%%
%% rubikexamples.tex
%% February 25, 2018
%% Requires ALL the rubik bundle packages and also the tikz package
%% Part of the Rubik bundle v5.0 (www.ctan.org/pkg/rubik)
%% authors: RWD Nickalls & A Syropoulos
%%--------NOTE-------------------------------------------
%% USAGE: (pdf)latex --shell-escape rubikexamples.tex
%% run three times to get all labels and fig nos correct
%%-------------------------------------------------------
\documentclass[a4paper]{article}
%------rubik----------
\usepackage{tikz} %% load tikz BEFORE rubikcube
\usepackage{rubikcube,rubikrotation,rubikpatterns,rubiktwocube}
%----------------------
\usepackage{url}
\usepackage{ifpdf}
\usepackage{lmodern}
%---------------------
\ifpdf
\usepackage[verbose]{microtype}
\usepackage{cmap}
\usepackage[pdfencoding=auto]{hyperref}
\hypersetup{%
pdftitle={rubikexamples.pdf},
pdfsubject={Rubik cube, Rubik bundle},
pdfkeywords={Rubik cube LaTeX}
}
\fi
%%-------------------
\pagestyle{myheadings}
\markright{\texttt{rubikexamples.pdf} \ \
(Rubik bundle v5.0, 2018) \ \ \texttt{www.ctan.org/pkg/rubik}}
%% increase text height and width
\addtolength{\oddsidemargin}{-1.5cm}
\addtolength{\textwidth}{3cm}
\addtolength{\textheight}{1cm}
%----------------------
%==========Rubik stuff================
%----edge sequences for 3x3x3 cubes--------------------------
\newcommand{\CycleThreeEdgesFlipTwo}{[CycleThreeEdgesFlipTwo],F,R,U,Rp,Up,Fp}%
\newcommand{\cyclethreeedgesfliptwo}{\CycleThreeEdgesFlipTwo}%
%
%----corner sequences--------------------------
\newcommand{\AllYellow}{[allyellow],R,U,Rp,U,R,Up,Up,Rp}% = SUNE %cross -->allyellow
\newcommand{\allyellow}{\AllYellow}%
\newcommand{\CycleThreeCorners}{[cyclethreecorners],Lp,U,R,Up,L,U,Rp,Up}%
\newcommand{\cyclethreecorners}{\CycleThreeCorners}%
\newcommand{\SwapTwoCorners}{[swaptwocorners],Rp,F,Rp,B2,R,Fp,Rp,B2,R2,Up}
\newcommand{\swaptwocorners}{\SwapTwoCorners}
%
%% brace and bracket macros
\newcommand{\Rubikbracket}[1]{$\left(\mbox{#1}\right)$}
\newcommand{\Rubikbrace}[1]{$\left\{\mbox{#1}\right\}$}
\newcommand{\cubenumber}[1]{\strut\raisebox{1cm}{#1}}
%-----------------
%
\begin{document}
%
\ifpdf\pdfbookmark[1]{Title}{Title}\fi
\title{Rubik examples\\
\smallskip\normalsize\texttt{www.ctan.org/tex-archives/macros/latex/contrib/rubik/rubikexamples.pdf}
\,\footnote{This file is part of the Rubik bundle v5.0. To generate this file,
use the following command: \newline \texttt{\$ pdflatex --shell-escape rubikexamples.tex}}}
\author{RWD Nickalls\,\footnote{email: \textsf{dick@nickalls.org}}%
\ \ \& A Syropoulos\,\footnote{email: \textsf{asyropoulos@yahoo.com}}}
\date{25 February, 2018 (Rubik bundle v5.0)}
\maketitle
\tableofcontents
\pagebreak
%%----------------------
\section{Preliminaries}
These examples were generated using the \TeX\
Rubik bundle\,\footnote{\url{http://www.ctan.org/pkg/rubik}} v5.0.
They assume some familiarity with the four packages \textsc{rubikcube},
\textsc{rubikrotation}, \textsc{rubikpatterns} and \textsc{rubiktwocube}.
For~documentation see the following files:
\begin{quote}
\begin{verbatim}
rubikcube.pdf 3x3x3 documentation
rubikrotation.pdf commands for implementing rotation sequences
rubikrotationPL.pdf documentation of rubikrotation.pl (Perl program)
rubikpatterns.pdf small database of well-known 3x3x3 configurations
rubikpatternsLIST.pdf shows all configurations in the rubikpatterns database
rubiktwocube.pdf 2x2x2 documentation
\end{verbatim}
\end{quote}
This file requires the following packages: \texttt{tikz, rubikcube, rubikrotation,
rubikpatterns, rubiktwocube}; note that the \texttt{tikz} package must be loaded
\textit{before} the \texttt{rubikcube} package.
This file needs to be run using the \verb!--shell-escape! command-line option;
for example:
\begin{quote}
\begin{verbatim}
pdflatex --shell-escape rubikexample.tex
\end{verbatim}
\end{quote}
This is because nearly all the examples make use of the \verb!\RubikRotation!
command, which calls the Perl script \texttt{rubikrotation.pl}.
If you do forget to use the \verb!--shell-escape! command-line switch,
the file will still run, but all the cubes will remain in the initial
unprocessed configuration\,\footnote{As a quick check, run this file
(\texttt{rubikexamples.tex}) in a separate directory and look at Figure~1;
if this appears as a `solved' cube then your system has failed to call the Perl
script \texttt{rubikrotation.pl} correctly for some reason. The most likely
causes of this are (a)~failure to invoke the \texttt{--shell-escape} command-line
option correctly, or (b)~failure to use a configuration file correctly
(ie your system can't find the Perl program)---search the \texttt{.log} file
to see whether the file \texttt{rubikrotation.pl} has been used.}.
\subsection{3x3x3 vs 2x2x2 command names}
The introduction of the \textsc{rubiktwocube} package (Rubik bundle v5)
has necessitated a few command name changes in order to avoid confusion
(see the \textsc{rubikcube} package documentation \S\,3.1 for details).
Consequently, command names containing the word `\texttt{Rubik}' denote
a 3x3x3 cube action; similarly those containing the word `\texttt{Two}'
denote a 2x2x2 cube action.
\subsection{Environments}
When using the Rubik bundle one sometimes needs to be mindful of the various
\LaTeX\ environments in which Rubik commands are placed (e.g.,~the figure,
minipage and TikZ picture environments), since these environments restrict
the actions of commands they contain to the particular environment.
The \verb!\ShowCube! command is also relevant here, since it is a
minipage-wrapper for the TikZ picture environment. Only Rubik \verb!\Draw..!
commands and TikZ commands (e.g.,~\verb!\draw! and \verb!\node!)
actually need to be inside a TikZ picture environment, and hence
inside a \verb!\ShowCube! command.
This issue arises because the Rubik bundle allows you to create figures
showing different stages during a sequence of rotations. Consequently
the effects of commands executed inside an environment (especially
commands which determine the colour-state or rotations), may not be
apparent to subsequent commands outside that particular environment.
See Example~3 for an illustration of how to handle environments.
\pagebreak
%%------------------
\section{RubikCube examples (3x3x3)}
\subsection{Sixspot}
\typeout{---Example (sixspot)}
In Figure~\ref{fig:sixspot} we show the so-called ``sixspot'' configuration,
generated from a solved cube using the rotation sequence
\rr{U}, \rr{Dp}, \rr{R}, \rr{Lp}, \rr{F}, \rr{Bp}, \rr{U}, \rr{Dp}.
%%--------FIGURE 1 -----------
\begin{figure}[hbt]
\centering
\RubikCubeSolvedWY
\RubikRotation{\sixspot}
\ShowCube{7cm}{0.7}{\DrawRubikCubeSF}
\caption{\label{fig:sixspot}The `sixspot' configuration.}
\end{figure}
%----------------
{\noindent}Creating a macro to hold a rotation sequence greatly
facilitates their use, as follows:
\begin{quote}
\begin{verbatim}
\newcommand{\sixspot}{[sixspot],U,Dp,R,Lp,F,Bp,U,Dp}
\end{verbatim}
\end{quote}
We can now process this sequence using its macro name as an argument
for the \verb!\RubikRotation! command. The code used for the above
Figure uses the \verb!\ShowCube{}{}{}! command for which \verb!#1!
is the minipage width, \verb!#2! is the tikzpicture scale factor (0--1),
and \verb!#3! can include \textsc{rubikcube} package \verb!\Draw..!
commands, and any commands which are valid for use in a TikZ
\texttt{tikzpicture} environment. The code for the above figure is
as follows:
\begin{quote}
\begin{verbatim}
\begin{figure}[hbt]
\centering
\RubikCubeSolvedWY
\RubikRotation{\sixspot}
\ShowCube{7cm}{0.7}{\DrawRubikCubeSF}
\caption{...}
\end{figure}
\end{verbatim}
\end{quote}
Note that the sixspot sequence is a so-called `order~3' sequence, which
means that running the `sixspot' sequence 3 times returns the cube
to its original `solved' state. The~command for processing it three
times is \verb!\RubikRotation[3]{\sixspot}!.
Note that the semi-flat form of the cube here is generated by the
\verb!\DrawRubikCubeSF! command, where the terminal SF denotes
the Semi-Flat form. To draw it completely flat use a terminal F,
e.g.,~\verb!\DrawRubikCubeF! (see Figure~\ref{fig:sixspotE}).
\subsubsection{Log-file extract}
Users may find it instructive to inspect the the log-file and
follow the dynamic interaction between \LaTeX\ and the Perl script.
This is easy to follow, since output by \texttt{rubikrotation.sty}
is prefixed by 3 dashes (\verb!---!), while output by the Perl script
is prefixed by 3 dots (\verb!...!). Search for the keyword `Example',
as this is written to the log-file at the start of each example.
