1}\def\Lrm{#1}%
\def\Llb{#1}\def\Lmb{#1}\def\Lrb{#1}%
}
\newcommand{\RubikFaceDownAll}[1]{%
\def\Dlt{#1}\def\Dmt{#1}\def\Drt{#1}% 
\def\Dlm{#1}\def\Dmm{#1}\def\Drm{#1}%
\def\Dlb{#1}\def\Dmb{#1}\def\Drb{#1}%
}
\newcommand{\RubikFaceBackAll}[1]{%
\def\Blt{#1}\def\Bmt{#1}\def\Brt{#1}% 
\def\Blm{#1}\def\Bmm{#1}\def\Brm{#1}%
\def\Blb{#1}\def\Bmb{#1}\def\Brb{#1}%
}
%    \end{macrocode}
%
% {\noindent}We now use these commands to initialise all visible faces to  default colour grey (X)
%
% \begin{macrocode}
\RubikFaceUpAll{X}%
\RubikFaceDownAll{X}%
\RubikFaceLeftAll{X}%
\RubikFaceRightAll{X}%
\RubikFaceFrontAll{X}%
\RubikFaceBackAll{X}%
%    \end{macrocode}
%
%
%   \subsection{  RubikCubeSolved command}
%
%    \begin{macro}{\RubikCubeSolved}
% This command sets the face/colour configuration (state) of a  typical 
% solved Rubik cube. 
%    \begin{macrocode}
\newcommand{\RubikCubeSolved}{%
  \RubikFaceUpAll{W}%
  \RubikFaceDownAll{Y}%
  \RubikFaceLeftAll{B}%
  \RubikFaceRightAll{G}%
  \RubikFaceFrontAll{O}%
  \RubikFaceBackAll{R}%
}
%    \end{macrocode}
%    \end{macro}
%
%
% \subsection{    Slice macros}
%
% \begin{macro}{\RubikSliceTopR} 
% \begin{macro}{\RubikSliceTopL} 
% \begin{macro}{\RubikSliceEquatorR} 
% \begin{macro}{\RubikSliceEquatorL} 
% \begin{macro}{\RubikSliceBottomR} 
% \begin{macro}{\RubikSliceBottomL} 
% These 6 commands allocate colours for the 6 visible ordered 
% cubiesquares along the side  of  a horizontal slice  
% ordered 1--6 from left to right. We require  both  L view  
% and R view versions.
%    \begin{macrocode}
\newcommand{\RubikSliceTopR}[6]{% 
\def\Flt{#1}\def\Fmt{#2}\def\Frt{#3}% 
\def\Rlt{#4}\def\Rmt{#5}\def\Rrt{#6}%
} 
\newcommand{\RubikSliceTopL}[6]{% 
\def\Llt{#1}\def\Lmt{#2}\def\Lrt{#3}% 
\def\Flt{#4}\def\Fmt{#5}\def\Frt{#6}%
} 
\newcommand{\RubikSliceEquatorR}[6]{% 
\def\Flm{#1}\def\Fmm{#2}\def\Frm{#3}%
\def\Rlm{#4}\def\Rmm{#5}\def\Rrm{#6}%
} 
\newcommand{\RubikSliceEquatorL}[6]{% 
\def\Llm{#1}\def\Lmm{#2}\def\Lrm{#3}%
\def\Flm{#4}\def\Fmm{#5}\def\Frm{#6}%
} 
\newcommand{\RubikSliceBottomR}[6]{% 
\def\Flb{#1}\def\Fmb{#2}\def\Frb{#3}%
\def\Rlb{#4}\def\Rmb{#5}\def\Rrb{#6}%
} 
\newcommand{\RubikSliceBottomL}[6]{% 
\def\Llb{#1}\def\Lmb{#2}\def\Lrb{#3}%
\def\Flb{#4}\def\Fmb{#5}\def\Frb{#6}%
}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
%
%      \subsection{    Cube drawing macros}
%
% Since the three visible sides of a Rubik cube have up to  27  non-grey colours, 
% and \TeX\ has only 9 macro  parameters available, we are forced to draw 
% Rubik cubes  by first specifying the colours on each of the three faces, 
% and then  using a `DrawRubikCubeXY' command, where the trailing XY code
%  defines the  view direction (X = either R, L; Y = either U, D). 
% The order of the XY code is important: X first, Y second.
%
% On each face the cubies are drawn in the following order: Top row 
% (left to right), Middle row (left to right), Bottom row (left to right).
%
% The TikZ  draw cycle for each cubie square on a Rubik cube face cycles 
% through the  four corners of the cubie in the following order: 
% lb $\rightarrow$ lt $\rightarrow$ rt $\rightarrow$ rb;  the code being lb (LeftBottom), 
% lt (LeftTop), rt (RightTop), rb (RightBottom)
% (only need four coords) origin at bottom left of front face.
%
%  \begin{macro}{\DrawRubikCubeFrontFace}
% This `FrontFace' command  is an `internal' command  which draws and paints 
% all the cubiesquares on the front face of a cube.  It is used by all 
% of the  cube drawing macros which display the front face. 
% The 9 colours are allocated by an earlier \cmd{\RubikFaceFront} command.
% These Face macros are based, in part, on those of Peter Bartal (2011).
%    \begin{macrocode}
\newcommand{\DrawRubikCubeFrontFace}{%
%  ---top row left to right
\draw[line join=round,line cap=round,ultra thick,fill=\Flt]%
(0,2) -- (0, 3) -- (1,3) -- (1,2)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Fmt]%
(1,2) -- (1, 3) -- (2,3) -- (2,2) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Frt]%
(2,2) -- (2, 3) -- (3,3) -- (3,2) -- cycle;
% -----middle row left to right
\draw[line join=round,line cap=round,ultra thick,fill=\Flm]%
(0,1) -- (0, 2) -- (1,2) -- (1,1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Fmm]%
(1,1) -- (1, 2) -- (2,2) -- (2,1) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Frm]%
(2,1) -- (2, 2) -- (3,2) -- (3,1) -- cycle;
% ----bottom row left to right
\draw[line join=round,line cap=round,ultra thick,fill=\Flb]%
(0,0) -- (0, 1) -- (1,1) -- (1,0)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Fmb]%
(1,0) -- (1, 1) -- (2,1) -- (2,0) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Frb]%
(2,0) -- (2, 1) -- (3,1) -- (3,0) -- cycle;
}
%    \end{macrocode}
% \end{macro}
%
%      \subsubsection{    Viewing direction}
%
%  The command  `DrawRubikCubeXY' command uses a trailing XY code
%  to specify  the  view direction (X = either R, L; Y = either U, D). 
%  The order of the XY code is important: X first, Y second (so its easy
%  to remember).
%
%  \begin{macro}{\DrawRubikCubeRU}
% This command draws and paints a Rubik cube  as viewed from 
% the Right Upper (RU) viewpoint. It starts by using the internal 
% command \cmd{\DrawRubikCubeFrontFace} to draw the Front face, 
% and then draws the Up face  followed by the Right face.
% The colours are allocated by  the last \cmd{\RubikFaceUp}
% and \cmd{\RubikFaceRight}  commands. 
%    \end{macro}
%    \begin{macrocode}
\newcommand{\DrawRubikCubeRU}{%
\DrawRubikCubeFrontFace %% frontface
%%-----------Up face----------
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Ult]%
(0.66,3.66) -- (1,4) -- (2,4) -- (1.66,3.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Umt]%
(1.66,3.66) -- (2,4) -- (3,4) -- (2.66,3.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Urt]%
(2.66,3.66) -- (3,4) -- (4,4) -- (3.66,3.66)  -- cycle;
%%---middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Ulm]%
(0.33,3.33) -- (0.66,3.66) -- (1.66,3.66) -- (1.33,3.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Umm]%
(1.33,3.33) -- (1.66,3.66) -- (2.66,3.66) -- (2.33,3.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Urm]%
(2.33,3.33) -- (2.66,3.66) -- (3.66,3.66) -- (3.33,3.33)  -- cycle;
%%---bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Ulb]%
(0,3) -- (0.33,3.33) -- (1.33,3.33) -- (1,3)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Umb]%
(1,3) -- (1.33,3.33) -- (2.33,3.33) -- (2,3)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Urb]%
(2,3) -- (2.33,3.33) -- (3.33,3.33) -- (3,3)  -- cycle;
%%-----------Right face----------
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Rlt]%
(3,2) -- (3, 3) -- (3.33,3.33) -- (3.33,2.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rmt]%
(3.33,2.33) -- (3.33, 3.33) -- (3.66,3.66) -- (3.66,2.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rrt]%
(3.66,2.66) -- (3.66, 3.66) -- (4,4) -- (4,3)  -- cycle;
%%---middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Rlm]%
(3,1) -- (3, 2) -- (3.33,2.33) -- (3.33,1.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rmm]%
(3.33,1.33) -- (3.33, 2.33) -- (3.66,2.66) -- (3.66,1.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rrm]%
(3.66,1.66) -- (3.66, 2.66) -- (4,3) -- (4,2)  -- cycle;
%%---bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Rlb]%
(3,0) -- (3, 1) -- (3.33,1.33) -- (3.33,0.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rmb]%
(3.33,0.33) -- (3.33, 1.33) -- (3.66,1.66) -- (3.66,0.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rrb]%
(3.66,0.66) -- (3.66, 1.66) -- (4,2) -- (4,1)  -- cycle;
}
%    \end{macrocode}
%
% \begin{macro}{\DrawRubikCube}
% This command  is equivalent to the previous \cmd{\DrawRubikCubeRU}
% and hence is the default form (ie if one omits the trailing XY 
% viewpoint code).
% \end{macro}
%    \begin{macrocode}
\newcommand{\DrawRubikCube}{\DrawRubikCubeRU}
%    \end{macrocode}
%
%
% \begin{macro}{\DrawRubikCubeRD}
% This command draws and paints a Rubik cube  as viewed from 
% the Right Down (RD) viewpoint.
% \end{macro}
%    \begin{macrocode}
\newcommand{\DrawRubikCubeRD}{%
\DrawRubikCubeFrontFace %% frontface
%%----------Right face--------
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Rlt]%
(3,2) -- (3, 3) -- (3.33,2.66) -- (3.33,1.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rmt]%
(3.33,1.66) -- (3.33, 2.66) -- (3.66,2.33) -- (3.66,1.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rrt]%
(3.66,1.33) -- (3.66, 2.33) -- (4,2) -- (4,1)  -- cycle;
%%---middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Rlm]%
(3,1) -- (3, 2) -- (3.33,1.66) -- (3.33,0.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rmm]%
(3.33,0.66) -- (3.33, 1.66) -- (3.66,1.33) -- (3.66,0.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rrm]%
(3.66,0.33) -- (3.66, 1.33) -- (4,1) -- (4,0)  -- cycle;
%%---bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Rlb]%
(3,0) -- (3, 1) -- (3.33,0.66) -- (3.33,-0.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rmb]%
(3.33,-0.33) -- (3.33, 0.66) -- (3.66,0.33) -- (3.66,-0.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rrb]%
(3.66,-0.66) -- (3.66, 0.33) -- (4,0) -- (4,-1)  -- cycle;
%%-----------Down face---------
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Dlt]%
(0.33,-0.33) -- (0, 0) -- (1,0) -- (1.33,-0.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Dmt]%
(1.33,-0.33) -- (1, 0) -- (2,0) -- (2.33,-0.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Drt]%
(2.33,-0.33) -- (2, 0) -- (3,0) -- (3.33,-0.33)  -- cycle;
%%---middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Dlm]%
(0.66,-0.66) -- (0.33, -0.33) -- (1.33,-0.33) -- (1.66,-0.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Dmm]%
(1.66,-0.66) -- (1.33, -0.33) -- (2.33,-0.33) -- (2.66,-0.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Drm]%
(2.66,-0.66) -- (2.33, -0.33) -- (3.33,-0.33) -- (3.66,-0.66)  -- cycle;
%%---bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Dlb]%
(1,-1) -- (0.66, -0.66) -- (1.66,-0.66) -- (2,-1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Dmb]%
(2,-1) -- (1.66, -0.66) -- (2.66,-0.66) -- (3,-1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Drb]%
(3,-1) -- (2.66, -0.66) -- (3.66,-0.66) -- (4,-1)  -- cycle;
}
%    \end{macrocode}
%
% \begin{macro}{\DrawRubikCubeLD}
% This command draws and paints a Rubik cube  as viewed from 
% the Left Down (LD) viewpoint.
% \end{macro}
%    \begin{macrocode}
\newcommand{\DrawRubikCubeLD}{%
\DrawRubikCubeFrontFace %% frontface
%%------------Left face--------
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Llt]%
(-1,1) -- (-1, 2) -- (-0.66,2.33) -- (-0.66,1.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lmt]%
(-0.66,1.33) -- (-0.66, 2.33) -- (-0.33,2.66) -- (-0.33,1.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lrt]%
(-0.33,1.66) -- (-0.33, 2.66) -- (0,3) -- (0,2)  -- cycle;
%%---middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Llm]%
(-1,0) -- (-1, 1) -- (-0.66,1.33) -- (-0.66,0.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lmm]%
(-0.66,0.33) -- (-0.66, 1.33) -- (-0.33,1.66) -- (-0.33,0.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lrm]%
(-0.33,0.66) -- (-0.33, 1.66) -- (0,2) -- (0,1)  -- cycle;
%%---bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Llb]%
(-1,-1) -- (-1, 0) -- (-0.66,0.33) -- (-0.66,-0.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lmb]%
(-0.66,-0.66) -- (-0.66, 0.33) -- (-0.33,0.66) -- (-0.33,-0.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lrb]%
(-0.33,-0.33) -- (-0.33, 0.66) -- (0,1) -- (0,0)  -- cycle;
%%------------Down face----------
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Dlt]%
(-0.33,-0.33) -- (0, 0) -- (1,0) -- (0.66,-0.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Dmt]%
(0.66,-0.33) -- (1, 0) -- (2,0) -- (1.66,-0.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Drt]%
(1.66,-0.33) -- (2, 0) -- (3,0) -- (2.66,-0.33)  -- cycle;
%%---middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Dlm]%
(-0.66,-0.66) -- (-0.33, -0.33) -- (0.66,-0.33) -- (0.33,-0.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Dmm]%
(0.33,-0.66) -- (0.66, -0.33) -- (1.66,-0.33) -- (1.33,-0.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Drm]%
(1.33,-0.66) -- (1.66, -0.33) -- (2.66,-0.33) -- (2.33,-0.66)  -- cycle;
%%---bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Dlb]%
(-1,-1) -- (-0.66, -0.66) -- (0.33,-0.66) -- (0,-1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Dmb]%
(0,-1) -- (0.33, -0.66) -- (1.33,-0.66) -- (1,-1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Drb]%
(1,-1) -- (1.33, -0.66) -- (2.33,-0.66) -- (2,-1)  -- cycle;
}
%    \end{macrocode}
%
%
% \begin{macro}{\DrawRubikCubeLU}
% This command draws and paints a Rubik cube  as viewed from 
% the Left Up (LU) viewpoint.
