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John Kerl \newline
Math. 473 \newline
University of Arizona \newline
Dec. 1, 1993 \newline
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\begin{center}
$\textbf{Some pretty patterns on the Rubik's Cube}$
\end{center}
The Rubik's Cube, also known as the Hungarian Magic Cube, has long been a
laboratory for mathematicians interested in group theory. While the great
number of patterns possible on the cube is legendary, there is a much smaller
(and personally defined) subset of patterns that are aesthetically pleasing.
Many of these patterns, sometimes known as ``pretty patterns'', are pleasing
because they embody some symmetries. This paper will present a particular
group of patterns that embody a certain symmetry and will provide techniques
for producing those patterns.
The patterns of the cube are \emph{permutations} of the pieces and
orientations. A permutation on a set is a function defined from a set to
itself such that every element has exactly one image. Thus, permutations are
defined on every member of the set, and can always be ``undone'' --- they are
\emph{invertible}. A permutation may be thought of as a rearranging --- for
example, shuffling a deck of cards is a permutation on the deck.
It was stated above that many of the ``pretty'' permutations of the cube may be
thought of as embodying certain symmetries. To make this vague statement more
clear, consider a square with the four vertices labeled A, B, C and D (fig. 1).
There are eight transformations, known as the \emph{symmetries of the square},
that carry the square back onto itself: rotations through $0^\circ$ (the
\emph{identity} transformation), $90^\circ$, $180^\circ$ and $270^\circ$; and
reflections through the vertical, horizontal, and either of the two diagonal
axes (fig. 2). The labels of the corners shown in fig. 2 represent the square
following a rotation right by $90^\circ$, or by a reflection through the
vertical axis followed by a reflection through the axis running from upper left
to lower right.
There are, of course, many more than eight \emph{symmetries of the cube}, by
which I mean the plain, geometric cube (fig. 3); one may have 24 reflections
and 24 rotations, rather than the four of each in the case of the square. The
symmetry that I will discuss is the rotation through an axis running through
opposite corners. (It is in this sense that it was stated above that a
particular group of patterns embodying a certain symmetry was to be
considered.) Consider fig. 3. A line runs through the corners labeled B and
E. The corners might be rotated as follows: A goes to G, G goes to C, C goes
to A; H goes to F, F goes to D, D goes to H. Now consider the Rubik's Cube
(fig. 4). It would be interesting to be able to \emph{cycle} the pieces
labeled with the same numbers on the Rubik's Cube in a way similar to the way
in which the corners A, G, C or H, F, D of the plain cube were rotated: for
example, to cycle all edge pieces labeled with a 5, just as the corners A, G, C
can be rotated. In the case of the corner piece, we would simply like to
rotate it (and its far-corner opposite) in place. In fact, all these patterns
are possible. I will not justify this claim, merely noting instead that,
whereas not all permutations of the pieces and their orientations can be
achieved by manipulating the cube (without disassembling and reassembling it),
the ones just described are possible. (The interested and/or mathematically
inclined reader is advised that only \emph{even} permutations of positions of
edges or corners are possible, and that flips of edges or twists or corners are
possible provided that the sum of the spins is zero, modulo 2 or 3
respectively.)
The pretty patterns to be considered are clockwise cycles of the pieces noted
in fig. 4. Before presenting the actual \emph{processes} (sequences of moves)
to achieve these patterns, a little notation must be devised. Pick a face of
the cube; consider this to be the top face. Of the remaining four side faces,
pick one to be the front. In this orientation, the faces may be labeled U, D,
F, B, L and R, for up, down, front, back, left and right (fig. 5). Denote a
clockwise (as viewed from outside the cube) quarter-turn of a given face by its
letter. Then a half turn of the face will be denoted by the exponent 2, and a
counterclockwise quarter turn will be denoted by the exponent $-1$. A process
will be denoted by the string of letters and their associated exponents, e.g.