The following is part of the log-file extract associated with Example~1.
\begin{quote}
\begin{verbatim}
---Example (sixspot)
---TeX process---------------------------
---script = rubikrotation.sty v5.0 (2018/02/25)
---NEW rotation command
---command = RubikRotation[1]{[SixSpot],U,Dp,R,Lp,F,Bp,U,Dp,<(8q*, 8f*)>}
---writing current cube state to file rubikstate.dat
\openout7 = `rubikstate.dat'.
\ourRRcounter=\count134
---CALLing Perl script (rubikrotation.pl)
runsystem(perl $HOME/RubikTeX/PERLstuff/dtx-rubikrotation/rubikrotation.pl
-i rubikstate.dat -o rubikstateNEW.dat)...executed.
---inputting NEW datafile (data written by Perl script)
(./rubikstateNEW.dat
...PERL process..................................
...script = rubikrotation.pl v5.0 (25 February 2018)
...reading the current cube state (from File: rubikstate.dat)
...
...command = cubesize,three
...cube = THREEcube
...
......up,W,W,W,W,W,W,W,W,W
....down,Y,Y,Y,Y,Y,Y,Y,Y,Y
....left,B,B,B,B,B,B,B,B,B
...right,G,G,G,G,G,G,G,G,G
...front,O,O,O,O,O,O,O,O,O
....back,R,R,R,R,R,R,R,R,R
...
...rotation keyword
...checking state of cube
...cubiesum = 54 (Red=9, Or=9, Ye=9, Gr=9, Bl=9, Wh=9, X=0)
...
...command = rotation,[SixSpot],U,Dp,R,Lp,F,Bp,U,Dp,<(8q*, 8f*)>
...dataline = rotation,[SixSpot],U,Dp,R,Lp,F,Bp,U,Dp,<(8q*; 8f*)>
...[SixSpot] is a label OK
...rotation U, OK
...rotation Dp, OK
...rotation R, OK
...rotation Lp, OK
...rotation F, OK
...rotation Bp, OK
...rotation U, OK
...rotation Dp, OK
...writing new cube state to file rubikstateNEW.dat
...SequenceName = SixSpot
...SequenceInfo = (8q*; 8f*)
...SequenceShort = [SixSpot],U,Dp,R,Lp,F,Bp,U,Dp
...SequenceLong = U,Dp,R,Lp,F,Bp,U,Dp
)
\end{verbatim}
\end{quote}
\pagebreak
\subsection{ShowErrors}
\typeout{---Example (ShowRubikErrors)}
In this example we demonstrate the use of the \verb!\ShowErrors!
command\,\footnote{ The \texttt{ {\textbackslash}ShowErrors} command
replaces the earlier \texttt{{\textbackslash}ShowRubikErrors} command
which is now deprecated. However, the earlier command is still
supported (for now) for backward compatibility.},
which places a copy of the Perl output file \verb!rubikstateERRORS.dat!
underneath the graphic so you can see a list of all the errors, if any.
Note that this example is similar to the previous one except that we have
introduced several errors---e.g.,~bad minipage width, typos, as well as
some animals---into the rotation sequence).
It is important to note that the \verb!\ShowErrors! command must be
placed \textit{after} the TikZ picture environment (i.e.,~in this case after
the \verb!\ShowCube! command), or even at the end of the document.
Note that full details of all errors are also included in the .log file (search for \texttt{*ERR}).
%%--------FIGURE 2 -----------
%
\RubikCubeSolvedWY
\RubikRotation{[sixspot],U,Dp,R,Lp,F,Bp,U,Dpppp,cat,dog}
\begin{figure}[hbt]
\centering
\ShowCubeF{4.5cm}{0.5}{\DrawRubikCubeF}
\ShowErrors
\parbox{0.8\textwidth}{%
\caption{\label{fig:sixspotE}The same `sixspot' sequence of rotations as shown
in Example~1, but now with some errors (wrong minipage width, typos and some
animals\,!) in the rotation sequence (it \textit{should} be just
\texttt{U,Dp,R,Lp,F,Bp,U,Dp}).}}
\end{figure}
%
{\noindent}In this example we have used the F version of the \verb!\ShowCube!
command (\verb!\ShowCubeF!) which places an fbox around the image so you can see the
extent of any white space etc. This reveals that the set minipage-width ($4.5$cm) in
the \verb!\ShowCubeF! command---see code below---is too narrow: it should be
5cm ($10 \times 0.5$) to just include the whole image (i.e.,~$10 \times$ the TikZ
scale-factor in this case).
Once fixed, we can remove the F from the \verb!\ShowCubeF! command.
Note also that only `\verb!\Draw...!' commands really need to be inside the TikZ
picture environment (i.e.,~inside the \verb!\ShowCube! command). The above figure
was generated by the following code.
%-----------------
\begin{quote}
\begin{verbatim}
\RubikCubeSolvedWY
\RubikRotation{[sixspot],U,Dp,R,Lp,F,Bp,U,Dpppp,cat,dog}
\begin{figure}[hbt]
\centering
\ShowCubeF{4.5cm}{0.5}{\DrawRubikCubeF}
\ShowErrors
\caption{....}
\end{figure}
\end{verbatim}
\end{quote}
Even if the \verb!\ShowErrors! command is not used, it is always a good idea
to check the file \verb!rubikstateERRORS.dat! after a \LaTeX\ run, since this
file will also reveal any errors.
\pagebreak
%----------------------
\subsection{Environments}
\typeout{---Example (Environments)}
In this example we highlight the fact that Rubik commands used inside a \LaTeX\
environment remain local to that environment, and how this can sometimes be
problematic. Commands whose reach is meant to be more global need to be
executed outside such environments, where they can implement global colour
settings, which will then be accessible to subsequent \verb!\Draw..! commands.
Since we are drawing images, this is primarily an issue
with the \verb!minipage!, \verb!figure!, and TikZ picture environments.
Consequently, it is generally best when drawing a sequence of cubes to reserve
the TikZ picture environment only for \verb!\Draw..! commands and TikZ
commands. Importantly, this also applies to our main display tool the \verb!\ShowCube!
command, since this is a minipage-wrapper for the TikZ picture environment
(see the \textsc{rubikcube} package documentation).
In this example the first cube (9a) uses a \verb!\RubikCubeSolvedWY! command
\textit{inside} a \verb!\ShowCube! environment. However, if we now perform the
rotation \textRubik{R}\ (using the command \verb!\RubikRotation{R}!) this results
in a quite unexpected effect on cube (9b) (and obviously not correct).
This is because the action of the initial \verb!\RubikCubeSolved! command
(setting a new colour-state) is not visible outside its \verb!\ShowCube!
environment, and hence the subsequent \verb!\RubikRotation{R}! command is unaware
of this earlier attempt to update the global colour-state information.
It turns out that the internal colour-state was actually last updated following
the action of the \textbackslash\texttt{RubikRotation\{[sixspot],....\}}
command used in Example~2, being the last colour-state command executed
\textit{outside} an environment (a \texttt{figure} environment in that
example). Consequently, the strange cube 9b is not what we expected.
\medskip
\noindent%
\cubenumber{9a}%
\ShowCube{2cm}{0.5}{%
\RubikCubeSolvedWY%
\DrawRubikCubeRU}%
\quad\Rubik{R}%
\RubikRotation{R}%
\cubenumber{9b}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
%---------------
\hspace{5mm}
\begin{minipage}{0.5\textwidth}
\begin{verbatim}
\usepackage{tikz,rubikcube,rubikrotation}
\newcommand{\cubenumber}[1]{\strut\raisebox{1cm}{#1}}
...
\cubenumber{9a}%
\ShowCube{2cm}{0.5}{%
\RubikCubeSolvedWY%
\DrawRubikCubeRU%
}%
\quad\Rubik{R}%
\RubikRotation{R}%
\cubenumber{9b}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\end{verbatim}
\end{minipage}
\bigskip
If we now bring the \verb!\RubikCubeSolvedWY! command out of the \verb!\ShowCube!
environment and place it \textit{before} the \verb!\ShowCube! command (see cubes 9c, 9d),
then its colour `state' information becomes globally accessible (i.e.,~colour-state
gets updated) and therefore gets used by the subsequent \verb!\RubikRotation{R}!
command, and hence the cube (9d) is now rendered correctly.
\bigskip
\noindent%
\cubenumber{9c}%
\RubikCubeSolvedWY%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\quad\Rubik{R}%
\RubikRotation{R}%
\cubenumber{9d}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
%---------------
\hspace{5mm}
\begin{minipage}{0.5\textwidth}
\begin{verbatim}
\cubenumber{9c}%
\RubikCubeSolvedWY%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\quad\Rubik{R}%
\RubikRotation{R}%
\cubenumber{9d}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\end{minipage}
\end{verbatim}
\end{minipage}
\bigskip
Note that the horizontal space occupied by \texttt{9d} is almost exactly
the same as \verb!\quad!, and so we don't need to include any additional
space before the \verb!\cubenumber{9d}! command.