% \end{macro}
%    \begin{macrocode}
\newcommand{\DrawRubikCubeLU}{%
\DrawRubikCubeFrontFace %% frontface
%%-----------Left face-----------
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Llt]%
(-1,3) -- (-1, 4) -- (-0.66,3.66) -- (-0.66,2.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lmt]%
(-0.66,2.66) -- (-0.66, 3.66) -- (-0.33,3.33) -- (-0.33,2.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lrt]%
(-0.33,2.33) -- (-0.33, 3.33) -- (0,3) -- (0,2)  -- cycle;
%%---middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Llm]%
(-1,2) -- (-1, 3) -- (-0.66,2.66) -- (-0.66,1.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lmm]%
(-0.66,1.66) -- (-0.66, 2.66) -- (-0.33,2.33) -- (-0.33,1.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lrm]%
(-0.33,1.33) -- (-0.33, 2.33) -- (0,2) -- (0,1)  -- cycle;
%%---bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Llb]%
(-1,1) -- (-1, 2) -- (-0.66,1.66) -- (-0.66,0.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lmb]%
(-0.66,0.66) -- (-0.66, 1.66) -- (-0.33,1.33) -- (-0.33,0.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lrb]%
(-0.33,0.33) -- (-0.33, 1.33) -- (0,1) -- (0,0)  -- cycle;
%%-----------Up face---------
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Ult]%
(-0.66,3.66) -- (-1, 4) -- (0,4) -- (0.33,3.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Umt]%
(0.33,3.66) -- (0, 4) -- (1,4) -- (1.33,3.66)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Urt]%
(1.33,3.66) -- (1, 4) -- (2,4) -- (2.33,3.66)  -- cycle;
%%---middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Ulm]%
(-0.33,3.33) -- (-0.66, 3.66) -- (0.33,3.66) -- (0.66,3.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Umm]%
(0.66,3.33) -- (0.33, 3.66) -- (1.33,3.66) -- (1.66,3.33)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Urm]%
(1.66,3.33) -- (1.33, 3.66) -- (2.33,3.66) -- (2.66,3.33)  -- cycle;
%%---bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Ulb]%
(0,3) -- (-0.33, 3.33) -- (0.66,3.33) -- (1,3)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Umb]%
(1,3) -- (0.66, 3.33) -- (1.66,3.33) -- (2,3)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Urb]%
(2,3) -- (1.66, 3.33) -- (2.66,3.33) -- (3,3)  -- cycle;%
\ %%trailing space 
}
%    \end{macrocode}
%
%
%   \subsection{    LayerFace commands}
%
%  \begin{macro}{\DrawRubikLayerFace}
%  \begin{macro}{\DrawRubikLayerFaceAll}
%  [Nov 2, 2013] These two LayerFace commands draw and paint a single  
%  9-cubie face. The first command takes 9 ordered colour parameters,
%  (ordered in layers from top left to bottom right, so \texttt{\#1} 
%  is the placeholder for the colour of the TopLeft cubie etc.)
%  The second takes only one colour parameter (since all the 
%  colours are the same).
%  The drawing origin (0,0) = bottom left corner. Cubies are drawn
%  from left to right.
%  NOTE: this macro is SAME as the  internal command 
%  \cmd{\DrawRubikCubeFrontFace}
%    which is used for drawing the front face of a  cube.
% \end{macro}
% \end{macro}
%    \begin{macrocode}
\newcommand{\DrawRubikLayerFace}[9]{%
%%-----------FRONT face---------
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
(0,2) -- (0, 3) -- (1,3) -- (1,2)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#2]%
(1,2) -- (1, 3) -- (2,3) -- (2,2) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#3]%
(2,2) -- (2, 3) -- (3,3) -- (3,2) -- cycle;
%%-----middle row
\draw[line join=round,line cap=round,ultra thick,fill=#4]%
(0,1) -- (0, 2) -- (1,2) -- (1,1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#5]%
(1,1) -- (1, 2) -- (2,2) -- (2,1) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#6]%
(2,1) -- (2, 2) -- (3,2) -- (3,1) -- cycle;
%%----bottom row
\draw[line join=round,line cap=round,ultra thick,fill=#7]%
(0,0) -- (0, 1) -- (1,1) -- (1,0)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#8]%
(1,0) -- (1, 1) -- (2,1) -- (2,0) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#9]%
(2,0) -- (2, 1) -- (3,1) -- (3,0) -- cycle;
}
\newcommand{\DrawRubikLayerFaceAll}[1]{%
%%----------FRONT face-----------
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
(0,2) -- (0, 3) -- (1,3) -- (1,2)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
(1,2) -- (1, 3) -- (2,3) -- (2,2) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
(2,2) -- (2, 3) -- (3,3) -- (3,2) -- cycle;
%%-----middle row
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
(0,1) -- (0, 2) -- (1,2) -- (1,1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
(1,1) -- (1, 2) -- (2,2) -- (2,1) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
(2,1) -- (2, 2) -- (3,2) -- (3,1) -- cycle;
%%----bottom row
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
(0,0) -- (0, 1) -- (1,1) -- (1,0)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
(1,0) -- (1, 1) -- (2,1) -- (2,0) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
(2,0) -- (2, 1) -- (3,1) -- (3,0) -- cycle;
}
%    \end{macrocode}
%
%
%   \subsection{    Flat commands}
%
%  \begin{macro}{\FlatUp}
%  \begin{macro}{\FlatDown}
%  \begin{macro}{\FlatLeft}
%  \begin{macro}{\FlatRight}
%  \begin{macro}{\FlatFront}
%  \begin{macro}{\FlatBack}
%  Each of these six internal `Flat'  commands draws a separate face.
%  Each command (except \cmd{\FlatFront}) takes two arguments, 
%  namely an  X-coordinate  and Y-coordinate of the bottom left 
%  corner of the  face.  This (X,Y) pair allows the user to position
%  the face. Note that the \cmd{\FlatFront} command takes no arguments,
%  since by definition the bottom left corner of this face is (0,0).
%
%  \textsc{example}: The following command positions the Up face so 
%  that its bottom left corner is located at (0,3): 
%  \begin{quote}
%    \cmd{\FlatUp\{0\}\{3\}}
%  \end{quote}
%  These internal commands are used  by the commands \cmd{\DrawRubikFlat}  
%  and \cmd{\DrawRubikCubeFlat} to draw various flat representations 
%  of a Rubik's cube.
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%    \begin{macrocode}
\newcommand{\FlatUp}[2]{%
\pgfmathsetmacro{\ux}{#1}%
\pgfmathsetmacro{\uy}{#2}%
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Ult]%
(\ux + 0,\uy + 2) -- (\ux + 0,\uy + 3) -- (\ux + 1,\uy + 3)%
 -- (\ux + 1,\uy + 2)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Umt]%
(\ux + 1,\uy + 2) -- (\ux + 1,\uy + 3) -- (\ux + 2,\uy + 3)%
 -- (\ux + 2,\uy + 2) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Urt]%
(\ux + 2,\uy + 2) -- (\ux + 2,\uy + 3) -- (\ux + 3,\uy + 3)%
 -- (\ux + 3,\uy + 2) -- cycle;
%%-----middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Ulm]%
(\ux + 0,\uy + 1) -- (\ux + 0,\uy + 2) -- (\ux + 1,\uy + 2)%
 -- (\ux + 1,\uy + 1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Umm]%
(\ux + 1,\uy + 1) -- (\ux + 1,\uy + 2) -- (\ux + 2,\uy + 2)%
 -- (\ux + 2,\uy + 1) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Urm]%
(\ux + 2,\uy + 1) -- (\ux + 2,\uy + 2) -- (\ux + 3,\uy + 2)%
 -- (\ux + 3,\uy + 1) -- cycle;
%%----bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Ulb]%
(\ux + 0,\uy + 0) -- (\ux + 0,\uy + 1) -- (\ux + 1,\uy + 1)%
 -- (\ux + 1,\uy + 0)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Umb]%
(\ux + 1,\uy + 0) -- (\ux + 1,\uy + 1) -- (\ux + 2,\uy + 1)%
 -- (\ux + 2,\uy + 0) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Urb]%
(\ux + 2,\uy + 0) -- (\ux + 2,\uy + 1) -- (\ux + 3,\uy + 1)%
 -- (\ux + 3,\uy + 0) -- cycle;
}
%%-------------------------
\newcommand{\FlatDown}[2]{%
\pgfmathsetmacro{\ddx}{#1}%
\pgfmathsetmacro{\ddy}{#2}%
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Dlt]%
(\ddx + 0,\ddy + 2) -- (\ddx + 0,\ddy + 3) -- (\ddx + 1,\ddy + 3)%
 -- (\ddx + 1,\ddy + 2)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Dmt]%
(\ddx + 1,\ddy + 2) -- (\ddx + 1,\ddy + 3) -- (\ddx + 2,\ddy + 3)%
 -- (\ddx + 2,\ddy + 2) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Drt]%
(\ddx + 2,\ddy + 2) -- (\ddx + 2,\ddy + 3) -- (\ddx + 3,\ddy + 3)%
 -- (\ddx + 3,\ddy + 2) -- cycle;
%%-----middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Dlm]%
(\ddx + 0,\ddy + 1) -- (\ddx + 0,\ddy + 2) -- (\ddx + 1,\ddy + 2)%
 -- (\ddx + 1,\ddy + 1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Dmm]%
(\ddx + 1,\ddy + 1) -- (\ddx + 1,\ddy + 2) -- (\ddx + 2,\ddy + 2)%
 -- (\ddx + 2,\ddy + 1) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Drm]%
(\ddx + 2,\ddy + 1) -- (\ddx + 2,\ddy + 2) -- (\ddx + 3,\ddy + 2)%
 -- (\ddx + 3,\ddy + 1) -- cycle;
%%----bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Dlb]%
(\ddx + 0,\ddy + 0) -- (\ddx + 0,\ddy + 1) -- (\ddx + 1,\ddy + 1)%
 -- (\ddx + 1,\ddy + 0)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Dmb]%
(\ddx + 1,\ddy + 0) -- (\ddx + 1,\ddy + 1) -- (\ddx + 2,\ddy + 1)%
 -- (\ddx + 2,\ddy + 0) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Drb]%
(\ddx + 2,\ddy + 0) -- (\ddx + 2,\ddy + 1) -- (\ddx + 3,\ddy + 1)%
 -- (\ddx + 3,\ddy + 0) -- cycle;
}
%%-----------------------
\newcommand{\FlatFront}{%
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Flt]%
(0,2) -- (0, 3) -- (1,3) -- (1,2)  -- cycle;
%%
\draw[line join=round,line cap=round,ultra thick,fill=\Fmt]%
(1,2) -- (1, 3) -- (2,3) -- (2,2) -- cycle;
%%
\draw[line join=round,line cap=round,ultra thick,fill=\Frt]%
(2,2) -- (2, 3) -- (3,3) -- (3,2) -- cycle;
%%-----middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Flm]%
(0,1) -- (0, 2) -- (1,2) -- (1,1)  -- cycle;
%%
\draw[line join=round,line cap=round,ultra thick,fill=\Fmm]%
(1,1) -- (1, 2) -- (2,2) -- (2,1) -- cycle;
%%
\draw[line join=round,line cap=round,ultra thick,fill=\Frm]%
(2,1) -- (2, 2) -- (3,2) -- (3,1) -- cycle;
%%----bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Flb]%
(0,0) -- (0, 1) -- (1,1) -- (1,0)  -- cycle;
%%
\draw[line join=round,line cap=round,ultra thick,fill=\Fmb]%
(1,0) -- (1, 1) -- (2,1) -- (2,0) -- cycle;
%%
\draw[line join=round,line cap=round,ultra thick,fill=\Frb]%
(2,0) -- (2, 1) -- (3,1) -- (3,0) -- cycle;
}
%%-------------------------
\newcommand{\FlatBack}[2]{%
\pgfmathsetmacro{\bx}{#1}%
\pgfmathsetmacro{\by}{#2}%
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Blt]%
(\bx + 0,\by + 2) -- (\bx + 0,\by + 3) -- (\bx + 1,\by + 3)%
 -- (\bx + 1,\by + 2)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Bmt]%
(\bx + 1,\by + 2) -- (\bx + 1,\by + 3) -- (\bx + 2,\by + 3)%
 -- (\bx + 2,\by + 2) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Brt]%
(\bx + 2,\by + 2) -- (\bx + 2,\by + 3) -- (\bx + 3,\by + 3)%
 -- (\bx + 3,\by + 2) -- cycle;
%%-----middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Blm]%
(\bx + 0,\by + 1) -- (\bx + 0,\by + 2) -- (\bx + 1,\by + 2)%
 -- (\bx + 1,\by + 1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Bmm]%
(\bx + 1,\by + 1) -- (\bx + 1,\by + 2) -- (\bx + 2,\by + 2)%
 -- (\bx + 2,\by + 1) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Brm]%
(\bx + 2,\by + 1) -- (\bx + 2,\by + 2) -- (\bx + 3,\by + 2)%
 -- (\bx + 3,\by + 1) -- cycle;
%%----bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Blb]%
(\bx + 0,\by + 0) -- (\bx + 0,\by + 1) -- (\bx + 1,\by + 1)%
 -- (\bx + 1,\by + 0)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Bmb]%
(\bx + 1,\by + 0) -- (\bx + 1,\by + 1) -- (\bx + 2,\by + 1)%
 -- (\bx + 2,\by + 0) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Brb]%
(\bx + 2,\by + 0) -- (\bx + 2,\by + 1) -- (\bx + 3,\by + 1)%
 -- (\bx + 3,\by + 0) -- cycle;
}
%%-------------------------
\newcommand{\FlatLeft}[2]{%
\pgfmathsetmacro{\lx}{#1}%
\pgfmathsetmacro{\ly}{#2}%
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Llt]%
(\lx + 0,2) -- (\lx + 0, 3) -- (\lx + 1,3) -- (\lx + 1,2)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lmt]%
(\lx + 1,2) -- (\lx + 1, 3) -- (\lx + 2,3) -- (\lx + 2,2) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lrt]%
(\lx + 2,2) -- (\lx + 2, 3) -- (\lx + 3,3) -- (\lx + 3,2) -- cycle;
%%-----middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Llm]%
(\lx + 0,1) -- (\lx + 0, 2) -- (\lx + 1,2) -- (\lx + 1,1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lmm]%
(\lx + 1,1) -- (\lx + 1, 2) -- (\lx + 2,2) -- (\lx + 2,1) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lrm]%
(\lx + 2,1) -- (\lx + 2, 2) -- (\lx + 3,2) -- (\lx + 3,1) -- cycle;
%%----bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Llb]%
(\lx + 0,0) -- (\lx + 0, 1) -- (\lx + 1,1) -- (\lx + 1,0)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lmb]%
(\lx + 1,0) -- (\lx + 1, 1) -- (\lx + 2,1) -- (\lx + 2,0) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Lrb]%
(\lx + 2,0) -- (\lx + 2, 1) -- (\lx + 3,1) -- (\lx + 3,0) -- cycle;
}
%%--------------------------
\newcommand{\FlatRight}[2]{%
\pgfmathsetmacro{\rx}{#1}% %3
\pgfmathsetmacro{\ry}{#2}% %0
%%---top row
\draw[line join=round,line cap=round,ultra thick,fill=\Rlt]%
(\rx + 0,2) -- (\rx + 0, 3) -- (\rx + 1,3) -- (\rx + 1,2)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rmt]%
(\rx + 1,2) -- (\rx + 1, 3) -- (\rx + 2,3) -- (\rx + 2,2) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rrt]%
(\rx + 2,2) -- (\rx + 2, 3) -- (\rx + 3,3) -- (\rx + 3,2) -- cycle;
%%-----middle row
\draw[line join=round,line cap=round,ultra thick,fill=\Rlm]%
(\rx + 0,1) -- (\rx + 0, 2) -- (\rx + 1,2) -- (\rx + 1,1)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rmm]%
(\rx + 1,1) -- (\rx + 1, 2) -- (\rx + 2,2) -- (\rx + 2,1) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rrm]%
(\rx + 2,1) -- (\rx + 2, 2) -- (\rx + 3,2) -- (\rx + 3,1) -- cycle;
%%----bottom row
\draw[line join=round,line cap=round,ultra thick,fill=\Rlb]%
(\rx + 0,0) -- (\rx + 0, 1) -- (\rx + 1,1) -- (\rx + 1,0)  -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rmb]%
(\rx + 1,0) -- (\rx + 1, 1) -- (\rx + 2,1) -- (\rx + 2,0) -- cycle;
\draw[line join=round,line cap=round,ultra thick,fill=\Rrb]%
(\rx + 2,0) -- (\rx + 2, 1) -- (\rx + 3,1) -- (\rx + 3,0) -- cycle;
}
%    \end{macrocode}
%
%  \begin{macro}{\DrawRubikFlat}
%  Draws a standard flat representation of the Rubik's 
%  cube (colours only)
%    \begin{macrocode}
\newcommand{\DrawRubikFlat}{%
  \FlatUp{0}{3}%
  \FlatDown{0}{-3}%
  \FlatLeft{-3}{0}%
  \FlatFront%
  \FlatRight{3}{0}%
  \FlatBack{6}{0}%
}
%    \end{macrocode}
%  \end{macro}
%
%  \begin{macro}{\DrawRubikCubeFlat}
%  Draws a Rubik's  cube together with the three hidden faces
%   (colours only)
%    \begin{macrocode}
\newcommand{\DrawRubikCubeFlat}{%
  \DrawRubikCube%
  \FlatDown{0}{-3}%
  \FlatLeft{-3}{0}%
  \FlatBack{4}{1}%
}
%    \end{macrocode}
%  \end{macro}
%
%
%   \subsection{    SideBar commands}
%
%  SideBar commands draw narrow bars of colour indicating  the 
%  side colours of each of the cubies forming the side of
%  a given layer (face).