$UF^2R^{-1}$ means turn the upper face a quarter-turn clockwise, then turn the
front face a half turn, then turn the right face a quarter-turn
counterclockwise. Spaces or dots have no significance other than to separate
mnemonic units. Processes are given in the following table.
\begin{tabular}{ll}
\emph{Pieces to be cycled (fig. 4)} & \emph{String} \\
\\
1 & $B^2 \cdot R^{-1} D R \; FDF^{-1} \cdot U \cdot FD^{-1}F^{-1} \;
R^{-1} D^{-1} R \cdot U^{-1} \cdot B^2$ \\
2 (do parts $a$ and $b$)
& $(a) \; R \cdot U^{-1} \; R^{-1}F^{-1} L^{-1}B^{-1} \cdot U^{-1} BLFR
\cdot U^2 \cdot R^{-1}$ \\
& $(b) \; L^{-1} \cdot D \; LBRF \cdot D \; F^{-1}R^{-1}B^{-1}L^{-1} \cdot D^2
\cdot L$ \\
3 & $F^2U^2F^{-1}R^2F \cdot 2a \cdot F^{-1} R^2 F U^2 F^2$ \\
4 & $F L^2 F^{-1} \cdot LFR^{-1}F^{-1}L^{-1}FRF^{-1} \cdot FL^2F^{-1}$ \\
5 & $F^2 R^2 F U^2 F^{-1} \cdot 2a \cdot F U^2 F^{-1} R^2 F^2$ \\
6 & $R^{-1} F \cdot L^{-1} U R U^{-1} L U R^{-1} U^{-1} \cdot F^{-1} R$ \\
\end{tabular}
It is clear that any of the above processes may be done once, twice, or no
times, and that each may be done independently of the rest. Thus there are
$3^6=729$ different patterns in this small group alone, a very small corner of
the group of the more than $4.3 \times 10^{19}$ permutations of the Rubik's
Cube.
\newpage
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a a another
diagonal vertical diagonal
axis axis axis H +-----------+ G
A +-----------+ B D +-----|-----+ A / | / |
| | | \ | / | A +-----------+ B|
| | | \ | / | | | | |
| | -------X------- horizontal | | | |
| | | / | \ | axis | | | |
| | | / | \ | | E+--------|--+ F
D +-----------+ C C +-----|-----+ B | / | /
Fig. 1 Fig. 2 D +-----------+ C
Fig. 3
+-----+-----+-----+ +-----+-----+-----+
/ 6 / 3 / 4| / / / /|
+-----+-----+-----+ | +-----+-----+-----+ |
/ 5 / / 2| | / / U / /| |
+-----+-----+-----+ |/| +-----+-----+-----+ |/|
/ / / /| / | / / / /| / | <--- B
+--4--+--2--+-----1 |/|5| +-----+-----+-----+ |/| |
| | | | / |/| | | | | /R|/|
| | | |/| / | L--> | | | |/| / |
+-----+-----+-----+ |/|6/ +-----+-----+-----+ |/| /
| 3 | | 2 / |/ | | F | | / |/
| | | |/|3/ | | | |/| /
+-----+-----+-----+ |/ +-----+-----+-----+ |/ <--- D
| 6 | 5 | 4 / | | | | /
| | | |/ | | | |/
+-----+-----+-----+ +-----+-----+-----+
Fig. 4 Fig. 5
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%% +-----+-----+-----+ +-----+-----+-----+
%% / 6 / 3 / 4| / / / /|
%% +-----+-----+-----+ | +-----+-----+-----+ |
%% / 5 / / 2| | / / U / /| |
%% +-----+-----+-----+ |/| +-----+-----+-----+ |/|
%% / / / /| / | / / / /| / | <--- B
%% +--4--+--2--+-----1 |/|5| +-----+-----+-----+ |/| |
%% | | | | / |/| | | | | /R|/|
%% | | | |/| / | L--> | | | |/| / |
%% +-----+-----+-----+ |/|6/ +-----+-----+-----+ |/| /
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%% | | | |/|3/ | | | |/| /
%% +-----+-----+-----+ |/ +-----+-----+-----+ |/ <--- D
%% | 6 | 5 | 4 / | | | | /
%% | | | |/ | | | |/
%% +-----+-----+-----+ +-----+-----+-----+
%% Fig. 4 Fig. 5
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