\pagebreak
%----------------------
\subsection{Coordinates}
\typeout{---Example (coordinates)}
For all cubes the origin of coordinates is defined as the bottom left corner of
the \textsc{front} face. Consequently, it is easy to determine the coordinates
of points and hence draw lines, circles, and place lettering or other objects
using the standard TikZ \verb!\draw..! and \verb!\node..! commands. Note that for
convenience point $P$ (bottom back left corner) is designed to be $(-1,-1)$
on the 2D view. (The~following diagram is Fig~1 from the \textsc{rubikcube}
package documentation).
\begin{figure}[hbt]
\centering
\RubikFaceFrontAll{X} % X = default = grey
\ShowCube{5.6cm}{0.7}{%
\DrawRubikFaceFront
\draw[line join=round,line cap=round,ultra thick] (0,0) -- (0,4); % Y
\draw[line join=round,line cap=round,ultra thick] (0,0) -- (4,0); % X
%draw labels
\node (Ylabel) at (-0.35, 3.8) {$Y$};
\node (Xlabel) at ( 3.8, -0.4) {$X$};
%%----------draw the Rubik faces--------
%% outline Front face
\draw[line join=round,line cap=round,ultra thick]%
(0,3) -- (-1,2) -- (-1,-1) -- (2,-1) -- (3,0);
%%
%% outline Down face
\draw[line join=round,line cap=round,ultra thick]%
(-1,-1) -- (0, 0);
%%
%%
%% draw labels
\node (Y0) at (-0.4, 0) [blue]{$0$};
\node (Y1) at (-0.4, 1) [blue]{$1$};
\node (Y2) at (-0.4, 2) [blue]{$2$};
\node (Y3) at (-0.4, 3) [blue]{$3$};
%%
\node (X0) at (0, -0.5) [blue]{$0$};
\node (X1) at (1, -0.5) [blue]{$1$};
\node (X2) at (2, -0.5) [blue]{$2$};
\node (X3) at (3, -0.5) [blue]{$3$};
%%
\node (P) at (-2.4, -1) [blue]{$P\,(-1,-1)$};
}
\end{figure}
{\noindent}The code for the figure is given below.
We draw everything in the \verb!\ShowCube! environment; the \textsc{front} face
in grey (colour code = \texttt{X}) using the Rubik command
\verb!\DrawRubikFlatFront!, and then draw all the lines and text using standard
TikZ commands. The correct minipage-width argument (5.6cm) for the
\verb!\ShowCube! command is determined by trial-and-error, using the `fbox'
form of the command (\verb!\ShowCubeF!), and then the `F' is removed
($\rightarrow$ \verb!\ShowCube!). In order to avoid confusion, all Rubik commands
start with a capital letter (e.g.,~\verb!\Draw..!), while all TikZ commands
start with a lower-case letter (e.g.,~\verb!\draw..!).
\begin{quote}
\begin{verbatim}
\begin{figure}[hbt]
\centering
\RubikFaceFrontAll{X}% X = default non-colour (grey)
\ShowCube{5.6cm}{0.7}{%
\DrawRubikFaceFront
\draw[line join=round,line cap=round,ultra thick] (0,0) -- (0,4);% Yaxis
\draw[line join=round,line cap=round,ultra thick] (0,0) -- (4,0);% Xaxis
\node (Ylabel) at (-0.35, 3.8) {$Y$};
\node (Xlabel) at ( 3.8, -0.4) {$X$};
%% outline Left and Down faces
\draw[line join=round,line cap=round,ultra thick]%
(0,3) -- (-1,2) -- (-1,-1) -- (2,-1) -- (3,0);
\draw[line join=round,line cap=round,ultra thick]%
(-1,-1) -- (0, 0);
\node (Y0) at (-0.4, 0) [blue]{$0$};
\node (Y1) at (-0.4, 1) [blue]{$1$};
\node (Y2) at (-0.4, 2) [blue]{$2$};
\node (Y3) at (-0.4, 3) [blue]{$3$};
\node (X0) at (0, -0.5) [blue]{$0$};
\node (X1) at (1, -0.5) [blue]{$1$};
\node (X2) at (2, -0.5) [blue]{$2$};
\node (X3) at (3, -0.5) [blue]{$3$};
\node (P) at (-2.4, -1) [blue]{$P\,(-1,-1)$};
}
\end{figure}
\end{verbatim}
\end{quote}
\pagebreak
%----------------------
\subsection{Face notation}
\typeout{---Example (face notation)}
The following diagram is Fig~2 from the \textsc{rubikcube} package documentation.
\begin{figure}[htb]
\centering%
\RubikFaceUpAll{Y}
\RubikFaceFrontAll{G}
\RubikFaceRightAll{O}
\ShowCubeF{5cm}{0.6}{%
\DrawRubikCubeRU%
%%-----------------
%% Right
\draw[line width=2pt,color=blue,->] (3.5,2) -- (5.3, 2);
\node (R) at (4.6, 2.5) [blue]{\textbf{\textsf{R}}};
\node (x) at (5.8, 2) [blue]{\textbf{\textsf{x}}};
%%Left
\draw[line width=2pt,color=blue] (-0.2,2) -- (-1.5, 2);
\node (L) at (-1.1, 2.5) [blue]{\textbf{\textsf{L}}};
%%Up
\draw[line width=2pt,color=blue,->] (2, 3.5) -- (2, 5.5);
\node (U) at (1.4, 4.7) [blue]{\textbf{\textsf{U}}};
\node (y) at (2, 6.1) [blue]{\textbf{\textsf{y}}};
%%Down
\draw[line width=2pt,color=blue] (2, -0.2) -- (2, -1.5);
\node (D) at (2.6, -1.1) [blue]{\textbf{\textsf{D}}};
%%Front
\draw[line width=2pt,color=blue,->] (1.5, 1.5) -- (0, -1);
\node (F) at (0.7, -0.7) [blue]{\textbf{\textsf{F}}};
\node (z) at (-0.3, -1.4) [blue]{\textbf{\textsf{z}}};
%%Back
\draw[line width=2pt,color=blue] (3.2, 4.2) -- (4, 5.5);
\node (B) at (4.4, 5) [blue]{\textbf{\textsf{B}}};
%%
}
\end{figure}
{\noindent}The code for the figure is as follows
(the origin $(0,0)$ is at the bottom left corner of the \textsc{front} (green) face).
\begin{quote}
\begin{verbatim}
\begin{figure}[htb]
\centering%
\RubikFaceUpAll{Y}
\RubikFaceFrontAll{G}
\RubikFaceRightAll{O}
\ShowCubeF{5cm}{0.6}{%
\DrawRubikCubeRU%
%% Right
\draw[line width=2pt,color=blue,->] (3.5,2) -- (5.3, 2);
\node (R) at (4.6, 2.5) [blue]{\textbf{\textsf{R}}};
\node (x) at (5.8, 2) [blue]{\textbf{\textsf{x}}};
%%Left
\draw[line width=2pt,color=blue] (-0.2,2) -- (-1.5, 2);
\node (L) at (-1.1, 2.5) [blue]{\textbf{\textsf{L}}};
%%Up
\draw[line width=2pt,color=blue,->] (2, 3.5) -- (2, 5.5);
\node (U) at (1.4, 4.7) [blue]{\textbf{\textsf{U}}};
\node (y) at (2, 6.1) [blue]{\textbf{\textsf{y}}};
%%Down
\draw[line width=2pt,color=blue] (2, -0.2) -- (2, -1.5);
\node (D) at (2.6, -1.1) [blue]{\textbf{\textsf{D}}};
%%Front
\draw[line width=2pt,color=blue,->] (1.5, 1.5) -- (0, -1);
\node (F) at (0.7, -0.7) [blue]{\textbf{\textsf{F}}};
\node (z) at (-0.3, -1.4) [blue]{\textbf{\textsf{z}}};
%%Back
\draw[line width=2pt,color=blue] (3.2, 4.2) -- (4, 5.5);
\node (B) at (4.4, 5) [blue]{\textbf{\textsf{B}}};
}
\end{figure}
\end{verbatim}
\end{quote}
\pagebreak
%----------------------
\subsection{Grey cube}
\label{sec:greycube}
\typeout{---Example (grey cube)}
When explaining elementary layer-1 moves, it can be useful to use the `grey cube'
(\verb!\RubikCubeGreyWY!) ---White opposite Yellow--- as this sets up only
the central cubie on each face.
We have shown the first cube here in Semi-Flat (SF) mode simply to show how the grey
cube is configured (note that a WB version (\verb!\RubikCubeGreyWB!)
and an `all-grey' version (\verb!\RubikCubeAllGrey!) are also available.
Note that all these grey cube commands will also accept
the word `gray' (to be consistent with TikZ).
In this example, we show how to position a single `flipped' white/orange edge
cubie in the top layer.
\begin{figure}[hbt]
\centering%
\RubikCubeGreyWY%
\RubikFaceUp{X}{X}{X}%
{X}{W}{O}%
{X}{X}{X}%
\RubikFaceRight{X}{W}{X}
{X}{G}{X}
{X}{X}{X}
%
\cubenumber{1}%
\ShowCube{5cm}{0.5}{\DrawRubikCubeSF}%
%------------------
\quad\Rubik{Rp}\RubikRotation{Rp}
\cubenumber{2}%
\ShowCube{2cm}{0.5}{\DrawRubikCube}%
%-----------------------
\quad\Rubik{Fp}\RubikRotation{Fp}
\cubenumber{3}%
\ShowCube{2cm}{0.5}{\DrawRubikCube}%
%-----------------------
\end{figure}
{\noindent}The code for the figure is given below.
After setting up the first cube, we then just use the \verb!\RubikRotation!
command to generate the remaining cubes.
The colours are coded as follows: R~(red), O~(orange), Y~(yellow),
G~(green), B~(blue), W~(white), and X~(grey).
\begin{quote}
\begin{verbatim}
\usepackage{tikz,rubikcube,rubikrotation}
\newcommand{\cubenumber}[1]{\strut\raisebox{1cm}{#1}}
...
\begin{figure}[hbt]
\centering
% set up the first cube
\RubikCubeGreyWY%
\RubikFaceUp{X}{X}{X}%
{X}{W}{O}%
{X}{X}{X}%
\RubikFaceRight{X}{W}{X}
{X}{G}{X}
{X}{X}{X}
\cubenumber{1}%
\ShowCube{5cm}{0.5}{\DrawRubikCubeSF}%
%
\quad\Rubik{Rp}\RubikRotation{Rp}
\cubenumber{2}%
\ShowCube{2cm}{0.5}{\DrawRubikCube}%
%
\quad\Rubik{Fp}\RubikRotation{Fp}
\cubenumber{3}%
\ShowCube{2cm}{0.5}{\DrawRubikCube}%
\end{figure}
\end{verbatim}
\end{quote}
\pagebreak
%-------------------------------
\subsection{Scramble a cube}
\label{sec:scrambleacube}
\typeout{---Example (scramble a cube)}
In this example we use the \verb!\RubikRotation! command to scramble a `solved'
Rubik cube via a sequence of 120 random rotations, using the following command
(the details of the process can be seen in the \texttt{.log} file):
\begin{quote}
\begin{verbatim}
\RubikRotation{random,120}
\end{verbatim}
\end{quote}
On this occasion we draw the cube Flat (F) using the command \verb!\DrawRubikCubeF!.