%
%  \begin{macro}{\RubikSideBarWidth}
%  \begin{macro}{\RubikSideBarLength} 
%  \begin{macro}{\RubikSideBarSep}
% These three commands allow the user to set the Width, Length 
% and Separation parameters for the sidebar (in decimal values, 
% where 1 is equivalent to the length of the side of a cubie).
%  \end{macro}
%  \end{macro}
%  \end{macro}
%    \begin{macrocode}
\newcommand{\RubikSideBarWidth}[1]{\pgfmathsetmacro{\bw}{#1}}
\newcommand{\RubikSideBarLength}[1]{\pgfmathsetmacro{\bl}{#1}}
\newcommand{\RubikSideBarSep}[1]{\pgfmathsetmacro{\bs}{#1}}
%    \end{macrocode}
% We first set some default values
%    \begin{macrocode}
\RubikSideBarWidth{0.3}%
\RubikSideBarLength{1}%
\RubikSideBarSep{0.3}%
%    \end{macrocode}
%
%
% \subsubsection{    Allocating  a colour to a single cubie sidebar}
% 
% \begin{macro}{\side@barT}
% \begin{macro}{\side@barB}
% \begin{macro}{\side@barL}
% \begin{macro}{\side@barR}
% Internal commands.
% Full length face SideBars are really multiple instances of 
% single cubie bars, 
% each of which is drawn  using one of four internal SideBar 
% commands---one for each of the sides which we shall 
% call Top, Bottom,  Left, Right.
% Each SideBar command  takes two arguments: one for cubie position 
% \marg{$ 1 \mid 2 \mid 3$} and  one for 
% the colour-code \marg{$ R \mid O \mid Y \mid G \mid B \mid W \mid X $}.
%
% \textsc{example}: the following command allocates a colour to a single 
% cubie sidebar on the Left of a Rubik face: 
%  \begin{quote}
%     \cmd{\side@barL}\marg{cubie-position}\marg{colour-code}
%  \end{quote}
% There are three cubie positions on  each of the four sides 
% of a face, and these 
% are numbered 1 to 3 starting from the bottom left corner (1,1). 
% The SideBar command  also implements the set (or default) 
% Length (\cmd{\bl}), Width (\cmd{\bw}) and Separation (\cmd{\bs}) 
% values mentioned above. \cmd{\blh} = Half \cmd{\bl} = \cmd{\bl}/2.
% Note that the TikZ \cmd{\pgfmathsetmacro} commands (which do the maths)
%  must be inside  the TeX sidebar command in order to work.
% The start point of Draw command for  each bar rectangle 
%    is botton Left corner of the bar =(\cmd{\dx},\cmd{\dy})
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
%    \begin{macrocode}
\newcommand{\side@barL}[2]{%
%% #1 = cubie possn no, #2 = colour
\pgfmathsetmacro{\blh}{\bl*(0.5)}%
\pgfmathsetmacro{\dx}{0 - \bs - \bw}%
\pgfmathsetmacro{\dy}{#1-1+0.5-\blh}%
\draw[fill=#2] (\dx,\dy) -- (\dx,\dy + \bl) 
  -- (\dx+\bw,\dy+\bl) -- (\dx+\bw,\dy) -- cycle;
}
\newcommand{\side@barR}[2]{%
%% #1 = cubie possn no, #2 = colour
\pgfmathsetmacro{\blh}{\bl*(0.5)}%
\pgfmathsetmacro{\dx}{3 + \bs}%
\pgfmathsetmacro{\dy}{#1 -1+0.5-\blh}%
\draw[fill=#2] (\dx,\dy) -- (\dx,\dy + \bl)
  -- (\dx+\bw,\dy+\bl) -- (\dx+\bw,\dy) -- cycle;
}
\newcommand{\side@barT}[2]{%
%% #1 = cubie possn no, #2 = colour
\pgfmathsetmacro{\blh}{\bl*(0.5)}%
\pgfmathsetmacro{\dx}{#1 -1+0.5-\blh}%
\pgfmathsetmacro{\dy}{3 +\bs}%
\draw[fill=#2] (\dx,\dy) -- (\dx,\dy + \bw)
  -- (\dx+\bl,\dy+\bw) -- (\dx+\bl,\dy) -- cycle;
}
\newcommand{\side@barB}[2]{%
%% #1 = cubie possn no, #2 = colour
\pgfmathsetmacro{\blh}{\bl*(0.5)}%
\pgfmathsetmacro{\dx}{#1 -1+0.5-\blh}%
\pgfmathsetmacro{\dy}{0 -\bs-\bw}%
\draw[fill=#2] (\dx,\dy) -- (\dx,\dy + \bw)
  -- (\dx+\bl,\dy+\bw) -- (\dx+\bl,\dy) -- cycle;
}
%    \end{macrocode}
%
%
% \subsubsection{    Drawing a single cubie sidebar}
% 
%  \begin{macro}{\DrawRubikLayerSideX$_1$X$_2$X$_3$}
% This command draws the sidebars using the above \cmd{\sidebar} command.
% The X$_1$X$_2$X$_2$ parameters refer to the  options Left, Middle, 
% Right, Top, Equator, Bottom, x, y, as follows:
% \begin{quote}
%  X$_1$  is an $x$ parameter: either $\langle L\mid M \mid  R\rangle$
%  {\newline}X$_2$  is an $y$ parameter: either $\langle T\mid E \mid  B\rangle$
%  {\newline}X$_3$  is an extra parameter: either $\langle x\mid y \rangle$,
%  required by corner sidebars to indicate whether the sidebar was either 
%  above or below $\langle y \rangle$, or to the left or 
%  right $\langle x \rangle$ of the associated  cubie.
% An $X_3$ parameter is not required for the sidebar of an edge cubie, since 
% only one location is posible in these cases.
% \end{quote}
% For example, the following command
% \begin{quote}
%   \cmd{\DrawRubikLayerSideLTy\{G\}}
% \end{quote}
% draws a Green sidebar above the Top Left cubie. 
% \end{macro}
% 
% \begin{macrocode}
%%---Left side
\newcommand{\DrawRubikLayerSideLTx}[1]{\side@barL{3}{#1}}
\newcommand{\DrawRubikLayerSideLEx}[1]{\side@barL{2}{#1}}
\newcommand{\DrawRubikLayerSideLE}[1]{\side@barL{2}{#1}}
\newcommand{\DrawRubikLayerSideLBx}[1]{\side@barL{1}{#1}}
%---Right side
\newcommand{\DrawRubikLayerSideRTx}[1]{\side@barR{3}{#1}}
\newcommand{\DrawRubikLayerSideREx}[1]{\side@barR{2}{#1}}
\newcommand{\DrawRubikLayerSideRE}[1]{\side@barR{2}{#1}}
\newcommand{\DrawRubikLayerSideRBx}[1]{\side@barR{1}{#1}}
%---Top side
\newcommand{\DrawRubikLayerSideLTy}[1]{\side@barT{1}{#1}}
\newcommand{\DrawRubikLayerSideMTy}[1]{\side@barT{2}{#1}}
\newcommand{\DrawRubikLayerSideMT}[1]{\side@barT{2}{#1}}
\newcommand{\DrawRubikLayerSideRTy}[1]{\side@barT{3}{#1}}
%---Bottom side
\newcommand{\DrawRubikLayerSideLBy}[1]{\side@barB{1}{#1}}
\newcommand{\DrawRubikLayerSideMBy}[1]{\side@barB{2}{#1}}
\newcommand{\DrawRubikLayerSideMB}[1]{\side@barB{2}{#1}}
\newcommand{\DrawRubikLayerSideRBy}[1]{\side@barB{3}{#1}}
%    \end{macrocode}
%
%
%  \subsubsection{    Drawing multiple cubie  sidebars}
%
%    \begin{macro}{\DrawRubikLayerSideT}
%    \begin{macro}{\DrawRubikLayerSideB}
%    \begin{macro}{\DrawRubikLayerSideL}
%    \begin{macro}{\DrawRubikLayerSideR}
%  These commands allow the drawing of  3 small sidebars along 
%  one particular side (Top, Bottom, Left, Right), as indicated 
%  by the appended T, B, L, R letter code.
%  Each command takes three ordered colour arguments, which are ordered 
%  either from left to right (the T and B forms), or from top to bottom
%  (the L and R forms)
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%   \begin{macrocode}
%%--Top side---
\newcommand{\DrawRubikLayerSideT}[3]{%
  \DrawRubikLayerSideLTy{#1}%
  \DrawRubikLayerSideMTy{#2}%
  \DrawRubikLayerSideRTy{#3}%
}
%%--Bottom side---
\newcommand{\DrawRubikLayerSideB}[3]{%
  \DrawRubikLayerSideLBy{#1}%
  \DrawRubikLayerSideMBy{#2}%
  \DrawRubikLayerSideRBy{#3}%
}
%%--Left side--------
%% colours run vertically DOWN
\newcommand{\DrawRubikLayerSideL}[3]{%
  \DrawRubikLayerSideLTx{#1}%
  \DrawRubikLayerSideLEx{#2}%
  \DrawRubikLayerSideLBx{#3}%
}
%%--Right side--------
%% colours run vertically DOWN
\newcommand{\DrawRubikLayerSideR}[3]{%
  \DrawRubikLayerSideRTx{#1}%
  \DrawRubikLayerSideREx{#2}%
  \DrawRubikLayerSideRBx{#3}%
}
%    \end{macrocode}
%
%  \begin{macro}{\DrawRubikLayerSideLR}
% This command draws six cubie sidebars, three on each side, drawn in (L, R)
%  pairs. The command takes six colour arguments, ordered in pairs, 
%  as shown in the following example.
% \begin{quote}
%\begin{verbatim}
%  \DrawRubikLayerSideLR{G}   {G}
%                       {R}   {B}
%                       {Y}   {B}
%\end{verbatim}
%  \end{quote}
%  \begin{macrocode}
\newcommand{\DrawRubikLayerSideLR}[6]{%
  \DrawRubikLayerSideLTx{#1}%
  \DrawRubikLayerSideRTx{#2}%
  \DrawRubikLayerSideLEx{#3}%
  \DrawRubikLayerSideREx{#4}%
  \DrawRubikLayerSideLBx{#5}%
  \DrawRubikLayerSideRBx{#6}%
}
%    \end{macrocode}
%  \end{macro}
%
%
%    \subsection{    NCube command}
%
%  \textsc{history}: The essence of this command was originally developed by Peter Bartal 
%  as his command \cmd{\rubikcube} (see Bartal,
%  2011). We have modified it, as follows (June 2012):
%  \newline\strut\hspace{\parindent}(1) adjusted  to use the TikZ \cmd{\pgfmathsetmacro\{\}\{\}} command,
%  \newline\strut\hspace{\parindent}(2)  renamed to \cmd{\DrawNCubeAll}.
%
%    \begin{macro}{\DrawNCubeAll}
% {\noindent}This command draws a solved NxNxN Rubik's cube from the RightUp viewpoint.
% All cubies on a given face have the same colour. 
% The command takes four ordered  arguments, as follows:
% \newline\strut\hspace{\parindent}\#1 = number of cubies ($n>0$) along each side,
% \newline\strut\hspace{\parindent}\#2, \#3, \#4 = colors of the visible faces (in X,Y,Z order);
%  X=Right face  colour, Y=Up face colour, Z=Front face colour.
%
% \begin{macrocode}
\newcommand{\DrawNCubeAll}[4]{%
   \pgfmathsetmacro{\ncubes}{#1-1}%
%% need to subtract 1 from the given number of cubies per side 
%% to avoid the origin of the initial cube to be displaced
    \foreach \x in {0,...,\ncubes}{%
      \foreach \y in {0,...,\ncubes}{%
        \foreach \z in {0,...,\ncubes}{%
          \cube@dxdydz{1}{#2}{#3}{#4}{\x}{\y}{\z}% 
  }}}}
%    \end{macrocode}
%    \end{macro}
%
%   \begin{macro}{\cube@dxdydz}
% This internal command is  used only by the \cmd{\DrawNCubeAll} command.
% The original version of this command was developed  by Peter Bartal (see Bartal, 2011).
% It was modified (2012) by RWD Nickalls (to implement  a more 
% intuitive X, Y, Z ordering of parameters), and also by Peter Grill (see below).
%
%  The cube need not be in the origin, the distances of 
% the Down-behind [L] corner from 
% the origin are taken as parameters.
% The command takes 7 ordered arguments:
% \newline\strut\hspace{\parindent}1 - length of an edge 
% \newline\strut\hspace{\parindent}2 - color of the X face (Right) (RWDN)
% \newline\strut\hspace{\parindent}3 - color of the Y face (Up)    (RWDN)
% \newline\strut\hspace{\parindent}4 - color of the Z face (Front) (RWDN)
% \newline\strut\hspace{\parindent}5 - x-position in space
% \newline\strut\hspace{\parindent}6 - y-position in space
% \newline\strut\hspace{\parindent}7 - z-position in space
%
% \textsc{usage}:  |\cube@dxdydz{1}{cX}{cY}{cZ}{x}{y}{z}|
%
% {\noindent}The original code
%   |\pgfmathparse{#1+#5}\let\dy\pgfmathresult| 
% {\newline} was changed  to the more intuitive  |\pgfmathsetmacro{\dx}{#1+#5}|
% (suggested by Peter Grill 2011).
%
% \textsc{changes}: RWD Nickalls (2012): 
% (1) added the [line join=round,line cap=round] 
% options to  each of the TikZ  `draw' commands, in order to improve 
% the line joining (first two options); (2) adjusted  the \cmd{\cube@dxdydz} macro 
% to adopt the ordered  XYZ face colour notation
% (by reassigning \#2, \#3, \#4).
%    \begin{macrocode}
\newcommand{\cube@dxdydz}[7]{%
   \pgfmathsetmacro{\dx}{#1+#5}% 
%% calculates the 'displacement' (distance from the origin) of the 
%% far corners of the cube along the x axis from the arguments
   \pgfmathsetmacro{\dy}{#1+#6}%
%% calculates the 'displacement' (distance from the origin) of the 
%% far corners of the cube along the y axis from the arguments
   \pgfmathsetmacro{\dz}{#1+#7}%
%% calculates the 'displacement' (distance from the origin) of the 
%% far corners of the cube along the z axis from the arguments
%% Draw FRONT face   (using the X colour = #4)
    \draw[line join=round,line cap=round,ultra thick,fill=#4]%
 (#5,#6,\dz) -- (\dx,#6,\dz) -- (\dx,\dy,\dz) -- (#5,\dy,\dz) -- cycle;
%% The 'rectangle' command does not work with 3D coordinates, 
%% so this is the way to draw the squres with space coordinates
%% Draw UP face (using the Y colour = #3)
   \draw[line join=round,line cap=round,ultra thick,fill=#3]%
 (#5,\dy,\dz) -- (\dx,\dy,\dz) -- (\dx,\dy,#7) -- (#5,\dy,#7) -- cycle;
%% Draw RIGHT face   (using the X colour = #2)
 \draw[line join=round,line cap=round,ultra thick,fill=#2]%
 (\dx,#6,\dz) -- (\dx,#6,#7) -- (\dx,\dy,#7) -- (\dx,\dy,\dz) -- cycle;
    }
%    \end{macrocode}
%    \end{macro}
%
%
%  \subsection{    Drawing single cubies}
%
%  \begin{macro}{\Cubiedx}
%  \begin{macro}{\Cubiedy}
%  These two commands set the value of the two length parameters
%  \texttt{cx} and \texttt{cy}, and allow the user to vary the size 
%  (adjust \texttt{cy}) and horizontal viewpoint  (adjust \texttt{cx})
%  of a single cubie (described in more detail in the \rubikcube\ 
%   package  documentation).
%  Note that we cannot use the names \texttt{dx}, \texttt{dy} since these have been 
% used before (see above). However we use \texttt{dx}, \texttt{dy} in the  command names
% since these will be more  readily understood by the user.
%    \begin{macrocode}
\newcommand{\Cubiedx}[1]{\pgfmathsetmacro{\cx}{#1}}
\newcommand{\Cubiedy}[1]{\pgfmathsetmacro{\cy}{#1}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
% We now set the default values (cx=cy=0.4)
%    \begin{macrocode}
\Cubiedx{0.4}
\Cubiedy{0.4}
%    \end{macrocode}
%
%
%  \begin{macro}{\DrawCubieRU}
%  \begin{macro}{\DrawCubieRD}
%  \begin{macro}{\DrawCubieLU}
%  \begin{macro}{\DrawCubieLD}
% These four commands draw a single cubie from the RightUp, RightDown, 
% LeftUp, LeftDown viewpoint. The viewpoint is specified using an appended 
% two-letter XY ordered viewpoint code: either RU, RD, LU, LD.
% These commands take three arguments, namely three different XYZ 
% ordered colour codes (R,O,Y,G,B,W,X).
% \newline\textsc{format}: \cmd{\DrawCubieRU}\marg{colourX}\marg{colourY}\marg{colourZ}
% \newline\textsc{usage}: |\DrawCubieRU{G}{B}{W}|
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \begin{macrocode}
\newcommand{\DrawCubieRU}[3]{%
%% Front face (z)
\draw[line join=round,line cap=round,ultra thick,fill=#3]%
 (0,0) -- (0, 1) -- (1, 1) -- (1,0) -- cycle;
%% Up face(y)
\draw[line join=round,line cap=round,ultra thick,fill=#2]%
 (0,1) -- (\cx, 1+\cy) -- (1+\cx,1+\cy) -- (1,1) -- cycle;
%% Right face(x)
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
 (1,0) -- (1,1) -- (1+\cx,1+\cy) -- (1+\cx, \cy) -- cycle;
}
\newcommand{\DrawCubieRD}[3]{%
%% Front face (z)
\draw[line join=round,line cap=round,ultra thick,fill=#3]%
 (0,0) -- (0, 1) -- (1, 1) -- (1,0) -- cycle;
%% Down face (y)
\draw[line join=round,line cap=round,ultra thick,fill=#2]%
 (\cx,-\cy) -- (0, 0) -- (1,0) -- (1+\cx,-\cy) -- cycle;
%% Right face (x)
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
 (1,0) -- (1,1) -- (1+\cx,-\cy+1) -- (1+\cx, -\cy) -- cycle;
}
\newcommand{\DrawCubieLD}[3]{%
%% Front face (z)
\draw[line join=round,line cap=round,ultra thick,fill=#3]%
 (0,0) -- (0, 1) -- (1, 1) -- (1,0) -- cycle;
%% Down face (y)
\draw[line join=round,line cap=round,ultra thick,fill=#2]%
 (-\cx,-\cy) -- (0, 0) -- (1,0) -- (1-\cx,-\cy) -- cycle;
%% Left face (x)
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
 (-\cx,-\cy) -- (-\cx,-\cy+1) -- (0,1) -- (0,0) -- cycle;
}
\newcommand{\DrawCubieLU}[3]{%
%% Front face (z)
\draw[line join=round,line cap=round,ultra thick,fill=#3]%
 (0,0) -- (0, 1) -- (1, 1) -- (1,0) -- cycle;
%% Up face (y)
\draw[line join=round,line cap=round,ultra thick,fill=#2]%
 (-\cx,1+\cy) -- (1-\cx, 1+\cy) -- (1,1) -- (0,1) -- cycle;
%% Left face (x)
\draw[line join=round,line cap=round,ultra thick,fill=#1]%
 (-\cx, \cy) -- (-\cx,\cy+1) -- (0,1) -- (0,0) -- cycle;
}
%    \end{macrocode}
%
%
%  \subsection{    Text cubies}
% 
%  \begin{macro}{\textCubieRU}
%  \begin{macro}{\textCubieRD}
%  \begin{macro}{\textCubieLU}
%  \begin{macro}{\textCubieLD}
%  These four commands draw a single `text' cubie from the RightUp, RightDown, 
%  LeftUp, LeftDown viewpoint. 
%  They are `text' forms of the DrawCubie' versions described above.
%  Their size is scaled to be suitable for use with  10--12 point fonts.  
%
%  As before, the viewpoint is specified using an appended 
% two-letter XY ordered viewpoint code: either RU, RD, LU, LD.
% These commands take three arguments, namely three different XYZ 
% ordered colour codes (R,O,Y,G,B,W,X).
% \newline\textsc{format}: \cmd{\textCubieRU}\marg{colourX}\marg{colourY}\marg{colourZ}
% \newline\textsc{usage}: |\textCubieRU{G}{B}{W}|
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \begin{macrocode}
\newcommand{\textCubieRU}[3]{%
\begin{minipage}{0.66cm}
\centering
\begin{tikzpicture}[scale=0.5]
\Cubiedx{0.4}\Cubiedy{0.4}
\DrawCubieRU{#1}{#2}{#3}
\end{tikzpicture}
\end{minipage}
}
\newcommand{\textCubieRD}[3]{%
\begin{minipage}{0.66cm}
\centering
\begin{tikzpicture}[scale=0.5]
\Cubiedx{0.4}\Cubiedy{0.4}
\DrawCubieRD{#1}{#2}{#3}
\end{tikzpicture}
\end{minipage}
}
\newcommand{\textCubieLD}[3]{%
\begin{minipage}{0.66cm}
\centering
\begin{tikzpicture}[scale=0.5]
\Cubiedx{0.4}\Cubiedy{0.4}
\DrawCubieLD{#1}{#2}{#3}
\end{tikzpicture}
\end{minipage}
}
\newcommand{\textCubieLU}[3]{%
\begin{minipage}{0.66cm}
\centering
\begin{tikzpicture}[scale=0.5]
\Cubiedx{0.4}\Cubiedy{0.4}
\DrawCubieLU{#1}{#2}{#3}
\end{tikzpicture}
\end{minipage}
}
%    \end{macrocode}
%
%
%  \subsection{    Rotation commands --- face and slice}
%
% \textsc{notation}: We require a special notation to denote the 
% various Rubik rotations (denoted 
% by the stem rr). These rotations can be denoted either  using a letter code
% say, \cmd{\rrR} for a Right  face rotation, or by the equivalent hieroglyph
% denoted as \cmd{\rrhR}  (Rubik rotation hieroglyph Right) etc.
% Commands such as \cmd{\RubikXXXX} typeset a pair of rr and rrh vertically.
% Commands such as \cmd{\textRubikXXXX} typeset  the pair rr and rrh horizontally 
% (ie  a sort of `text' mode).
%
% The square hieroglyphs are built up in stages. We first create an internal 
% command  for drawing the square (\cmd{\DrawNotationBox}), and then draw the 
% contents (lines, arrows, arcs of circles, and sometimes just text).
% (for the TikZ  ARC command see  TikZ pgfmanual2012  page p146 (\S 14.8)).
%
%  The presence of small overfilled hboxes associated with these squares
% were originally checked using the ltugboat.cls, and all fixed mainly
% by setting the associated  minipages $\rightarrow$ width = 0.6cm, 
%  and using TKZ scale=0.5.
% NOTE: do not use a trailing |%|  after each |\end{tikzpicture}| 
% as we actually need a small space to appear when used in text.
%
%
%   \begin{macro}{\DrawNotationBox}
% This command draws the surrounding square box of  all the hieroglyphs.
% Note that we start at (0,0) and draw to the final point 
% in order to make a nice corner join.
%
%   \begin{macrocode}
\newcommand{\DrawNotationBox}{%
  \draw [thick] (0,0) -- (0,1) -- (1,1) -- (1,0) -- (0,0) -- (0,1)%
}
%    \end{macrocode}
%    \end{macro}
%
% We now define  a number of points and line-segments inside the square 
% (eg.; \cmd{\sd}, \cmd{\sh} \ldots\ etc.) which will be required
% for use in drawing the various lines and arrows.
% Some hieroglyphs contain either one circular arc, or two concentric arcs, 
% and these arcs require both a center and a start point.
% 
%\begin{macrocode}
\pgfmathsetmacro{\sd}{0.25} % horiz space 
\pgfmathsetmacro{\sdd}{2*\sd} 
\pgfmathsetmacro{\sddd}{3*\sd}
\pgfmathsetmacro{\sh}{0.6} % height
\pgfmathsetmacro{\sb}{0.2} % base
\pgfmathsetmacro{\sbh}{\sb + \sh} % UP
\pgfmathsetmacro{\scx}{\sdd+0.2} % Start of CircleX
\pgfmathsetmacro{\scy}{\sd*2/3}  % Start of CircleY
\pgfmathsetmacro{\sqcx}{\scx-0.13} %% SQuare CenterX
\pgfmathsetmacro{\sqcy}{\scy+0.25} %% SQuare CenterY
%    \end{macrocode}
%
%
%  \subsubsection{     Rotation B}
%
%  \begin{macro}{\rrB}
%  \begin{macro}{\SquareB}
%  \begin{macro}{\rrhB}
%  \begin{macro}{\RubikB}
%  \begin{macro}{\textRubikB}
%  These commands  all draw forms which denote the B rotation.
%  Not visible from the front.
% \begin{macrocode}
\newcommand{\rrB}{\@rr{B}}
\newcommand{\SquareB}{\@SquareLetter{\rrB}}
\newcommand{\rrhB}{\raisebox{-0.25mm}{\SquareB}\,}
\newcommand{\RubikB}{\raisebox{\@hRubik}{\SquareB}\,}
\newcommand{\textRubikB}{\rrhB\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Bp}
%
%  \begin{macro}{\rrBp}
%  \begin{macro}{\SquareBp}
%  \begin{macro}{\rrhBp}
%  \begin{macro}{\RubikBp}
%  \begin{macro}{\textRubikBp}
%  These commands  all draw forms which denote the Bp rotation.