In this example, we also make use of the \verb!\SaveRubikState{}! command to save
the configuration (state) displayed here in Figure~3 to a file
(\texttt{rubikexampfig3.tex}) using \verb!\SaveRubikState{rubikexampfig3.tex}!,
so we can display the same cube configuration later but in a different format
(we show it again in the following example (Example~\ref{sec:saverubikstate}).
Note that since we are using a random sequence, it follows that each
time this file is run not only will a visually different cube be generated,
but the same state will be shown both
here and in Example~\ref{sec:saverubikstate}.
%%--------FIGURE 3 -----------
\begin{figure}[hbt]
\centering
\RubikCubeSolvedWY
\RubikRotation{random,120}
\SaveRubikState{rubikexampfig3.tex}
\ShowCubeF{6cm}{0.5}{\DrawRubikCubeF}
\parbox{0.7\textwidth}{%
\caption{\label{fig:randomone}This shows a cube generated by 120 random rotations}}
\end{figure}
%----------------
\begin{quote}
\begin{verbatim}
\usepackage{tikz,rubikcube,rubikrotation}
...
\begin{figure}[hbt]
\centering\RubikCubeSolvedWY
\RubikRotation{random,120}
\SaveRubikState{rubikexampfig3.tex}
\ShowCubeF{6cm}{0.5}{\DrawRubikCubeF}
\caption{....}
\end{figure}
\end{verbatim}
\end{quote}
{\noindent}Q: How do we determine the minipage-width ($7.2$cm) in the
\verb!\ShowCube! command?
\smallskip
{\noindent}A: The object is 12~cubie squares wide. Since the TikZ scale-factor
argument of the \verb!\ShowCube! command (cms/unit length) in this
case is set to~$0.5$, then the true width of the image will
be $12 \times 0.5 = 6$\,cm. Note that here we have used the \verb!\ShowCubeF!
command and so we can see that this is correct. Changing the scale-factor will
change the image size and hence a new width argument will be required to just fit
the image.
Note that in this particular case (where there is only a single image in the
`figure' environment), since the \verb!\ShowCube! command places the image
(in a TikZ picture environment) centrally inside a minipage, the image will
in fact be centrally placed in the \verb!\textwidth! provided the image
is \textit{smaller} than the fbox---i.e.,~if we used instead a minipage-width of,
say, 12\,cm the image would still appear centred in the \verb!\textwidth! in
this case. However, when there are several images in the `figure', then the spacing
may appear strange unless each image closely fits its own minipage-width etc.
It is often useful, therefore, to check the size of the fbox (using the
\verb!\ShowCubeF! command) as we have done here.
\pagebreak
%----------------------
\subsection{SaveRubikState}
\label{sec:saverubikstate}
\typeout{---Example (SaveRubikState)}
In this example we display a cube having the same colour state as that shown in
the previous example (Example~\ref{sec:scrambleacube}) ---see Figure~\ref{fig:randomone}.
The cube's colour state was saved from the previous example using the command
\verb!\SaveRubikState{rubikexampfig3.tex}!, and then input here
using the command \verb!\input{rubikexampfig3.tex}!. These commands therefore allow the
state of a previous cube to be saved to a file, and then displayed again
later in a different format.
%%-----------FIGURE 4 -----------
\begin{figure}[hbt]
\centering
\input{rubikexampfig3.tex}
\ShowCube{4cm}{0.8}{\DrawRubikCubeLD}
\parbox{0.7\textwidth}{%
\caption{\label{fig:randomtwo} This shows a Rubik cube in exactly the same
colour state as the one shown in Figure~\ref{fig:randomone}, but here displayed
as a cube seen from the LD (Left-Down) viewpoint.}%
}%
\end{figure}
%%---------------------------------
\begin{quote}
\begin{verbatim}
\usepackage{tikz,rubikcube,rubikrotation}
...
\begin{figure}[hbt]
\centering
\input{rubikexampfig3.tex}
\ShowCube{4cm}{0.8}{\DrawRubikCubeLD}
\parbox{0.7\textwidth}{%
\caption{....}%
}%
\end{figure}
\end{verbatim}
\end{quote}
\noindent\textsc{note}: The command \verb!\SaveRubikState{}! is part of the
\textsc{rubikrotation} package; see the documentation for further details.
\pagebreak
%----------------------
\subsection{Series of cubes}
\typeout{---Example (series of cubes)}
Here we show a convenient way of displaying a series of small cubes showing
the action of a sequence of rotations (\rr{U}, \rr{R}, \rr{F}) on a solved cube.
\begin{figure}[hbt]
\centering
\RubikCubeSolvedWY%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\quad\Rubik{U}\quad%
\RubikRotation{U}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\quad\Rubik{R}\quad%
\RubikRotation{R}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\quad\Rubik{F}\quad%
\RubikRotation{F}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\caption{The rotations \rr{U}, \rr{R}, \rr{F}\ on a solved cube.}
\end{figure}
{\noindent}The code for the above sequence is as follows:
\begin{quote}
\begin{verbatim}
\usepackage{tikz,rubikcube,rubikrotation}
...
\begin{figure}[hbt]
\centering%
\RubikCubeSolvedWY%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\quad\Rubik{U}\quad%
\RubikRotation{U}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\quad\Rubik{R}\quad%
\RubikRotation{R}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\quad\Rubik{F}\quad%
\RubikRotation{F}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\caption{The rotations \rr{U}, \rr{R}, \rr{F}\ on a solved cube.}
\end{figure}
\end{verbatim}
\end{quote}
{\noindent}\textsc{Note} 1: We start with the White-opposite-Yellow (WY)
solved cube, using the command \verb!\RubikCubeSolvedWY!.
A White-opposite-Blue (WB) solved cube is available as \verb!\RubikCubeSolvedWB!.
\medskip
{\noindent}\textsc{Note} 2: There are four different commands for typesetting
3x3x3 rotation codes in various formats (a slightly different set of four
commands are used for the equivalent 2x2x2 rotation codes).
In~this (3x3x3) example, we have used two of them; for example:
{\newline}\verb!\rr{U}! $\rightarrow$ \rr{U} \ (here the \verb!\rr{}! stands
for `rubik rotation'), and also
{\newline}\verb!\Rubik{U}! $\rightarrow$ \Rubik{U}. This command typesets
both the hieroglyph and the rotation code vertically one above the other.
The other two commands are as follows:
{\newline}\verb!\rrh{U}! $\rightarrow$ \rrh{U} \ (here the \verb!\rrh{}!
stands for `rubik rotation hieroglyph'), and also
{\newline}\verb!\textRubik{U}! $\rightarrow$ \textRubik{U}. This command
typesets both the rotation code and the hieroglyph on the same line one
after the other.
\pagebreak
%----------------------
\subsection{Rotation sequence}
\typeout{---Example (rotation sequence)}
We now explore using the named Rubik cube rotation sequences and associated
patterns available in the \textsc{rubikpatterns} package---a small macro database
(see the \textsc{rubikpatterns} documentation, and also its companion file \texttt{rubikpatternsLIST.pdf}). Having the sequences available as macros is
very convenient since (a)~it avoids making errors when typing them out,
and (b)~allows the easy application of software tools.
A Rubik pattern is the configuration generated by a sequence of rotations
(or `moves') from some initial starting configuration (typically a `solved'
configuration). For example, FourSpot is a well known pattern which we can
generate from a solved Rubik cube using the macro \verb!\FourSpot!; it is
defined in the \textsc{rubikpatterns} package as follows:
\begin{quote}
\begin{verbatim}
\newcommand{\FourSpot}{[FourSpot],F2,B2,U,Dp,R2,L2,U,Dp,<(12q*, 8f*)>}
\newcommand{\fourspot}{\FourSpot}
\end{verbatim}
\end{quote}
Note that for convenience the macros names in the \textsc{rubikpatterns} package
are defined in both upper and lower-case (i.e.,~the macros \verb!\FourSpot!
and \verb!\fourspot! are identical). The following figure shows the FourSpot
sequence and pattern.
\bigskip
\noindent%
\RubikCubeSolvedWY%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRD}
\RubikRotation{\FourSpot}
\quad%
\SequenceBraceA{FourSpot}{%
\ShowSequence{}{\Rubik}{\SequenceLong}%
}
\quad%
\ShowCube{5cm}{0.5}{\DrawRubikCubeSF}
\bigskip
{\noindent}The code for the above figure is as follows:
\begin{quote}
\begin{verbatim}
\usepackage{tikz,rubikcube,rubikrotation,rubikpatterns}
...
\noindent%
\RubikCubeSolvedWY%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRD}
\RubikRotation{\FourSpot}
\quad%
\SequenceBraceA{FourSpot}{%
\ShowSequence{}{\Rubik}{\SequenceLong}%
}
\quad%
\ShowCube{5cm}{0.5}{\DrawRubikCubeSF}
\end{verbatim}
\end{quote}
Note that we have spread the code slightly here in order to emphasise that
the \verb!\ShowSequence! command is being used as an argument for the
\verb!\SequenceBraceA! command (the `A' in the command \verb!\SequenceBraceA!
denotes that the annotation is placed Above the sequence.)