%  Not visible from the front.
% \begin{macrocode}
\newcommand{\rrBp}{\@rrp{B}}
\newcommand{\SquareBp}{\@SquareLetter{\rrBp}}
\newcommand{\rrhBp}{\raisebox{-0.25mm}{\SquareBp}\,}
\newcommand{\RubikBp}{\raisebox{\@hRubik}{\SquareBp}\,}
\newcommand{\textRubikBp}{\rrhBp\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Bw}
%
%  \begin{macro}{\rrBw}
%  \begin{macro}{\SquareBw}
%  \begin{macro}{\rrhBw}
%  \begin{macro}{\RubikBw}
%  \begin{macro}{\textRubikBw}
%  These commands  all draw forms which denote the Bw rotation.
%  Not visible from the front.
% \begin{macrocode}
%\newcommand{\rrBw}{\textbf{\textsf{Bw}}}
\newcommand{\rrBw}{\@rrw{B}}
\newcommand{\SquareBw}{\@SquareLetter{\rrBw}}
\newcommand{\rrhBw}{\raisebox{-0.25mm}{\SquareBw}\,}
\newcommand{\RubikBw}{\raisebox{\@hRubik}{\SquareBw}\,}
\newcommand{\textRubikBw}{\rrhBw\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Bwp}
%
%  \begin{macro}{\rrBwp}
%  \begin{macro}{\SquareBwp}
%  \begin{macro}{\rrhBwp}
%  \begin{macro}{\RubikBwp}
%  \begin{macro}{\textRubikBwp}
%  These commands  all draw forms which denote the Bwp rotation.
% Not visible from the front.
% \begin{macrocode}
%\newcommand{\rrBwp}{\textbf{\textsf{Bwp}}}
\newcommand{\rrBwp}{\@rrwp{B}}
\newcommand{\SquareBwp}{\@SquareLetter{\rrBwp}}
\newcommand{\rrhBwp}{\raisebox{-0.25mm}{\SquareBwp}\,}
\newcommand{\RubikBwp}{\raisebox{\@hRubik}{\SquareBwp}\,}
\newcommand{\textRubikBwp}{\rrhBwp\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Bs}
%
%  \begin{macro}{\rrBs}
%  \begin{macro}{\SquareBs}
%  \begin{macro}{\rrhBs}
%  \begin{macro}{\RubikBs}
%  \begin{macro}{\textRubikBs}
%  These commands  all draw forms which denote the Bs rotation.
%  Not visible from the front.
% \begin{macrocode}
%\newcommand{\rrBs}{\textbf{\textsf{Bs}}}
\newcommand{\rrBs}{\@rrs{B}}
\newcommand{\SquareBs}{\@SquareLetter{\rrBs}}
\newcommand{\rrhBs}{\raisebox{-0.25mm}{\SquareBs}\,}
\newcommand{\RubikBs}{\raisebox{\@hRubik}{\SquareBs}\,}
\newcommand{\textRubikBs}{\rrhBs\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Bsp}
%
%  \begin{macro}{\rrBsp}
%  \begin{macro}{\SquareBsp}
%  \begin{macro}{\rrhBsp}
%  \begin{macro}{\RubikBsp}
%  \begin{macro}{\textRubikBsp}
%  These commands  all draw forms which denote the Bsp rotation.
%  Not visible from the front.
% \begin{macrocode}
%\newcommand{\rrBsp}{\textbf{\textsf{Bsp}}}
\newcommand{\rrBsp}{\@rrsp{B}}
\newcommand{\SquareBsp}{\@SquareLetter{\rrBsp}}
\newcommand{\rrhBsp}{\raisebox{-0.25mm}{\SquareBsp}\,}
\newcommand{\RubikBsp}{\raisebox{\@hRubik}{\SquareBsp}\,}
\newcommand{\textRubikBsp}{\rrhBsp\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Ba}
%
%  \begin{macro}{\rrBa}
%  \begin{macro}{\SquareBa}
%  \begin{macro}{\rrhBa}
%  \begin{macro}{\RubikBa}
%  \begin{macro}{\textRubikBa}
%  These commands  all draw forms which denote the Ba rotation.
% Not visible from the front.
% \begin{macrocode}
%\newcommand{\rrBa}{\textbf{\textsf{Ba}}}
\newcommand{\rrBa}{\@rra{B}}
\newcommand{\SquareBa}{\@SquareLetter{\rrBa}}
\newcommand{\rrhBa}{\raisebox{-0.25mm}{\SquareBa}\,}
\newcommand{\RubikBa}{\raisebox{\@hRubik}{\SquareBa}\,}
\newcommand{\textRubikBa}{\rrhBa\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Bap}
%
%  \begin{macro}{\rrBap}
%  \begin{macro}{\SquareBap}
%  \begin{macro}{\rrhBap}
%  \begin{macro}{\RubikBap}
%  \begin{macro}{\textRubikBap}
%  These commands  all draw forms which denote the Bap rotation.
% Not visible from the front.
% \begin{macrocode}
%\newcommand{\rrBap}{\textbf{\textsf{Bap}}}
\newcommand{\rrBap}{\@rrap{B}}
\newcommand{\SquareBap}{\@SquareLetter{\rrBap}}
\newcommand{\rrhBap}{\raisebox{-0.25mm}{\SquareBap}\,}
\newcommand{\RubikBap}{\raisebox{\@hRubik}{\SquareBap}\,}
\newcommand{\textRubikBap}{\rrhBap\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation D}
%
%  \begin{macro}{\rrD}
%  \begin{macro}{\SquareD}
%  \begin{macro}{\rrhD}
%  \begin{macro}{\RubikD}
%  \begin{macro}{\textRubikD}
%  These commands  all draw forms which denote the D rotation.
% \begin{macrocode}
\newcommand{\rrD}{\textbf{\textsf{D}}}
%%
\newcommand{\SquareD}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick, ->]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%%
}
\newcommand{\rrhD}{\raisebox{-0.333\height}{\SquareD}\,}
%%
\newcommand{\RubikD}{%
\begin{minipage}{0.6cm}
\centering
\SquareD\\
\rrD
\end{minipage}%
}
\newcommand{\textRubikD}{\rrD\,\rrhD}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Dp}
%
%  \begin{macro}{\rrDp}
%  \begin{macro}{\SquareDp}
%  \begin{macro}{\rrhDp}
%  \begin{macro}{\RubikDp}
%  \begin{macro}{\textRubikDp}
%  These commands  all draw forms which denote the Dp rotation.
% \begin{macrocode}
\newcommand{\rrDp}{\textbf{\textsf{D}$^\prime$}}
%%
\newcommand{\SquareDp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick]     (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd)  -- (\sbh, \sdd);
\draw [thick, <-] (\sb,\sd)   -- (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhDp}{\raisebox{-0.333\height}{\SquareDp}\,}
%%
\newcommand{\RubikDp}{%
\begin{minipage}{0.6cm}
\centering
\SquareDp\\
\rrDp
\end{minipage}%
}
\newcommand{\textRubikDp}{\rrDp\,\rrhDp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Dw}
%
%  \begin{macro}{\rrDw}
%  \begin{macro}{\SquareDw}
%  \begin{macro}{\rrhDw}
%  \begin{macro}{\RubikDw}
%  \begin{macro}{\textRubikDw}
%  These commands  all draw forms which denote the Dw rotation.
% \begin{macrocode}
\newcommand{\rrDw}{\textbf{\textsf{D\footnotesize{w}}}}
%%
\newcommand{\SquareDw}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick]     (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick, ->] (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick, ->] (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhDw}{\raisebox{-0.333\height}{\SquareDw}\,}
%%
\newcommand{\RubikDw}{%
\begin{minipage}{0.6cm}
\centering
\SquareDw\\
\rrDw
\end{minipage}%
}
\newcommand{\textRubikDw}{\rrDw\,\rrhDw}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Dwp}
%
%  \begin{macro}{\rrDwp}
%  \begin{macro}{\SquareDwp}
%  \begin{macro}{\rrhDwp}
%  \begin{macro}{\RubikDwp}
%  \begin{macro}{\textRubikDwp}
%  These commands  all draw forms which denote the Dwp rotation.
% \begin{macrocode}
\newcommand{\rrDwp}{\textbf{\textsf{D\footnotesize{w}}$^\prime$}}
%%
\newcommand{\SquareDwp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick]     (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick, <-] (\sb,\sdd)  -- (\sbh, \sdd);
\draw [thick, <-] (\sb,\sd)   -- (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhDwp}{\raisebox{-0.333\height}{\SquareDwp}\,}
%%
\newcommand{\RubikDwp}{%
\begin{minipage}{0.6cm}
\centering
\SquareDwp\\
\rrDwp
\end{minipage}%
}
\newcommand{\textRubikDwp}{\rrDwp\,\rrhDwp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Ds}
%
%  \begin{macro}{\rrDs}
%  \begin{macro}{\SquareDs}
%  \begin{macro}{\rrhDs}
%  \begin{macro}{\RubikDs}
%  \begin{macro}{\textRubikDs}
%  These commands  all draw forms which denote the Ds rotation.
% \begin{macrocode}
\newcommand{\rrDs}{\@rrs{D}}
%%
\newcommand{\SquareDs}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, ->] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick, ->]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhDs}{\raisebox{-0.333\height}{\SquareDs}\,}
%%
\newcommand{\RubikDs}{%
\begin{minipage}{0.6cm}
\centering
\SquareDs\\
\rrDs
\end{minipage}%
}
\newcommand{\textRubikDs}{\rrDs\,\rrhDs}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Dsp}
%
%  \begin{macro}{\rrDsp}
%  \begin{macro}{\SquareDsp}
%  \begin{macro}{\rrhDsp}
%  \begin{macro}{\RubikDsp}
%  \begin{macro}{\textRubikDsp}
%  These commands  all draw forms which denote the Dsp rotation.
% \begin{macrocode}
\newcommand{\rrDsp}{\@rrsp{D}}
%%
\newcommand{\SquareDsp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick, <-]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhDsp}{\raisebox{-0.333\height}{\SquareDsp}\,}
%%
\newcommand{\RubikDsp}{%
\begin{minipage}{0.6cm}
\centering
\SquareDsp\\
\rrDsp
\end{minipage}%
}
\newcommand{\textRubikDsp}{\rrDsp\,\rrhDsp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Da} 
%
%  \begin{macro}{\rrDa}
%  \begin{macro}{\SquareDa}
%  \begin{macro}{\rrhDa}
%  \begin{macro}{\RubikDa}
%  \begin{macro}{\textRubikDa}
%  These commands  all draw forms which denote the Da rotation.
% \begin{macrocode}
\newcommand{\rrDa}{\@rra{D}}
%%
\newcommand{\SquareDa}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick, ->]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhDa}{\raisebox{-0.333\height}{\SquareDa}\,}
%%
\newcommand{\RubikDa}{%
\begin{minipage}{0.6cm}
\centering
\SquareDa\\
\rrDa
\end{minipage}%
}
\newcommand{\textRubikDa}{\rrDa\,\rrhDa}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Dap} 
%
%  \begin{macro}{\rrDap}
%  \begin{macro}{\SquareDap}
%  \begin{macro}{\rrhDap}
%  \begin{macro}{\RubikDap}
%  \begin{macro}{\textRubikDap}
%  These commands  all draw forms which denote the Dap rotation.
% \begin{macrocode}
\newcommand{\rrDap}{\@rrap{D}}
%%
\newcommand{\SquareDap}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, ->] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick, <-]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhDap}{\raisebox{-0.333\height}{\SquareDap}\,}
%%
\newcommand{\RubikDap}{%
\begin{minipage}{0.6cm}
\centering
\SquareDap\\
\rrDap
\end{minipage}%
}
\newcommand{\textRubikDap}{\rrDap\,\rrhDap}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation E}
%
%  \begin{macro}{\rrE}
%  \begin{macro}{\SquareE}
%  \begin{macro}{\rrhE}
%  \begin{macro}{\RubikE}
%  \begin{macro}{\textRubikE}
%  These commands  all draw forms which denote the E rotation.
% \begin{macrocode}
\newcommand{\rrE}{\textbf{\textsf{E}}}
%%
\newcommand{\SquareE}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick, ->]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhE}{\raisebox{-0.333\height}{\SquareE}\,}
%%
\newcommand{\RubikE}{%
\begin{minipage}{0.6cm}
\centering
\SquareE\\
\rrE
\end{minipage}%
}
\newcommand{\textRubikE}{\rrE\,\rrhE}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Ep}
%
%  \begin{macro}{\rrEp}
%  \begin{macro}{\SquareEp}
%  \begin{macro}{\rrhEp}
%  \begin{macro}{\RubikEp}
%  \begin{macro}{\textRubikEp}
%  These commands  all draw forms which denote the Ep rotation.
% \begin{macrocode}
\newcommand{\rrEp}{\textbf{\textsf{E}$^\prime$}}
%%
\newcommand{\SquareEp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick, <-]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhEp}{\raisebox{-0.333\height}{\SquareEp}\,}
%%
\newcommand{\RubikEp}{%
\begin{minipage}{0.6cm}
\centering
\SquareEp\\
\rrEp
\end{minipage}%
}
\newcommand{\textRubikEp}{\rrEp\,\rrhEp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation F}
%
%  \begin{macro}{\rrF}
%  \begin{macro}{\SquareF}
%  \begin{macro}{\rrhF}
%  \begin{macro}{\RubikF}
%  \begin{macro}{\textRubikF}
%  These commands  all draw forms which denote the F rotation.
%  \begin{macrocode}
\newcommand{\rrF}{\textbf{\textsf{F}}}
%%
\newcommand{\SquareF}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\scx, \scy) arc[radius=0.35, start angle=-60, delta angle=290];
\end{tikzpicture}%
}
\newcommand{\rrhF}{\raisebox{-0.333\height}{\SquareF}\,}
%%
\newcommand{\RubikF}{%
\begin{minipage}{0.6cm}
\centering
\SquareF\\
\rrF
\end{minipage}%
}
\newcommand{\textRubikF}{\rrF\,\rrhF}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Fp}
%
%  \begin{macro}{\rrFp}
%  \begin{macro}{\SquareFp}
%  \begin{macro}{\rrhFp}
%  \begin{macro}{\RubikFp}
%  \begin{macro}{\textRubikFp}
%  These commands  all draw forms which denote the Fp rotation.
% \begin{macrocode}
\newcommand{\rrFp}{\textbf{\textsf{F}$^\prime$}}
%%
\newcommand{\SquareFp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, ->] (\scx, \scy) arc[radius=0.35, start angle=-60, delta angle=290];
\end{tikzpicture}%
}
\newcommand{\rrhFp}{\raisebox{-0.333\height}{\SquareFp}\,}
%%
\newcommand{\RubikFp}{%
\begin{minipage}{0.6cm}
\centering
\SquareFp\\
\rrFp
\end{minipage}%
}
\newcommand{\textRubikFp}{\rrFp\,\rrhFp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Fw}
%
%  \begin{macro}{\rrFw}
%  \begin{macro}{\SquareFw}
%  \begin{macro}{\rrhFw}
%  \begin{macro}{\RubikFw}
%  \begin{macro}{\textRubikFw}
%  These commands  all draw forms which denote the Fw rotation.
% \begin{macrocode}
\newcommand{\rrFw}{\textbf{\textsf{F\footnotesize{w}}}}
%%
\newcommand{\SquareFw}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\scx, \scy) arc[radius=0.35, start angle=-60, delta angle=290];
\draw [thick] (\sqcx,\sqcy) arc[radius=0.1, start angle=-60, delta angle=360];
%\node (squareLab) at (0.5,0.5)  {$o$};
\end{tikzpicture}%
}
\newcommand{\rrhFw}{\raisebox{-0.333\height}{\SquareFw}\,}
%%
\newcommand{\RubikFw}{%
\begin{minipage}{0.6cm}
\centering
\SquareFw\\
\rrFw
\end{minipage}%
}
\newcommand{\textRubikFw}{\rrFw\,\rrhFw}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Fwp}
%
%  \begin{macro}{\rrFwp}
%  \begin{macro}{\SquareFwp}
%  \begin{macro}{\rrhFwp}
%  \begin{macro}{\RubikFwp}
%  \begin{macro}{\textRubikFwp}
%  These commands  all draw forms which denote the Fwp rotation.
% \begin{macrocode}
\newcommand{\rrFwp}{\textbf{\textsf{F\footnotesize{w}}$^\prime$}}
%%
\newcommand{\SquareFwp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, ->] (\scx, \scy) arc[radius=0.35, start angle=-60, delta angle=290];
\draw [thick] (\sqcx,\sqcy) arc[radius=0.1, start angle=-60, delta angle=360];
\end{tikzpicture}%
}
\newcommand{\rrhFwp}{\raisebox{-0.333\height}{\SquareFwp}\,}
%%
\newcommand{\RubikFwp}{%
\begin{minipage}{0.6cm}
\centering
\SquareFwp\\
\rrFwp
\end{minipage}%
}
\newcommand{\textRubikFwp}{\rrFwp\,\rrhFwp}
%%
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Fs}
%
%  \begin{macro}{\rrFs}
%  \begin{macro}{\rrhFs}
%  \begin{macro}{\RubikFs}
%  \begin{macro}{\textRubikFs}
%  These commands  draw forms of the Singmaster Fs slice rotation.
%  We need to just make square with Fs in square; 
%  adjust box height using a \cmd{\rule};
%  adjust \cmd{\fboxsep} (default=3pt);
%  adjust \cmd{\fboxrule} (default=0.4pt);
%  bounded by \{\} so no need to reset to defaults.
%  Not visible from the front.
% \begin{macrocode}
\newcommand{\rrFs}{\@rrs{F}}
\newcommand{\SquareFs}{\@SquareLetter{\rrFs}} 
\newcommand{\rrhFs}{\raisebox{-0.25mm}{\SquareFs}\,}
\newcommand{\RubikFs}{\raisebox{\@hRubik}{\SquareFs}\,}
\newcommand{\textRubikFs}{\rrhFs\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Fsp}
%
%  \begin{macro}{\rrFsp}
%  \begin{macro}{\rrhFsp}
%  \begin{macro}{\RubikFsp}
%  \begin{macro}{\textRubikFsp}
%  These commands  draw forms of the Singmaster Fsp slice rotation.
%  We need to just make square with Fsp in square; 
%  adjust box height using a \cmd{\rule};
%  adjust \cmd{\fboxsep} (default=3pt);
%  adjust \cmd{\fboxrule} (default=0.4pt);
%  bounded by \{\} so no need to reset to defaults.
%  Not visible from the front.
% \begin{macrocode}
\newcommand{\rrFsp}{\@rrsp{F}}
\newcommand{\SquareFsp}{\@SquareLetter{\rrFsp}}
\newcommand{\rrhFsp}{\raisebox{-0.25mm}{\SquareFsp}\,}
\newcommand{\RubikFsp}{\raisebox{\@hRubik}{\SquareFsp}\,}
\newcommand{\textRubikFsp}{\rrhFsp\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Fa}
%
%  \begin{macro}{\rrFa}
%  \begin{macro}{\rrhFa}
%  \begin{macro}{\RubikFa}
%  \begin{macro}{\textRubikFa}
%  These commands  draw forms of the Singmaster Fa slice rotation.
%  We need to just make square with Fa in square; 
%  adjust box height using a \cmd{\rule};
%  adjust \cmd{\fboxsep} (default=3pt);
%  adjust \cmd{\fboxrule} (default=0.4pt);
%  bounded by \{\} so no need to reset to defaults.
%  Not visible from the front.
% \begin{macrocode}
\newcommand{\rrFa}{\@rra{F}}
\newcommand{\SquareFa}{\@SquareLetter{\rrFa}}
\newcommand{\rrhFa}{\raisebox{-0.25mm}{\SquareFa}\,}
\newcommand{\RubikFa}{\raisebox{\@hRubik}{\SquareFa}\,}
\newcommand{\textRubikFa}{\rrhFa\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Fap}
%
%  \begin{macro}{\rrFap}
%  \begin{macro}{\rrhFap}
%  \begin{macro}{\RubikFap}
%  \begin{macro}{\textRubikFap}
%  These commands  draw forms of the Singmaster Fap slice rotation.
%  We need to just make square with Fap in square; 
%  adjust box height using a \cmd{\rule};
%  adjust \cmd{\fboxsep} (default=3pt);
%  adjust \cmd{\fboxrule} (default=0.4pt);
%  bounded by \{\} so no need to reset to defaults.
%  Not visible from the front.
% \begin{macrocode}
\newcommand{\rrFap}{\@rrap{F}}
\newcommand{\SquareFap}{\@SquareLetter{\rrFap}}
\newcommand{\rrhFap}{\raisebox{-0.25mm}{\SquareFap}\,}
\newcommand{\RubikFap}{\raisebox{\@hRubik}{\SquareFap}\,}
\newcommand{\textRubikFap}{\rrhFap\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation L}
%
%  \begin{macro}{\rrL}
%  \begin{macro}{\SquareL}
%  \begin{macro}{\rrhL}
%  \begin{macro}{\RubikL}
%  \begin{macro}{\textRubikL}
%  These commands  all draw forms which denote the L rotation.
% \begin{macrocode}
\newcommand{\rrL}{\textbf{\textsf{L}}}
%%
\newcommand{\SquareL}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhL}{\raisebox{-0.333\height}{\SquareL}\,}
%%
\newcommand{\RubikL}{%
\begin{minipage}{0.6cm}
\centering
\SquareL\\
\rrL
\end{minipage}%
}
\newcommand{\textRubikL}{\rrL\,\rrhL}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Lp}
%
%  \begin{macro}{\rrLp}
%  \begin{macro}{\SquareLp}
%  \begin{macro}{\rrhLp}
%  \begin{macro}{\RubikLp}
%  \begin{macro}{\textRubikLp}
%  These commands  all draw forms which denote the Lp rotation.
% \begin{macrocode}
\newcommand{\rrLp}{\textbf{\textsf{L}$^\prime$}}
%%
\newcommand{\SquareLp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick,->] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhLp}{\raisebox{-0.333\height}{\SquareLp}\,}
%%
\newcommand{\RubikLp}{%
\begin{minipage}{0.6cm}
\centering
\SquareLp\\
\rrLp
\end{minipage}%
}
\newcommand{\textRubikLp}{\rrLp\,\rrhLp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Lw}
%
%  \begin{macro}{\rrLw}
%  \begin{macro}{\SquareLw}
%  \begin{macro}{\rrhLw}
%  \begin{macro}{\RubikLw}
%  \begin{macro}{\textRubikLw}
%  These commands  all draw forms which denote the Lw rotation.
% \begin{macrocode}
\newcommand{\rrLw}{\textbf{\textsf{L\footnotesize{w}}}}
%%
\newcommand{\SquareLw}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\sd, \sb) -- (\sd, \sbh);
\draw [thick, <-] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhLw}{\raisebox{-0.333\height}{\SquareLw}\,}
%%
\newcommand{\RubikLw}{%
\begin{minipage}{0.6cm}
\centering
\SquareLw\\
\rrLw
\end{minipage}%
}
\newcommand{\textRubikLw}{\rrLw\,\rrhLw}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Lwp}
%
%  \begin{macro}{\rrLwp}
%  \begin{macro}{\SquareLwp}
%  \begin{macro}{\rrhLwp}
%  \begin{macro}{\RubikLwp}
%  \begin{macro}{\textRubikLwp}
%  These commands  all draw forms which denote the Lwp rotation.
% \begin{macrocode}
\newcommand{\rrLwp}{\textbf{\textsf{L\footnotesize{w}}$^\prime$}}
%%
\newcommand{\SquareLwp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick,->] (\sd, \sb) -- (\sd, \sbh);
\draw [thick,->] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhLwp}{\raisebox{-0.333\height}{\SquareLwp}\,}
%%
\newcommand{\RubikLwp}{%
\begin{minipage}{0.6cm}
\centering
\SquareLwp\\
\rrLwp
\end{minipage}%
}
\newcommand{\textRubikLwp}{\rrLwp\,\rrhLwp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{    Rotation Ls}
%
%  \begin{macro}{\rrLs}
%  \begin{macro}{\SquareLs}
%  \begin{macro}{\rrhLs}
%  \begin{macro}{\RubikLs}
%  \begin{macro}{\textRubikLs}
%  These commands  all draw forms which denote the Ls rotation.
% \begin{macrocode}
\newcommand{\rrLs}{\@rrs{L}}
%%
\newcommand{\SquareLs}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick, <-] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhLs}{\raisebox{-0.333\height}{\SquareLs}\,}
%%
\newcommand{\RubikLs}{%
\begin{minipage}{0.6cm}
\centering
\SquareLs\\
\rrLs
\end{minipage}%
}
\newcommand{\textRubikLs}{\rrLs\,\rrhLs}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{    Rotation Lsp}
%
%  \begin{macro}{\rrLsp}
%  \begin{macro}{\SquareLsp}
%  \begin{macro}{\rrhLsp}
%  \begin{macro}{\RubikLsp}
%  \begin{macro}{\textRubikLsp}
%  These commands  all draw forms which denote the Lsp rotation.
% \begin{macrocode}
\newcommand{\rrLsp}{\@rrsp{L}}
%%
\newcommand{\SquareLsp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, ->] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick, ->] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhLsp}{\raisebox{-0.333\height}{\SquareLsp}\,}
%%
\newcommand{\RubikLsp}{%
\begin{minipage}{0.6cm}
\centering
\SquareLsp\\
\rrLsp
\end{minipage}%
}
\newcommand{\textRubikLsp}{\rrLsp\,\rrhLsp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{    Rotation La}
%
%  \begin{macro}{\rrLa}
%  \begin{macro}{\SquareLa}
%  \begin{macro}{\rrhLa}
%  \begin{macro}{\RubikLa}
%  \begin{macro}{\textRubikLa}
%  These commands  all draw forms which denote the La rotation.
% \begin{macrocode}
\newcommand{\rrLa}{\@rra{L}}
%%
\newcommand{\SquareLa}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick, ->] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhLa}{\raisebox{-0.333\height}{\SquareLa}\,}
%%
\newcommand{\RubikLa}{%
\begin{minipage}{0.6cm}
\centering
\SquareLa\\
\rrLa
\end{minipage}%
}
\newcommand{\textRubikLa}{\rrLa\,\rrhLa}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Lap}
%
%  \begin{macro}{\rrLap}
%  \begin{macro}{\SquareLap}
%  \begin{macro}{\rrhLap}
%  \begin{macro}{\RubikLap}
%  \begin{macro}{\textRubikLap}
%  These commands  all draw forms which denote the Lap rotation.
% \begin{macrocode}
\newcommand{\rrLap}{\@rrap{L}}
%%
\newcommand{\SquareLap}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, ->] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick, <-] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhLap}{\raisebox{-0.333\height}{\SquareLap}\,}
%%
\newcommand{\RubikLap}{%
\begin{minipage}{0.6cm}
\centering
\SquareLap\\
\rrLap
\end{minipage}%
}
\newcommand{\textRubikLap}{\rrLap\,\rrhLap}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{    Rotation M}
%
%  \begin{macro}{\rrM}
%  \begin{macro}{\SquareM}
%  \begin{macro}{\rrhM}
%  \begin{macro}{\RubikM}
%  \begin{macro}{\textRubikM}
%  These commands  all draw forms which denote the M rotation.
% \begin{macrocode}
\newcommand{\rrM}{\textbf{\textsf{M}}}
%%
\newcommand{\SquareM}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick] (\sd, \sb) -- (\sd, \sbh);
\draw [thick, <-] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhM}{\raisebox{-0.333\height}{\SquareM}\,}
%%
\newcommand{\RubikM}{%
\begin{minipage}{0.6cm}
\centering
\SquareM\\
\rrM
\end{minipage}%
}
\newcommand{\textRubikM}{\rrM\,\rrhM}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Mp}
%
%  \begin{macro}{\rrMp}
%  \begin{macro}{\SquareMp}
%  \begin{macro}{\rrhMp}
%  \begin{macro}{\RubikMp}
%  \begin{macro}{\textRubikMp}
%  These commands  all draw forms which denote the Mp rotation.
% \begin{macrocode}
\newcommand{\rrMp}{\textbf{\textsf{M}$^\prime$}}
%%
\newcommand{\SquareMp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick] (\sd, \sb) -- (\sd, \sbh);
\draw [thick,->] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhMp}{\raisebox{-0.333\height}{\SquareMp}\,}
%%
\newcommand{\RubikMp}{%
\begin{minipage}{0.6cm}
\centering
\SquareMp\\
\rrMp
\end{minipage}%
}
\newcommand{\textRubikMp}{\rrMp\,\rrhMp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{    Rotation R}
%
%  \begin{macro}{\rrR}
%  \begin{macro}{\SquareR}
%  \begin{macro}{\rrhR}
%  \begin{macro}{\RubikR}
%  \begin{macro}{\textRubikR}
%  These commands  all draw forms which denote the R rotation.