We have used a solved cube with the WY (White opposite Yellow)
configuration (\verb!\RubikCubeSolvedWY!).
The first cube is drawn from the RD (Right-Down) viewpoint (\verb!\DrawRubikCubeRD!).
The second cube is drawn from the SF (Semi-Flat) viewpoint (\verb!\DrawRubikCubeSF!)
so we can see all the faces.
Note that we execute the \verb!\RubikRotation{}! command \textit{before} showing
the sequence itself. This is because the Perl program (CALLed by
\verb!\RubikRotation!) allocates the rotation sequence to the variable
\verb!\SequenceLong! which we then display using the \verb!\ShowSequence! command
(see the \textsc{rubikrotation} documentation for details).
\pagebreak
%----------------------
\subsection{SixTs}
\typeout{---Example (SixTs)}
A more interesting cube pattern is the SixTs configuration (from the
\textsc{rubikpatterns} package), which we now show in a slightly
different way (adding an extra face and also some cube sidebars), as follows:
\bigskip
\noindent\hfil%
\RubikCubeSolvedWB
\ShowCube{1.6cm}{0.4}{
\DrawRubikCubeRU
\DrawRubikFlatDown{0}{-3.3}}
\RubikRotation{\sixts}
\quad\SequenceBraceA{SixTs}{\ShowSequence{,\ }{\texttt}{\SequenceLong}}
\quad\ShowCube{3cm}{0.5}{
\DrawRubikCubeRU
\DrawRubikCubeSidebarFL{RU}
\DrawRubikCubeSidebarFD{RU}
\DrawRubikCubeSidebarBR{RU}
\DrawRubikCubeSidebarBU{RU}
}
\hfil
\bigskip
This time we have started with a solved cube having the WB configuration
(White opposite Blue), which is generated using the command
\verb!\RubikCubeSolvedWB!
(we have added the \textsc{down} face (blue) below to reveal the colour of this
face---see note below).
The rotation sequence is in `long-format' (expanded into
separate rotations), comma-separated and space, typewriter font,
using the command \verb!\ShowSequence{,\ }{\texttt}{\SequenceLong}!.
The final image shows the sixT cube together with the main cube sidebars
indicating the colours of the adjacent facelets,
using the \verb!\DrawRubikCubeSidebarXX! commands.
The code for the above figure is as follows:
\begin{quote}
\begin{verbatim}
\RubikCubeSolvedWB
\ShowCube{1.6cm}{0.4}{%
\DrawRubikCubeRU%
\DrawRubikFlatDown{0}{-3.3}%
}
\RubikRotation{\SixTs}
\quad\SequenceBraceA{SixTs}{%
\ShowSequence{,\ }{\texttt}{\SequenceLong}%
}
\quad\ShowCube{2cm}{0.5}{
\DrawRubikCubeRU
\DrawRubikCubeSidebarFL{RU}
\DrawRubikCubeSidebarFD{RU}
\DrawRubikCubeSidebarBR{RU}
\DrawRubikCubeSidebarBU{RU}
}
\end{verbatim}
\end{quote}
\subsubsection*{Notes}
\begin{enumerate}
\item We have drawn the \textsc{down} face of the first cube using the command
\verb!\DrawRubikFlatDown{0}{-3.3}! where the two arguments are the $x$ and
$y$ coordinates of the \textit{bottom left} corner of the \textsc{down} face
(blue); these $(x,y)$ arguments allow us to position the blue `flat' face in
relation to the cube. Note that the grid origin of a cube image (on the page)
coincides with the \textit{bottom left} corner of the \textsc{front} face
(green in this case).
\item The first image is really just 4 units wide. This is because the 2D width
of the \textit{side} face (red) is designed to measure 1~unit wide in the
oblique view (similarly, the 2D height of the \textit{top} face also measures
just 1~unit). Consequently, since the TikZ scale factor used is $0.4$, then
the (minimum) width argument for its \verb!\ShowCube{}{}{}! command is
$4\times 0.4 = 1.6$cm., hence we have \verb!\ShowCube{1.6cm}{0.4}{...}!.
\end{enumerate}
\pagebreak
%----------------------
\subsection[Cycle three edges]{Cycle three edges (flip two)}
\typeout{---Example (cycle three edges (flip two))}
The following example illustrates a sequence used for cycling three edges,
which is often used in generating the `cross' formation while solving the
final layer. However, it is a useful exercise to perform this sequence on
a solved cube, as this reveals quite clearly its rather complex actions
(it~also swaps two pairs of corners).
The black arrow (no flip) and magenta arrows (flip) indicate the \textsc{up}
face edge cubies which are cycled anti-clockwise by the sequence
\ShowSequence{,}{\rr}{F,R,U,Rp,Up,Fp}. The blue arrows indicate the associated
`collateral damage' actions (two pairs of corner cubies swap positions);
depending on the circumstances, these other actions may be an acceptable
cost since they can be fixed at a later stage. (This diagram is from
Section~13 in the \textsc{rubikcube} package documentation).
Note that we include the \textsc{up}-face sidebars to reveal the remaining
facelet colours of the top layer (using the command \verb!\DrawRubikFaceUpSide!)
so we can see how the sequence makes the cubies rotate and turn as they move.
\bigskip%
\noindent\hfil%
\RubikCubeSolvedWY%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\quad\ShowCube{2.3cm}{0.5}{%
\DrawRubikFaceUpSide%
\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);
}%
\RubikRotation{\CycleThreeEdgesFlipTwo}%
\quad\ShowSequence{}{\Rubik}{\SequenceLong}\quad$\longrightarrow$\quad%
\ShowCube{2.3cm}{0.5}{\DrawRubikFaceUpSide}%
\hfil%
\bigskip%
{\noindent}The code for the figure is as follows
\begin{verbatim}
\newcommand{\CycleThreeEdgesFlipTwo}{F,R,U,Rp,Up,Fp}%
...
...
\RubikCubeSolved%
\ShowCube{2cm}{0.4}{\DrawRubikCubeRU}%
\quad\ShowCube{2.3cm}{0.4}{%
\DrawRubikFaceUpSide%
\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);
}%
\RubikRotation{\CycleThreeEdgesFlipTwo}%
\quad\ShowSequence{}{\Rubik}{\SequenceLong}\quad$\longrightarrow$\quad%
\ShowCube{2.3cm}{0.4}{\DrawRubikFaceUpSide}%
\end{verbatim}
{\noindent}Note that the TikZ \verb!\draw! commands (for drawing the arrows
on the cube face) start with a lower-case letter, and also require a terminal
semicolon.
\pagebreak
%----------------------
\subsection{Superflip}
\typeout{---Example (superflip)}
Once you can solve Rubik's cube, then an interesting exercise is to generate
the so-called `superflip' configuration, in which all the
corners are correctly solved, while all the edges are
flipped\,\footnote{See the `superflip' entry in \textit{Wikipedia}, and also
the Kociemba website (\url{www.kociemba.org/cube.htm}); particularly the page
\url{http://kociemba.org/math/oh.htm}}.
%--------draw the images-------------
\begin{figure}[hbt]
\centering
\RubikCubeSolvedWY%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\qquad$\longrightarrow$\qquad%
\RubikRotation{\superflip}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeLD}%
\qquad%
\ShowCube{5cm}{0.5}{\DrawRubikCubeSF}%
\caption{Two representations of the superflip configuration.}
\end{figure}
%----------------------
{\noindent}A superflip sequence converts the solved cube on the left
into the form on the right, using the command \verb!\RubikRotation{\superflip}!.
The code for the above figure is shown below (the~\verb!\superflip! macro
used here is from the \textsc{rubikpatterns} package).
\begin{verbatim}
\usepackage{tikz,rubikcube,rubikrotation,rubikpatterns}
...
\begin{figure}[hbt]
\centering
\RubikCubeSolvedWY%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\qquad$\longrightarrow$\qquad%
\RubikRotation{\superflip}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeLD}
\qquad%
\ShowCube{5cm}{0.5}{\DrawRubikCubeSF}
\caption{...}
\end{figure}
\end{verbatim}
The following superflip sequence\,\footnote{This particular superflip
sequence (in the \textsc{rubikpatterns} package) is due to Reid (1995);
for details see the \textsc{rubikpatterns} package documentation, and
also \url{http://kociemba.org/math/oh.htm}.
Another 20-move superflip sequence (due to H Kociemba), is designated
as K32466 in \url{www.nickalls.org/dick/papers/tex/RUBIK20moves.zip}.}
has just 20 HTM rotations (Half Turn Metric: counting 180~degree turns
as just one `rotation'). Note that the \textsc{rubikpatterns} package
contains this particular superflip sequence as the macro \verb!\superflip!
(see \texttt{rubikpatterns.pdf}). Consequently the code
\verb!\ShowSequence{,}{\large\texttt}{\superflip}!, will typeset the sequence
as follows:
\begin{center}
%% show the Short code
\ShowSequence{,}{\texttt}{\superflip}
\end{center}
Note that for convenience, the \textsc{rubikpatterns} package includes
the sequence name (in square brackets) as the first element of the
associated macro. This is possible since the contents of a comma-separated
square bracket are not actioned as a rotation by the \verb!\RubikRotation!
command.
\pagebreak
Next we present the same superflip sequence but without commas and in
the form of hieroglyphs, using the \verb!\Rubik! font, for which we
require the `expanded' \verb!\SequenceLong! form (since the `short form'
includes trailing digits---see Section~11 in the \textsc{rubikcube}
documentation), using the following code:
\begin{quote}
\begin{verbatim}
\usepackage{tikz,rubikcube,rubikrotation,rubikpatterns}
...