% \begin{macrocode}
\newcommand{\rrR}{\textbf{\textsf{R}}}
%%
\newcommand{\SquareR}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
%% draw three lines in the square, one with an arrow
\draw [thick] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick, ->] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhR}{\raisebox{-0.333\height}{\SquareR}\,}
%%
\newcommand{\RubikR}{%
\begin{minipage}{0.6cm}
\centering
\SquareR\\
\rrR
\end{minipage}%
}
\newcommand{\textRubikR}{\rrR\,\rrhR}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{    Rotation Rp}
%
%  \begin{macro}{\rrRp}
%  \begin{macro}{\SquareRp}
%  \begin{macro}{\rrhRp}
%  \begin{macro}{\RubikRp}
%  \begin{macro}{\textRubikRp}
%  These commands  all draw forms which denote the Rp rotation.
% \begin{macrocode}
\newcommand{\rrRp}{\textbf{\textsf{R}$^\prime$}}
%%
\newcommand{\SquareRp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick, <-] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhRp}{\raisebox{-0.333\height}{\SquareRp}\,}
%%
\newcommand{\RubikRp}{%
\begin{minipage}{0.6cm}
\centering
\SquareRp\\
\rrRp
\end{minipage}%
}
\newcommand{\textRubikRp}{\rrRp\,\rrhRp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{    Rotation Rw}
%
%  \begin{macro}{\rrRw}
%  \begin{macro}{\SquareRw}
%  \begin{macro}{\rrhRw}
%  \begin{macro}{\RubikRw}
%  \begin{macro}{\textRubikRw}
%  These commands  all draw forms which denote the Rw rotation.
% \begin{macrocode}
\newcommand{\rrRw}{\textbf{\textsf{R\footnotesize{w}}}}
%%
\newcommand{\SquareRw}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick] (\sd, \sb) -- (\sd, \sbh);
\draw [thick, ->] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick, ->] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhRw}{\raisebox{-0.333\height}{\SquareRw}\,}
%%
\newcommand{\RubikRw}{%
\begin{minipage}{0.6cm}
\centering
\SquareRw\\
\rrRw
\end{minipage}%
}
\newcommand{\textRubikRw}{\rrRw\,\rrhRw}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{    Rotation Rwp}
%
%  \begin{macro}{\rrRwp}
%  \begin{macro}{\SquareRwp}
%  \begin{macro}{\rrhRwp}
%  \begin{macro}{\RubikRwp}
%  \begin{macro}{\textRubikRwp}
%  These commands  all draw forms which denote the Rwp rotation.
% \begin{macrocode}
\newcommand{\rrRwp}{\textbf{\textsf{R\footnotesize{w}}$^\prime$}}
%%
\newcommand{\SquareRwp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick] (\sd, \sb) -- (\sd, \sbh);
\draw [thick, <-] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick, <-] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhRwp}{\raisebox{-0.333\height}{\SquareRwp}\,}
%%
\newcommand{\RubikRwp}{%
\begin{minipage}{0.6cm}
\centering
\SquareRwp\\
\rrRwp
\end{minipage}%
}
\newcommand{\textRubikRwp}{\rrRwp\,\rrhRwp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Rs}
%
%  \begin{macro}{\rrRs}
%  \begin{macro}{\SquareRs}
%  \begin{macro}{\rrhRs}
%  \begin{macro}{\RubikRs}
%  \begin{macro}{\textRubikRs}
%  These commands  all draw forms which denote the Rs rotation.
% \begin{macrocode}
\newcommand{\rrRs}{\@rrs{R}}
%%
\newcommand{\SquareRs}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick,->] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick,->] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhRs}{\raisebox{-0.333\height}{\SquareRs}\,}
%%
\newcommand{\RubikRs}{%
\begin{minipage}{0.6cm}
\centering
\SquareRs\\
\rrRs
\end{minipage}%
}
\newcommand{\textRubikRs}{\rrRs\,\rrhRs}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Rsp}
%
%  \begin{macro}{\rrRsp}
%  \begin{macro}{\SquareRsp}
%  \begin{macro}{\rrhRsp}
%  \begin{macro}{\RubikRsp}
%  \begin{macro}{\textRubikRsp}
%  These commands  all draw forms which denote the Rsp rotation.
% \begin{macrocode}
\newcommand{\rrRsp}{\@rrsp{R}}
%%
\newcommand{\SquareRsp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick,<-] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick,<-] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhRsp}{\raisebox{-0.333\height}{\SquareRsp}\,}
%%
\newcommand{\RubikRsp}{%
\begin{minipage}{0.6cm}
\centering
\SquareRsp\\
\rrRsp
\end{minipage}%
}
\newcommand{\textRubikRsp}{\rrRsp\,\rrhRsp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Ra}
%
%  \begin{macro}{\rrRa}
%  \begin{macro}{\SquareRa}
%  \begin{macro}{\rrhRa}
%  \begin{macro}{\RubikRa}
%  \begin{macro}{\textRubikRa}
%  These commands  all draw forms which denote the Ra rotation.
% \begin{macrocode}
\newcommand{\rrRa}{\@rra{R}}
%%
\newcommand{\SquareRa}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick,<-] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick,->] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhRa}{\raisebox{-0.333\height}{\SquareRa}\,}
%%
\newcommand{\RubikRa}{%
\begin{minipage}{0.6cm}
\centering
\SquareRa\\
\rrRa
\end{minipage}%
}
\newcommand{\textRubikRa}{\rrRa\,\rrhRa}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Rap}
%
%  \begin{macro}{\rrRap}
%  \begin{macro}{\SquareRap}
%  \begin{macro}{\rrhRap}
%  \begin{macro}{\RubikRap}
%  \begin{macro}{\textRubikRap}
%  These commands  all draw forms which denote the Rap rotation.
% \begin{macrocode}
\newcommand{\rrRap}{\@rrap{R}}
%%
\newcommand{\SquareRap}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick,->] (\sd, \sb) -- (\sd, \sbh);
\draw [thick] (\sdd,\sb) -- (\sdd, \sbh);
\draw [thick,<-] (\sddd, \sb) -- (\sddd, \sbh);
\end{tikzpicture}%
}
\newcommand{\rrhRap}{\raisebox{-0.333\height}{\SquareRap}\,}
%%
\newcommand{\RubikRap}{%
\begin{minipage}{0.6cm}
\centering
\SquareRap\\
\rrRap
\end{minipage}%
}
\newcommand{\textRubikRap}{\rrRap\,\rrhRap}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation S}
%
%  \begin{macro}{\rrS}
%  \begin{macro}{\SquareS}
%  \begin{macro}{\rrhS}
%  \begin{macro}{\RubikS}
%  \begin{macro}{\textRubikS}
%  These commands  all draw forms which denote the S rotation.
% Not visible from the front.
% \begin{macrocode}
\newcommand{\rrS}{\@rr{S}}
\newcommand{\SquareS}{\@SquareLetter{\rrS}}
\newcommand{\rrhS}{\raisebox{-0.25mm}{\SquareS}\,}
\newcommand{\RubikS}{\raisebox{\@hRubik}{\SquareS}\,}
\newcommand{\textRubikS}{\rrhS\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Sp}
%
%  \begin{macro}{\rrSp}
%  \begin{macro}{\SquareSp}
%  \begin{macro}{\rrhSp}
%  \begin{macro}{\RubikSp}
%  \begin{macro}{\textRubikSp}
%  These commands  all draw forms which denote the Sp rotation.
% Not visible from the front.
% \begin{macrocode}
\newcommand{\rrSp}{\@rrp{S}}
\newcommand{\SquareSp}{\@SquareLetter{\rrSp}}
\newcommand{\rrhSp}{\raisebox{-0.25mm}{\SquareSp}\,}
\newcommand{\RubikSp}{\raisebox{\@hRubik}{\SquareSp}\,}
\newcommand{\textRubikSp}{\rrhSp\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Su}
%
%  \begin{macro}{\rrSu}
%  \begin{macro}{\rrhSu}
%  \begin{macro}{\RubikSu}
%  \begin{macro}{\textRubikSu}
%  These commands  draw forms of the Singmaster Su slice rotation.
%  We also need to finetune the spacing 
%  between these `slice' hieroglyphs (especially Fs  and Bs).
% \begin{macrocode}
\newcommand{\rrSu}{\textbf{\textsf{S\footnotesize{u}}}}
\newcommand{\rrhSu}{\rrhEp}%
\newcommand{\RubikSu}{%
\begin{minipage}{0.6cm}
\centering
\SquareEp\\
\rrSu
\end{minipage}%
}
\newcommand{\textRubikSu}{\rrSu\,\rrhEp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Sup}
%
%  \begin{macro}{\rrSup}
%  \begin{macro}{\rrhSup}
%  \begin{macro}{\RubikSup}
%  \begin{macro}{\textRubikSup}
%  These commands  draw forms of the Singmaster Sup slice rotation.
%  We also need to finetune the spacing 
%  between these `slice' hieroglyphs (especially Fs  and Bs).
% \begin{macrocode}
\newcommand{\rrSup}{\textbf{\textsf{S\footnotesize{u}}$^\prime$}}
\newcommand{\rrhSup}{\rrhE}%
\newcommand{\RubikSup}{%
\begin{minipage}{0.6cm}
\centering
\SquareE\\
\rrSup
\end{minipage}%
}
\newcommand{\textRubikSup}{\rrSup\,\rrhE}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Sd}
%
%  \begin{macro}{\rrSd}
%  \begin{macro}{\rrhSd}
%  \begin{macro}{\RubikSd}
%  \begin{macro}{\textRubikSd}
%  These commands  draw forms of the Singmaster Sd slice rotation.
% \begin{macrocode}
\newcommand{\rrSd}{\textbf{\textsf{S\footnotesize{d}}}}
\newcommand{\rrhSd}{\rrhE}%
\newcommand{\RubikSd}{%
\begin{minipage}{0.6cm}
\centering
\SquareE\\
\rrSd
\end{minipage}%
}
\newcommand{\textRubikSd}{\rrSd\,\rrhE}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Sdp}
%
%  \begin{macro}{\rrSdp}
%  \begin{macro}{\rrhSdp}
%  \begin{macro}{\RubikSdp}
%  \begin{macro}{\textRubikSdp}
%  These commands  draw forms of the Singmaster Sdp slice rotation.
% \begin{macrocode}
\newcommand{\rrSdp}{\textbf{\textsf{S\footnotesize{d}}$^\prime$}}
\newcommand{\rrhSdp}{\rrhEp}%
\newcommand{\RubikSdp}{%
\begin{minipage}{0.6cm}
\centering
\SquareEp\\
\rrSdp
\end{minipage}%
}
\newcommand{\textRubikSdp}{\rrSdp\,\rrhEp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Sl}
%
%  \begin{macro}{\rrSl}
%  \begin{macro}{\rrhSl}
%  \begin{macro}{\RubikSl}
%  \begin{macro}{\textRubikSl}
%  These commands  draw forms of the Singmaster Sl slice rotation.
% \begin{macrocode}
\newcommand{\rrSl}{\textbf{\textsf{S\footnotesize{l}}}}
\newcommand{\rrhSl}{\rrhM}%
\newcommand{\RubikSl}{%
\begin{minipage}{0.6cm}
\centering
\SquareM\\
\rrSl
\end{minipage}%
}
\newcommand{\textRubikSl}{\rrSl\,\rrhM}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Slp}
%
%  \begin{macro}{\rrSlp}
%  \begin{macro}{\rrhSlp}
%  \begin{macro}{\RubikSlp}
%  \begin{macro}{\textRubikSlp}
%  These commands  draw forms of the Singmaster Slp slice rotation.
% \begin{macrocode}
\newcommand{\rrSlp}{\textbf{\textsf{S\footnotesize{l}}$^\prime$}}
\newcommand{\rrhSlp}{\rrhMp}%
\newcommand{\RubikSlp}{%
\begin{minipage}{0.6cm}
\centering
\SquareMp\\
\rrSlp
\end{minipage}%
}
\newcommand{\textRubikSlp}{\rrSlp\,\rrhMp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Sr}
%
%  \begin{macro}{\rrSr}
%  \begin{macro}{\rrhSr}
%  \begin{macro}{\RubikSr}
%  \begin{macro}{\textRubikSr}
%  These commands  draw forms of the Singmaster Sr slice rotation.
% \begin{macrocode}
\newcommand{\rrSr}{\textbf{\textsf{S\footnotesize{r}}}}
\newcommand{\rrhSr}{\rrhMp}%
\newcommand{\RubikSr}{%
\begin{minipage}{0.6cm}
\centering
\SquareMp\\
\rrSr
\end{minipage}%
}
\newcommand{\textRubikSr}{\rrSr\,\rrhMp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Srp}
%
%  \begin{macro}{\rrSrp}
%  \begin{macro}{\rrhSrp}
%  \begin{macro}{\RubikSrp}
%  \begin{macro}{\textRubikSrp}
%  These commands  draw forms of the Singmaster Srp slice rotation.
% \begin{macrocode}
\newcommand{\rrSrp}{\textbf{\textsf{S\footnotesize{r}}$^\prime$}}
\newcommand{\rrhSrp}{\rrhM}%
\newcommand{\RubikSrp}{%
\begin{minipage}{0.6cm}
\centering
\SquareM\\
\rrSrp
\end{minipage}%
}
\newcommand{\textRubikSrp}{\rrSrp\,\rrhM}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Sf}
%
%  \begin{macro}{\rrSf}
%  \begin{macro}{\rrhSf}
%  \begin{macro}{\RubikSf}
%  \begin{macro}{\textRubikSf}
%  These commands  draw forms of the Singmaster Sf slice rotation.
%  We need to just make square with Sf in square; 
%  adjust box height using a \cmd{\rule};
%  adjust \cmd{\fboxsep} (default=3pt);
%  adjust \cmd{\fboxrule} (default=0.4pt);
%  bounded by \{\} so no need to reset to defaults.
% Not visible from the front.
% \begin{macrocode}
\newcommand{\rrSf}{\textbf{\textsf{S\footnotesize{f}}}}
\newcommand{\SquareSf}{\@SquareLetter{\rrSf}}
\newcommand{\rrhSf}{\raisebox{-0.25mm}{\SquareSf}\,}
\newcommand{\RubikSf}{\raisebox{\@hRubik}{\SquareSf}\,}
\newcommand{\textRubikSf}{\rrhSf\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Sfp}
%
%  \begin{macro}{\rrSfp}
%  \begin{macro}{\rrhSfp}
%  \begin{macro}{\RubikSfp}
%  \begin{macro}{\textRubikSfp}
%  These commands  draw forms of the Singmaster Sfp slice rotation.
%  We need to just make square with Sfp in square; 
%  adjust box height using a \cmd{\rule};
%  adjust \cmd{\fboxsep} (default=3pt);
%  adjust \cmd{\fboxrule} (default=0.4pt);
%  bounded by \{\} so no need to reset to defaults.
% Not visible from the front.
% \begin{macrocode}
\newcommand{\rrSfp}{\textbf{\textsf{S\footnotesize{f}}$^\prime$}}
\newcommand{\SquareSfp}{\@SquareLetter{\rrSfp}}
\newcommand{\rrhSfp}{\raisebox{-0.25mm}{\SquareSfp}\,}
\newcommand{\RubikSfp}{\raisebox{\@hRubik}{\SquareSfp}\,}
\newcommand{\textRubikSfp}{\rrhSfp\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Sb}
%
%  \begin{macro}{\rrSb}
%  \begin{macro}{\rrhSb}
%  \begin{macro}{\RubikSb}
%  \begin{macro}{\textRubikSb}
%  These commands  draw forms of the Singmaster Sb slice rotation.
%  We need to just make square with Sb in square; 
%  adjust box height using a \cmd{\rule};
%  adjust \cmd{\fboxsep} (default=3pt);
%  adjust \cmd{\fboxrule} (default=0.4pt);
%  bounded by \{\} so no need to reset to defaults.
% Not visible from the front.
% \begin{macrocode}
\newcommand{\rrSb}{\textbf{\textsf{S\footnotesize{b}}}}
\newcommand{\SquareSb}{\@SquareLetter{\rrSb}}
\newcommand{\rrhSb}{\raisebox{-0.25mm}{\SquareSb}\,}
\newcommand{\RubikSb}{\raisebox{\@hRubik}{\SquareSb}\,}
\newcommand{\textRubikSb}{\rrhSb\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Sbp}
%
%  \begin{macro}{\rrSbp}
%  \begin{macro}{\rrhSbp}
%  \begin{macro}{\RubikSbp}
%  \begin{macro}{\textRubikSbp}
%  These commands  draw forms of the Singmaster Sbp slice rotation.
%  We need to just make square with Sbp in square; 
%  adjust box height using a \cmd{\rule};
%  adjust \cmd{\fboxsep} (default=3pt);
%  adjust \cmd{\fboxrule} (default=0.4pt);
%  bounded by \{\} so no need to reset to defaults.
% Not visible from th front.
% \begin{macrocode}
\newcommand{\rrSbp}{\textbf{\textsf{S\footnotesize{b}}$^\prime$}}
\newcommand{\SquareSbp}{\@SquareLetter{\rrSbp}}
\newcommand{\rrhSbp}{\raisebox{-0.25mm}{\SquareSbp}\,}
\newcommand{\RubikSbp}{\raisebox{\@hRubik}{\SquareSbp}\,}
\newcommand{\textRubikSbp}{\rrhSbp\,}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation U}
%
%  \begin{macro}{\rrU}
%  \begin{macro}{\SquareU}
%  \begin{macro}{\rrhU}
%  \begin{macro}{\RubikU}
%  \begin{macro}{\textRubikU}
%  These commands  all draw forms which denote the U rotation.
% \begin{macrocode}
\newcommand{\rrU}{\textbf{\textsf{U}}}
%%
\newcommand{\SquareU}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhU}{\raisebox{-0.333\height}{\SquareU}\,}
%%
\newcommand{\RubikU}{%
\begin{minipage}{0.6cm}
\centering
\SquareU\\
\rrU
\end{minipage}%%
}
\newcommand{\textRubikU}{\rrU\,\rrhU}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Uw}
%
%  \begin{macro}{\rrUw}
%  \begin{macro}{\SquareUw}
%  \begin{macro}{\rrhUw}
%  \begin{macro}{\RubikUw}
%  \begin{macro}{\textRubikUw}
%  These commands  all draw forms which denote the Uw rotation.
% \begin{macrocode}
\newcommand{\rrUw}{\textbf{\textsf{U\footnotesize{w}}}}
%%
\newcommand{\SquareUw}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick, <-] (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
%
\newcommand{\rrhUw}{\raisebox{-0.333\height}{\SquareUw}\,}
%%
\newcommand{\RubikUw}{%
\begin{minipage}{0.6cm}
\centering
\SquareUw\\
\rrUw
\end{minipage}%%
}
%%
\newcommand{\textRubikUw}{\rrUw\,\rrhUw}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Up}
%
%  \begin{macro}{\rrUp}
%  \begin{macro}{\SquareUp}
%  \begin{macro}{\rrhUp}
%  \begin{macro}{\RubikUp}
%  \begin{macro}{\textRubikUp}
%  These commands  all draw forms which denote the Up rotation.
% \begin{macrocode}
\newcommand{\rrUp}{\textbf{\textsf{U}$^\prime$}}
%%
\newcommand{\SquareUp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, ->] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhUp}{\raisebox{-0.333\height}{\SquareUp}\,}
%%
\newcommand{\RubikUp}{%
\begin{minipage}{0.6cm}
\centering
\SquareUp\\
\rrUp
\end{minipage}%%
}
\newcommand{\textRubikUp}{\rrUp\,\rrhUp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Uwp}
%
%  \begin{macro}{\rrUwp}
%  \begin{macro}{\SquareUwp}
%  \begin{macro}{\rrhUwp}
%  \begin{macro}{\RubikUwp}
%  \begin{macro}{\textRubikUwp}
%  These commands  all draw forms which denote the Uwp rotation.
% \begin{macrocode}
\newcommand{\rrUwp}{\textbf{\textsf{U\footnotesize{w}}$^\prime$}}
%%
\newcommand{\SquareUwp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, ->] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick, ->] (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhUwp}{\raisebox{-0.333\height}{\SquareUwp}\,}
%%
\newcommand{\RubikUwp}{%
\begin{minipage}{0.6cm}
\centering
\SquareUwp\\
\rrUwp
\end{minipage}%%
}
\newcommand{\textRubikUwp}{\rrUwp\,\rrhUwp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Us} 
%
%  \begin{macro}{\rrUs}
%  \begin{macro}{\SquareUs}
%  \begin{macro}{\rrhUs}
%  \begin{macro}{\RubikUs}
%  \begin{macro}{\textRubikUs}
%  These commands  all draw forms which denote the Us rotation.
% \begin{macrocode}
\newcommand{\rrUs}{\@rrs{U}}
%%
\newcommand{\SquareUs}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick, <-]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhUs}{\raisebox{-0.333\height}{\SquareUs}\,}
%%
\newcommand{\RubikUs}{%
\begin{minipage}{0.6cm}
\centering
\SquareUs\\
\rrUs
\end{minipage}%
}
\newcommand{\textRubikUs}{\rrUs\,\rrhUs}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Usp} 
%
%  \begin{macro}{\rrUsp}
%  \begin{macro}{\SquareUsp}
%  \begin{macro}{\rrhUs}
%  \begin{macro}{\RubikUs}
%  \begin{macro}{\textRubikUsp}
%  These commands  all draw forms which denote the Usp rotation.
% \begin{macrocode}
\newcommand{\rrUsp}{\@rrsp{U}}
%%
\newcommand{\SquareUsp}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, ->] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick, ->]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhUsp}{\raisebox{-0.333\height}{\SquareUsp}\,}
%%
\newcommand{\RubikUsp}{%
\begin{minipage}{0.6cm}
\centering
\SquareUsp\\
\rrUsp
\end{minipage}%
}
\newcommand{\textRubikUsp}{\rrUsp\,\rrhUsp}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Ua} 
%
%  \begin{macro}{\rrUa}
%  \begin{macro}{\SquareUa}
%  \begin{macro}{\rrhUa}
%  \begin{macro}{\RubikUa}
%  \begin{macro}{\textRubikUa}
%  These commands  all draw forms which denote the Ua rotation.
% \begin{macrocode}
\newcommand{\rrUa}{\@rra{U}}
%%
\newcommand{\SquareUa}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, <-] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick, ->]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhUa}{\raisebox{-0.333\height}{\SquareUa}\,}
%%
\newcommand{\RubikUa}{%
\begin{minipage}{0.6cm}
\centering
\SquareUa\\
\rrUa
\end{minipage}%
}
\newcommand{\textRubikUa}{\rrUa\,\rrhUa}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotation Uap} 
%
%  \begin{macro}{\rrUap}
%  \begin{macro}{\SquareUap}
%  \begin{macro}{\rrhUap}
%  \begin{macro}{\RubikUap}
%  \begin{macro}{\textRubikUap}
%  These commands  all draw forms which denote the Uap rotation.
% \begin{macrocode}
\newcommand{\rrUap}{\@rrap{U}}
%%
\newcommand{\SquareUap}{%
\begin{tikzpicture}[scale=0.5]
\DrawNotationBox;
\draw [thick, ->] (\sb,\sddd) -- (\sbh, \sddd);
\draw [thick]     (\sb,\sdd) --  (\sbh, \sdd);
\draw [thick, <-]     (\sb,\sd) --   (\sbh, \sd);
\end{tikzpicture}%
}
\newcommand{\rrhUap}{\raisebox{-0.333\height}{\SquareUap}\,}
%%
\newcommand{\RubikUap}{%
\begin{minipage}{0.6cm}
\centering
\SquareUap\\
\rrUap
\end{minipage}%
}
\newcommand{\textRubikUap}{\rrUap\,\rrhUap}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsection{    Rotation commands --- axis rotations}
%
%  \subsubsection{    Rotations x and xp}
%
%  \begin{macro}{\rrx}
%  \begin{macro}{\rrhx}
%  \begin{macro}{\Rubikx}
%  These commands  all draw forms which denote the x rotation.
% \begin{macrocode}
\newcommand{\rrx}{\textbf{\textsf{x}}}
\newcommand{\Rubikx}{\@xyzRubik{x}}
\newcommand{\rrhx}{\@xyzh{x}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%  \begin{macro}{\rrxp}
%  \begin{macro}{\rrhxp}
%  \begin{macro}{\Rubikxp}
%  These commands  all draw forms which denote the xp rotation.
% \begin{macrocode}
\newcommand{\rrxp}{\textbf{\textsf{x}$^\prime$}}
\newcommand{\Rubikxp}{\@xyzRubikp{x}}
\newcommand{\rrhxp}{\@xyzhp{x}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotations y and yp}
%
%  \begin{macro}{\rry}
%  \begin{macro}{\rrhy}
%  \begin{macro}{\Rubiky}
%  These commands  all draw forms which denote the y rotation.
% \begin{macrocode}
\newcommand{\rry}{\textbf{\textsf{y}}}
\newcommand{\Rubiky}{\@xyzRubik{y}}
\newcommand{\rrhy}{\@xyzh{y}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%  \begin{macro}{\rryp}
%  \begin{macro}{\rrhyp}
%  \begin{macro}{\Rubikyp}
%  These commands  all draw forms which denote the yp rotation.
% \begin{macrocode}
\newcommand{\rryp}{\textbf{\textsf{y}$^\prime$}}
\newcommand{\Rubikyp}{\@xyzRubikp{y}}
\newcommand{\rrhyp}{\@xyzhp{y}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotations z and zp}
%
%  \begin{macro}{\rrz}
%  \begin{macro}{\rrhz}
%  \begin{macro}{\Rubikz}
%  These commands  all draw forms which denote the z rotation.
% \begin{macrocode}
\newcommand{\rrz}{\textbf{\textsf{z}}}
\newcommand{\Rubikz}{\@xyzRubik{z}}
\newcommand{\rrhz}{\@xyzh{z}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%  \begin{macro}{\rrzp}
%  \begin{macro}{\rrhzp}
%  \begin{macro}{\Rubikzp}
%  These commands  all draw forms which denote the zp rotation.
% \begin{macrocode}
\newcommand{\rrzp}{\textbf{\textsf{z}$^\prime$}}
\newcommand{\Rubikzp}{\@xyzRubikp{z}}
\newcommand{\rrhzp}{\@xyzhp{z}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotations u and d}
%
%  \begin{macro}{\rru}
%  \begin{macro}{\rrhu}
%  \begin{macro}{\Rubiku}
%  These commands  all draw forms which denote the u rotation.
% \begin{macrocode}
\newcommand{\rru}{\textbf{\textsf{u}}}
\newcommand{\Rubiku}{\@xyzRubik{u}}
\newcommand{\rrhu}{\@xyzh{u}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%  \begin{macro}{\rrd}
%  \begin{macro}{\rrhd}
%  \begin{macro}{\Rubikd}
%  These commands  all draw forms which denote the d rotation.
% \begin{macrocode}
\newcommand{\rrd}{\textbf{\textsf{d}}}
\newcommand{\Rubikd}{\@xyzRubik{d}}
\newcommand{\rrhd}{\@xyzh{d}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotations l and r}
%
%  \begin{macro}{\rrl}
%  \begin{macro}{\rrhl}
%  \begin{macro}{\Rubikl}
%  These commands  all draw forms which denote the l rotation.
% \begin{macrocode}
\newcommand{\rrl}{\textbf{\textsf{l}}}
\newcommand{\Rubikl}{\@xyzRubik{l}}
\newcommand{\rrhl}{\@xyzh{l}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%  \begin{macro}{\rrr}
%  \begin{macro}{\rrhr}
%  \begin{macro}{\Rubikr}
%  These commands  all draw forms which denote the r rotation.
% \begin{macrocode}
\newcommand{\rrr}{\textbf{\textsf{r}}}
\newcommand{\Rubikr}{\@xyzRubik{r}}
\newcommand{\rrhr}{\@xyzh{r}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
%  \subsubsection{     Rotations f and b}
%
%  \begin{macro}{\rrf}
%  \begin{macro}{\rrhf}
%  \begin{macro}{\Rubikf}
%  These commands  all draw forms which denote the f rotation.
% \begin{macrocode}
\newcommand{\rrf}{\textbf{\textsf{f}}}
\newcommand{\Rubikf}{\@xyzRubik{f}}
\newcommand{\rrhf}{\@xyzh{f}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%  \begin{macro}{\rrb}
%  \begin{macro}{\rrhb}
%  \begin{macro}{\Rubikb}
%  These commands  all draw forms which denote the b rotation.
% \begin{macrocode}
\newcommand{\rrb}{\textbf{\textsf{b}}}
\newcommand{\Rubikb}{\@xyzRubik{b}}
\newcommand{\rrhb}{\@xyzh{b}}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%
%
% --------------------------
%    End of this package
% --------------------------
%    \begin{macrocode}
%</rubikcube>
%    \end{macrocode}
%
%
%
%
% \Finale
%
\endinput