\RubikCubeSolvedWY
\RubikRotation{\superflip}
\noindent\strut\hspace{-8mm}\ShowSequence{}{\Rubik}{\SequenceLong}
\end{verbatim}
\end{quote}
which gives
\bigskip
\RubikCubeSolvedWY
\RubikRotation{\superflip}
\noindent\strut\hspace{-8mm}\ShowSequence{}{\Rubik}{\SequenceLong}
%%---------------------------------
\bigskip
\subsubsection*{Equivalent sequence}
Interestingly, the superflip sequence is actually equivalent to
\Rubikbrace{\Rubikbracket{\Rubik{Rm}\Rubik{U}}4, \Rubik{x},
\Rubik{yp}}3 \ \ (Harris (2008), p151).
Furthermore we can readily demonstrate this, as we can process
this novel form of the sequence using some useful features of
the \verb!\RubikRotation! command, as follows:
\begin{quote}
\begin{verbatim}
\RubikCubeSolvedWF
\RubikRotation[3]{[superflip],(Rm,U)4,x,yp,<Harris (2008), p151>}
\ShowCube{4cm}{0.4}{\DrawRubikCubeSF}
\end{verbatim}
\end{quote}
which generates the following
%-------------------
\begin{figure}[hbt]
\centering
\RubikCubeSolvedWY
%%% \RubikRotation[3]{[shortsuper],(Rmp,Up)4,yp,x}% ORG
\RubikRotation[3]{[shortsuper],(Rm,U)4,x,yp,<Harris (2008), p151>}
\ShowCube{5cm}{0.5}{\DrawRubikCubeSF}
\end{figure}
%----------------------
{\noindent}which is exactly the same configuration as before.
Note that to do this we made use of the `repeat' option \verb![3]!
as well as the \verb!(Rmp,Up)4! `repeat-block' in
the argument of the \verb!\RubikRotation! command above.
\pagebreak
%----------------------
\subsection{Inverse sequence}
\typeout{---Example (inverse sequence)}
Generating the inverse of a Rubik sequence involves (a)~reversing the
order of the sequence, and (b)~inverting each rotation in the sequence
(see Sections~5.1 and 5.1.1 in the \textsc{rubikrotation} package
documentation).
From the grey-cube example~(\ref{sec:greycube}) we saw that the sixspot
sequence is: \verb!U,Dp,R,Lp,F,Bp,U,Dp!; its inverse is therefore
readily determined as \verb!D,Up,B,Fp,L,Rp,D,Up!. Note that this is
easy to check since the sequence generated by the \verb!\RubikRotation!
command is held by the macro \verb!\SequenceLong!. For~example, the
output of the following commands
\begin{quote}
\begin{verbatim}
\fbox{\strut\rule{0pt}{10pt}%
The inverse of the sixspot sequence is:
\RubikRotation{\sixspot,<inverse>}
\ShowSequence{,}{\texttt}{\SequenceLong}.
}
\end{verbatim}
\end{quote}
is \quad\fbox{\strut\rule{0pt}{10pt} %
The inverse of the sixspot sequence is:
\RubikRotation{\sixspot,<inverse>}
\ShowSequence{,}{\texttt}{\SequenceLong}.
}
\bigskip
A sequence and its inverse will annihilate each other when applied
consecutively. For example, in the following figure we start with
a solved cube and apply the sixspot sequence. Applying the inverse
of the sixspot sequence then results in the solved cube configuration
again.
\newcommand{\sixspotarrow}{$\quad\overrightarrow{\strut\textsc{sixspot}}\quad$}
\newcommand{\sixspotINVarrow}{$\quad\overrightarrow{\strut\textsc{sixspotINV}}\quad$}
\begin{figure}[hbt]
\centering
\RubikCubeSolvedWY%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\RubikRotation{\sixspot}%
\sixspotarrow%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\RubikRotation{\sixspot,<inverse>}%
\sixspotINVarrow%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}
\end{figure}
{\noindent}The code for the above figure is as follows:
\begin{verbatim}
\newcommand{\sixspotarrow}{$\quad\overrightarrow{\strut\textsc{sixspot}}\quad$}
\newcommand{\sixspotINVarrow}{$\quad\overrightarrow{\strut\textsc{sixspotINV}}\quad$}
...
\RubikCubeSolvedWY\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\RubikRotation{\sixspot}%
\sixspotarrow%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}%
\RubikRotation{\sixspot,<inverse>}%
\sixspotINVarrow%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}
\end{verbatim}
A significant property of the superflip configuration is that it is its own
inverse. Consequently we can achieve a similar result simply by applying the
superflip sequence \textit{twice in succession}, as follows:
\newcommand{\superfliparrow}{$\quad\overrightarrow{\strut\textsc{superflip}}\quad$}
\begin{figure}[hbt]
\centering
\RubikCubeSolved%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}\superfliparrow%
\RubikRotation{\superflip}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}\superfliparrow%
\RubikRotation{\superflip}%
\ShowCube{2cm}{0.5}{\DrawRubikCubeRU}
\end{figure}
\pagebreak
%-------------------
\subsection{OLL example}
% ThreeCube YellowCross (from dnRUBIKsolution15)
\newcommand{\yellowcross}{[yellowcross],F,R,U,Rp,Up,Fp}
\newcommand{\arrow}[1]{$\overrightarrow{\strut\mbox{#1}}$}
\newcommand{\ShowYellowFace}{\ShowCube{2.6cm}{0.6}{\NoSidebar{X}\DrawRubikFaceUpSide}}
Here we illustrate the use of the \verb!\NoSidebar! command (see the
\textsc{rubikcube} package documentation) to generate nice displays
of so-called OLL (Orientate Lower Layer) configurations, which are
typically rendered using the yellow face.
For example, the following sequence shows an efficient
algorithm (sequence) for `solving' the OLL configuration shown
in cube~A (listed as OLL case~33 in Harris
(2008, page~61)\footnote{Harris D (2008). \textit{Speedsolving the cube},
(Stirling Publishing Co, New York) {\newline}(\texttt{www.stirlingpublishing.com})
[covers 2x2x2, 3x3x3, 4x4x4, 5x5x5 cubes].} and also on the Harris OLL webpage \url{http://www.cubestation/3x3x3/cfop/oll/ollprintablepage.php}).
%% Dan Harris case 33 (in his 2008 book)
\newcommand{\CaseXXXIII}{[caseHarris33],R, U, Rp, U, R, Up, Rp, Up, Rp, F, R, Fp}
% {\DH2008thth}
\bigskip
%% setup OLL position DH33 (Dan Harris)
\RubikCubeGreyAll
\RubikFaceUp{X}{Y}{Y}
{Y}{Y}{X}
{Y}{X}{X}
\RubikFaceBack XXY XXXXXX
\RubikFaceLeft XXX XXXXXX
\RubikFaceRight YYX XXXXXX
\RubikFaceFront XYX XXXXXX
%
\cubenumber{A}%
\ShowCube{2.6cm}{0.6}{\NoSidebar{X}\DrawRubikFaceUpSide}%
%
\RubikRotation{\CaseXXXIII}%
\quad\SequenceBraceA{OLLcase33Harris}{%
\ShowSequence{}{\Rubik}{\CaseXXXIII}%
}%
\quad$\longrightarrow$\quad%
\cubenumber{B}%
\ShowCube{2.6cm}{0.6}{\NoSidebar{X}\DrawRubikFaceUpSide}%
\bigskip
{\noindent}The code for the above figure is as follows (note that
the \verb!\NoSidebar{X}! command is placed inside the \verb!\ShowCube!
environment in order to limit its action).
\begin{quote}
\begin{verbatim}
\newcommand{\cubenumber}[1]{\strut\raisebox{1cm}{#1}}
\newcommand{\CaseXXXIII}{[caseHarris33],R,U,Rp,U,R,Up,Rp,Up,Rp,F,R,Fp}
%
\RubikCubeGreyAll
\RubikFaceUp{X}{Y}{Y}
{Y}{Y}{X}
{Y}{X}{X}
\RubikFaceBack XXY XXXXXX
\RubikFaceLeft XXX XXXXXX
\RubikFaceRight YYX XXXXXX
\RubikFaceFront XYX XXXXXX
\cubenumber{A}%
\ShowCube{2.6cm}{0.6}{\NoSidebar{X}\DrawRubikFaceUpSide}%
\RubikRotation{\CaseXXXIII}%
\quad\SequenceBraceA{OLLcase33Harris}{%
\ShowSequence{}{\Rubik}{\CaseXXXIII}%
}%
\quad$\longrightarrow$\quad%
\cubenumber{B}%
\ShowCube{2.6cm}{0.6}{\NoSidebar{X}\DrawRubikFaceUpSide}%
\end{verbatim}
\end{quote}
Note that since we originally defined the OLL case~33 algorithm
(Harris 2008) as a macro, it follows that we could instead have
recovered the OLL~33 configuration (cube~A above) using the
inverse algorithm, as follows:
\bigskip
\RubikCubeGreyAll
\RubikFaceUpAll{Y}
\RubikRotation{<inverse>,\CaseXXXIII}
\ShowCube{2.6cm}{0.6}{\NoSidebar{X}\DrawRubikFaceUS}%
\hspace{1cm}
\begin{minipage}{4cm}
\begin{verbatim}
\RubikCubeGreyAll
\RubikFaceUpAll{Y}
\RubikRotation{<inverse>,\CaseXXXIII}
\ShowCube{2.6cm}{0.6}{\NoSidebar{X}\DrawRubikFaceUS}
\end{verbatim}
\end{minipage}
\bigskip
{\noindent}Note that in this last example we have used the more convenient
short-hand US (UpSide) version of the \verb!\DrawRubikFaceUpSide! command,
namely \verb!\DrawRubikFaceUS!.
\pagebreak
%----------------------
\subsection{Cube patterns}
\typeout{---Example (Cube patterns)}
\newcommand{\CycleThreeUpEdges}{[CycleUpEdges],F2,U,Rm,Up,Up,Lm,U,Fp2}
\newcommand{\StripesOrtho}{[StripesOrtho],Lp,Fp,Rp,L,U,D, Rp,Dp,Lp,Bp,U2,Lp,Up,R,L,Up,F2,Rp,<Harris 2008, p150>}
We now look at how the \verb!\RubikRotation! command can be used to
investigate cube patterns, which are typically shown only as a
regular cube. For example, consider the following interesting sequence
\linebreak\texttt{L' F' R' L U D R' D' L' B' U2 L' U' R L U' F2 R'}
(Harris 2008, page~150), which is presented as a cube with orthogonal
stripes (see figure~A). In order to show the pattern using the Rubik
bundle, it is often useful to first express the sequence as a macro.
In this particular case we need to adjust the Harris (2008) notation
slightly to make it compatible (ie, insert commas, change trailing
`primes' into a trailing `p'), give it a suitable name
(say, \verb![StripesOrtho]!), and, for completion, include the citation
in an `infoblock' ($<$...$>$), as follows:
\begin{verbatim}
\newcommand{\StripesOrtho}{%
[StripesOrtho],Lp,Fp,Rp,L,U,D,Rp,Dp,Lp,Bp,U2,Lp,Up,R,L,Up,F2,Rp,<Harris 2008, p150>}
\end{verbatim}
We can now process it simply by using its macro-name as the argument for
the \verb!\RubikRotation! command; for example, the following code displays
it as figure~A.
\begin{verbatim}
\RubikCubeSolved
\RubikRotation{\StripesOrtho}
\ShowCube{2.6cm}{0.6}{\DrawRubikCubeRU}
\end{verbatim}
At first sight this looks like a nice example of orthogonal stripes,
but if we look at all the faces by rendering the cube in a Semi-Flat
(SF) view (using the command \verb!\DrawRubikCubeSF!)
---see~figure~B--- this reveals that unfortunately only five faces
actually have a center stripe.
\bigskip
\RubikCubeSolved
\RubikRotation{\StripesOrtho}
\cubenumber{A}
\ShowCube{2.6cm}{0.6}{\DrawRubikCubeRU}
\hspace{3cm}
\cubenumber{B}
\ShowCube{4cm}{0.4}{\DrawRubikCubeSF}
\bigskip
{\noindent}So the question is: can we improve on this and create a cube
with a fully orthogonal pattern, by swapping the relevant red/yellow
and blue/yellow edge cubies in the back layer? Since it is impossible
to swap just two edge cubies within the same layer in a 3x3x3 cube
without changing any other cubies, the answer is no. However, we are
able to cycle cleanly three edge cubies within one face, and a single
cycle could at least create the final stripe with only minimal collateral
damage.
For example, if we rotate the whole cube using the rotation denoted
as \rr{Rcp} (= \verb!\rr{Rcp}!)\,\footnote{There are several equivalent
notations for this: \rr{Rcp} $\equiv$ \rr{xp} $\equiv$ \rr{CRp};
the c and C denote a whole cube rotation.} to bring the \textsc{back} face to
the \textsc{up} position (as shown in figure~C), we can then cycle the three
\textsc{up} edge cubies UL, UR, UF clockwise using the following
`CycleThreeUpEdges' sequence:
\begin{verbatim}
\newcommand{\CycleThreeUpEdges}{F2,U,Rm,Up,Up,Lm,U,Fp2}
\end{verbatim}
We can combine these two actions using the command
\verb!\RubikRotation{Rcp,\CycleThreeUpEdges}!, which will then align the
three red facelets in the \textsc{up} face and form the final stripe
(see figure~D).
\RubikCubeSolved
\RubikRotation{\StripesOrtho,Rcp}
\cubenumber{C}
\ShowCube{2.6cm}{0.6}\DrawRubikCubeRU
\hspace{3cm}
\RubikRotation{\CycleThreeUpEdges}
\cubenumber{D}
\ShowCube{4cm}{0.4}\DrawRubikCubeSF
\medskip
The downside, of course, is that a yellow and a white facelet are also
swapped in the top layer, but at least we now have a cube with six
mutually adjacent orthogonal central stripes\,!
\pagebreak
%%=====================TWOcube stuff======================================
%%========================================================================
\section{TwoCube examples (2x2x2)}
\subsection{Commands \& notation}
Regarding the 2x2x2 cube as a `TwoCube' (instead of a RubikCube) makes
the 2x2x2 commands very intuitive (assuming you are familiar with the
3x3x3 commands), since they are mostly obtained just by exchanging
the word `Rubik' in 3x3x3 commands for the word `Two'. For example,
\medskip
\verb!3x3x3!\quad\hspace{1cm}$\rightarrow$\quad\hspace{1cm}\verb!2x2x2!
\smallskip
\verb!\RubikCubeSolvedWY! $\rightarrow$ \verb!\TwoCubeSolvedWY!
\verb!\RubikRotation!\qquad\ $\rightarrow$ \quad\verb!\TwoRotation!
\verb!\SaveRubikState!\qquad\ $\rightarrow$ \quad\verb!\SaveTwoState!
\medskip
Note that although the 2x2x2 cube does not have any middle slices, one can still
actually use the 3x3x3 commands \verb!\RubikRotation! and \verb!\SaveRubikState!
in the usual way (because the corner cubies are still processed in
the usual way). However, in the interests of consistency
both of these commands have equivalent `Two' versions: namely
\verb!\TwoRotation! and \verb!\SaveTwoState!.
Of course there are considerably fewer rotation codes and hieroglyphs
associated with the TwoCube (since there are no middle slices), and
their commands also mirror the 3x3x3 equivalents in the \textsc{rubikcube}
package; i.e.,~\verb!Rubik! $\rightarrow$ \verb!Two! and
\verb!r! $\rightarrow$ \verb!t!. Thus the changes associated with the
four key commands can be summarised as follows :
\medskip
\verb!3x3x3! \hphantom{$\rightarrow$} \verb!2x2x2!
\smallskip
\verb!\rr{}! $\rightarrow$ \verb!\tr{}!
\verb!\rrh{}! $\rightarrow$ \verb!\trh{}!
\verb!\Rubik{}! $\rightarrow$ \verb!\Two{}!
\verb!\textRubik{}! $\rightarrow$ \verb!\textTwo{}!
%-----------------------
\bigskip
%-------------------
\subsection{Visualising white-space}
This example shows how the F version of the \verb!\ShowCube{}{}{}! command can be useful for revealing excess white space.
\bigskip
\TwoCubeSolvedWB%
\ShowCubeF{3cm}{0.6}{\DrawTwoCubeRU}%
\quad\ShowCubeF{1.7cm}{0.6}{\DrawTwoFaceUS}%
\hspace{1cm}
\begin{minipage}{0.3\textwidth}
\begin{verbatim}
\TwoCubeSolvedWB%
\ShowCubeF{3cm}{0.6}{\DrawTwoCubeRU}%
\quad%
\ShowCubeF{1.7cm}{0.5}{\DrawTwoFaceUS}%
\end{verbatim}
\end{minipage}
\bigskip
Both images above have an F-box drawn around them (so we can see the
extent of any white space) using the F version of the \verb!\ShowCube!$\{\mbox{\textit{width}}\}\{\mbox{\textit{scale-factor}}\}\{\}$ command,
namely \verb!\ShowCubeF{}{}{}!. Clearly the \verb!3cm! width we have
used for the first cube above is too much, and so in the second pair
of images below we have reduced this width for the first image to
its correct value of \verb!1.6cm!, and have removed the F from the
\verb!\ShowCubeF! command accordingly.
\bigskip
\quad%
\TwoCubeSolvedWB%
\ShowCube{1.6cm}{0.6}{\DrawTwoCubeRU}%
\quad%
\ShowCube{1.7cm}{0.6}{\DrawTwoFaceUS}%
\hspace{1cm}
\begin{minipage}{0.3\textwidth}
\begin{verbatim}
\TwoCubeSolvedWB%
\ShowCube{1.6cm}{0.6}{\DrawTwoCubeRU}%
\quad%
\ShowCube{1.7cm}{0.5}{\DrawTwoFaceUS}%
\end{verbatim}
\end{minipage}
\bigskip
Thus use of the F version of the the \verb!ShowCube! command allows
the user to easily determine---by trial \& error---the correct minipage
width for a particular cube.
\pagebreak
%-------------------
\subsection{Flat and semi-flat formats}
We can view the hidden faces of a cube using the semi-flat (SF) and
flat (F) versions of the \verb!\DrawTwoCube..! command as follows.
In the first figure we have also used the F version of the
\verb!\ShowCube{}{}{}! command to reveal the size of the minipage
which contains the image. Notice that since the 2D width of the side
face (red) is approximately 1 unit, we can estimate that the total
width of the image will be 6 + 1 = 7 units, and hence as we have
specified a TikZ scale factor of 0.6, then the required width of
the minipage is approximately $4.2$ ($= 7 \times 0.6$)
---see Section~4.1 in the \textsc{rubikcube} package documentation
for details).
\bigskip
\begin{minipage}{0.3\textwidth}
\centering
\TwoCubeSolvedWB%
\ShowCubeF{4.2cm}{0.6}{\DrawTwoCubeSF}%
\end{minipage}
\hspace{2cm}
\begin{minipage}{0.3\textwidth}
\begin{verbatim}
\TwoCubeSolvedWB%
\ShowCubeF{4.2cm}{0.6}{\DrawTwoCubeSF}%
\end{verbatim}
\end{minipage}
\bigskip
\begin{minipage}{0.3\textwidth}
\centering
\TwoCubeSolvedWB%
\ShowCube{4.8cm}{0.6}{\DrawTwoCubeF}%
\end{minipage}
\hspace{2cm}
\begin{minipage}{0.3\textwidth}
\begin{verbatim}
\TwoCubeSolvedWB%
\ShowCube{4.8cm}{0.6}{\DrawTwoCubeF}%
\end{verbatim}
\end{minipage}
\pagebreak
%-------------------
\subsection{Grey cube}
A cube with all of the facelets coloured grey (a `grey' cube) can be
useful as a starting state for instruction. Such a cube is generated
by the command \verb!\TwoCubeGreyAll!.
We can then allocate colours to those facelets demonstrating
some feature of cube manipulation.
Here we are showing how to rotate the GWR corner cubie in the bottom layer
up into its `solved' position in the top layer. We also use two cube
sidebars to show the colours of the hidden facelets of the corner cubie.
\bigskip
\noindent\hfil%
\TwoCubeGrey
\ShowCube{2cm}{0.7}{\DrawTwoCube}%
%
\TwoFaceUp{X}{X}
{R}{W}%
\TwoFaceFront{G}{R}
{W}{X}%
\TwoFaceLeft XX
XG%
\TwoFaceDown{R}{X}
{X}{X}%
\quad\ShowCube{2cm}{0.7}{
\DrawTwoCube
\DrawTwoCubeSidebarFD{RU}
\DrawTwoCubeSidebarFL{RU}
}%
\quad\Two{Lp}\quad
\TwoRotation{Lp}
\ShowCube{2cm}{0.7}{\DrawTwoCube}%
\hfil%
\bigskip
{\noindent}The code for the above figure is as follows:
\begin{quote}
\begin{verbatim}
\noindent\hfil%
\TwoCubeGrey
\ShowCube{2cm}{0.6}{\DrawTwoCube}%
%
\TwoFaceUp{X}{X}
{R}{W}%
\TwoFaceFront{G}{R}
{W}{X}%
\TwoFaceLeft XX
XG%
\TwoFaceDown{R}{X}
{X}{X}%
\ShowCube{2cm}{0.6}{
\DrawTwoCube
\DrawTwoCubeSidebarFD{RU}
\DrawTwoCubeSidebarFL{RU}
}%
\quad\Two{Lp}\quad
\TwoRotation{Lp}
\ShowCube{2cm}{0.6}{\DrawTwoCube}%
\hfil%
\end{verbatim}
\end{quote}
Notice that the curly brackets associated with the \verb!\TwoFace..! commands
can be omitted providing at least one space is left before the first colour code
(see the \verb!\TwoFaceLeft..! command above).
\pagebreak
%-------------------
\subsection{Scrambled cube}
In the following example we scramble the cube using 20 random rotations using
$\langle$\texttt{random,20}$\rangle$ as the argument
for the \verb!\TwoRotation{}! command. We have also added a set of cube sidebars
to the second cube. The final semi-flat (SF) cube has the same colour configuration
and shows the hidden faces, allowing you to check the colours of the sidebars.
\bigskip
\noindent\hfil%
\TwoCubeSolvedWY%
\ShowCube{1.8cm}{0.7}{\DrawTwoCubeRU}%
\qquad$\longrightarrow$\qquad%
%---------------
\TwoRotation{random,20}
%----------------
\ShowCube{2.8cm}{0.7}{
\DrawTwoCubeRU
\DrawTwoCubeSidebarFL{RU} % Front-Left edge sidebar
\DrawTwoCubeSidebarFD{RU} % Front-Down edge
\DrawTwoCubeSidebarBR{RU} % Back-Right edge
\DrawTwoCubeSidebarBU{RU} % Back-Up edge
}%
\qquad\ShowCube{5cm}{0.7}{\DrawTwoCubeSF}%
\hfil%
\bigskip
\begin{quote}
\begin{verbatim}
\noindent\hfil%
\TwoCubeSolvedWY%
\ShowCube{1.8cm}{0.7}{\DrawTwoCubeRU}%
\qquad$\longrightarrow$\qquad%
%---------------
\TwoRotation{random,20}
%----------------
\ShowCube{2.8cm}{0.7}{
\DrawTwoCubeRU
\DrawTwoCubeSidebarFL{RU} % Front-Left edge sidebar
\DrawTwoCubeSidebarFD{RU} % Front-Down edge
\DrawTwoCubeSidebarBR{RU} % Back-Right edge
\DrawTwoCubeSidebarBU{RU} % Back-Up edge
}%
\qquad\ShowCube{5cm}{0.7}{\DrawTwoCubeSF}%
\hfil%
\end{verbatim}
\end{quote}
%-------------------
\subsubsection*{A simple scramble}
A simple scramble (from a solved position) which
makes each face have four different facelets is $\left\{\mbox{\tr{R},
\tr{y}}\right\}$5 as follows (note that the optional \verb!\TwoRotation!
square bracket holds the repeat number~$5$):
\bigskip
\begin{minipage}{0.3\textwidth}
\centering
\TwoCubeSolvedWB%
\TwoRotation[5]{R,y}%
\ShowCube{5cm}{0.6}{\DrawTwoCubeF}%
\end{minipage}
\hspace{2cm}
\begin{minipage}{0.3\textwidth}
\begin{verbatim}
\TwoCubeSolvedWB%
\TwoRotation[5]{R,y}%
\ShowCube{5cm}{0.6}{\DrawTwoCubeF}%
\end{verbatim}
\end{minipage}
\pagebreak
%-------------------
\subsection{Swap two corners}
The following example is the header figure for the \textsc{rubiktwocube}
package documentation, and shows the `SwapTwoCorners' sequence often
necessary in the solution of a 2x2x2 cube.
\bigskip
\noindent\hfil%
%% set up the cube with corners swapped
\TwoCubeSolvedWY%
\TwoRotation{Rc2}% turn upside down (= Rc2) x2
\TwoRotation{\swaptwocorners}%
%%
% now show the cube, before and after swapping the corners again
\ShowCube{1.6cm}{0.6}{\DrawTwoCubeRU}%
\quad\ShowCube{1.7cm}{0.5}{\DrawTwoFaceUS}%
\TwoRotation{\swaptwocorners}%
\quad\SequenceBraceA{SwapTwoCorners}{\ShowSequence{}{\Two}{\SequenceLong}}%
\quad$\longrightarrow$%
\quad\ShowCube{1.7cm}{0.5}{\DrawTwoFaceUS}%
\hfil%
\bigskip%
{\noindent}The code for the above figure is as follows:
\begin{verbatim}
\usepackage{tikz}
\usepackage{rubikcube,rubikrotation,rubikpatterns,rubiktwocube}
...
\newcommand{\SwapTwoCorners}{[swaptwocorners],Rp,F,Rp,B2,R,Fp,Rp,B2,R2,Up}
\newcommand{\swaptwocorners}{\SwapTwoCorners}
...
\noindent
\hfil%
%% set up the cube with corners swapped
\TwoCubeSolvedWY%
\TwoRotation{Rc2}% turn cube upside down about the R face
\TwoRotation{\swaptwocorners}%
%% now show the cube, before and after swapping the corners
\ShowCube{1.6cm}{0.6}{\DrawTwoCubeRU}%
\quad\ShowCube{1.7cm}{0.5}{\DrawTwoFaceUS}%
\TwoRotation{\swaptwocorners}%
\quad\SequenceBraceA{SwapTwoCorners}{\ShowSequence{}{\Two}{\SequenceLong}}%
\quad$\longrightarrow$%
\quad\ShowCube{1.7cm}{0.5}{\DrawTwoFaceUS}%
\hfil%
\end{verbatim}
\pagebreak
%-------------------
\subsection{OLL example}
Here we illustrate an OLL algorithm which orientates the last layer
starting from a single yellow facelet in the UFL position.
\bigskip
\noindent\hfil%
\TwoCubeGreyAll%
\TwoFaceUp{X}{X}%
{Y}{X}%
\TwoFaceBack XY XX
\TwoFaceRight XY XX%
\TwoFaceFront XY XX%
\ShowCube{2cm}{0.7}{\NoSidebar{X}\DrawTwoFaceUS}
\SaveTwoState{cubeTWO.tex} %***
\TwoRotation{\allyellow}%
\quad\SequenceBraceA{AllYellow}{\ShowSequence{}{\Two}{\SequenceLong}}\quad$\longrightarrow$\quad%
\ShowCube{1.5cm}{0.7}{\NoSidebar{X}\DrawTwoFaceUS}
\hfil%
\bigskip
The code is shown below. Note that we have used the \verb!\NoSidebar{X}!
command to disable the drawing of any grey sidebars by the \verb!\DrawTwoFaceUS!
command\footnote{Note that the \textbackslash\texttt{DrawTwoFaceUS} command is a short-hand
equivalent to the \textbackslash\texttt{DrawTwoFaceUpSide} command.}
and hence only yellow sidebars are drawn. Note that we have placed the
\verb!\NoSidebar{X}! command inside the \verb!\ShowCube! environment in order
to limit its action locally.
\begin{quote}
\begin{verbatim}
\noindent\hfil%
\TwoCubeGreyAll%
\TwoFaceUp{X}{X}%
{Y}{X}%
\TwoFaceBack XY XX
\TwoFaceRight XY XX%
\TwoFaceFront XY XX%
\ShowCube{2cm}{0.7}{\NoSidebar{X}\DrawTwoFaceUS}%
\TwoRotation{\allyellow}%
\quad\SequenceBraceA{AllYellow}{\ShowSequence{}{\Two}{\SequenceLong}}%
\quad$\longrightarrow$\quad%
\ShowCube{1.5cm}{0.7}{\NoSidebar{X}\DrawTwoFaceUS}%
\hfil%
\end{verbatim}
\end{quote}
\bigskip
\begin{center}
------ END ------
\end{center}
\end{document}