**THE RUBIK CUBE:**
**A THREE-STAGE APPROACH TO MASTERING IT**
**David McNaughton**

**Extracted from Junior News (Al Nisr, Dubai),
1st November 1989 to 14th February 1990**

If a Rubik Cube is scrambled *using only half-turns* (i.e., through
180º), then every face will show at most two colours - the original
plus the one belonging to the opposite side. A puzzle
in that state is not too difficult to restore. Furthermore, this is still
a challenging and enjoyable exercise. At the same time you will be learning
and practising techniques which remain useful when it comes to solving
a completely mixed up cube. These techniques may in turn be supplemented
by various problem-solving
games which should provide extra stimulation in that general context.

The approach described below was followed by our children, working up through the following three levels:

*[The notation employed here is widely recognised, and
is described in the *__Appendix__].

**LEVEL ONE: PATTERNS EXHIBITING TWO-WAY SYMMETRY**

Start by learning this **operator:**

** 1a. L²R²F².L²R²B²**
- exchanging two pairs of edge-pieces, and leaving an H-pattern on the
top and bottom of the cube.

You can of course keep changing the colours of the faces which get subjected
to that operator. While doing that, every now and again apply one or more
of the following three move-combinations:

(i)** U²** *followed immediately*
by **D²
**...
__and/or sometimes:__ (ii) **L²** followed
immediately by **R²
**...
__or perhaps:__ (iii) **F²** followed
immediately by **B²**.

Judicious applications of operator 1a in between one or other of the above three move-combinations, will produce a number of symmetrical patterns. Try varying the faces on which your operator is acting; (i.e., by turning the cube around, for example so as to bring the Right Face to the Front).

Eventually, you should come across (and be able to correct) the following
configurations:
**Chequers** on two, four or six faces,
**Crosses** on four faces,
**Dots** on four faces (i.e. the original centre-piece
surrounded by a different colour),
**Bars **on four faces - sometimes parallel
(like Pillars), sometimes skewed.

Several mixtures of those patterns are also feasible.

Alternatively, ask a cube-expert to make these patterns - so that you can then practise undoing them using operator 1a together with the three other permitted move-combinations.

*At this early stage, it is best not to try any extra exploratory
experiments other than these mentioned above.*

However, later on - when you feel confident - take a look at this other
operator:

** 1b. R²U².R²U².R²U²**-
which also exchanges two pairs of edge-pieces, and in addition it produces
what our youngsters called the "Eyes" pattern on opposite faces. This operator
can also be written as

You can then practise and utilize this second operator 1b for creating
or rectifying the *chequered pattern* on opposite faces. While in
Level One, however, it is essential to *preserve horizontal and vertical
symmetry,* so every application of 1b should be followed immediately
by another one on the opposite sides. For example, straightaway after a
**(R²U²)³**,
turn **(L²D²)³**...
(that yields a chequered or partial chessboard configuration on the front
and back faces). If you wish, such a pattern may then be corrected with
1a followed by another 1a after turning the cube around - which is essentially
the same as applying **L²R²F².L²R²B²**
and then **F²B²R².F²B²L²**.

**LEVEL TWO: A "PARTIALLY SCRAMBLED" PUZZLE**

First, you might like to jump ahead and practise the move-sequence recommended
for tackling Figure 1. When performed on a pristine, unscrambled cube,
that sequence *produces* Figure 1; repeating the same sequence then
corrects it. (And after that, before proceeding any further, it might be
worth doing the same with Figures 2 and 3 - examining the effects of applying
the move-sequences given below those diagrams - although these may be left
till later if you find them too complicated).

> Now take your cube and (partially) scramble it by subjecting it*
only to 180-degree turns;* (i.e., apart from that restriction, you may
turn all or any of its layers at random)...

Then try and rectify *just the corner-pieces* using trial and error.
Each corner-piece has three different colours, which have to match those
at the *centres *of the faces they are touching.

> Occasionally, the following sequences may be helpful:
**R²B²R²**
or **F²R²F²:**
they swap two corner-pieces on the top face together with another pair
on the bottom, where the two lines of exchange are parallel; (either sequence
will do, because in the long run it does not matter if you end up swapping
along the wrong diagonals). For illustration, it may help to look at Figure
8 - visualizing it after applying an **U²**-move.
Admittedly, the above sequences do disturb a few edge-pieces, but those
can all be dealt with later....

> ... __Once all the corners are correct__, tackle the edge-pieces
- interchanging them as necessary by employing operators 1a and 1b. However,
to manage that, it is essential to **master the art of the "preliminary
manoeuvre":**

For example, you will often find it necessary to perform a swap like that shown in Figure 1 below:

**FIGURE 1. **This configuration is easily adapted
for operator 1b.

In order to apply operator 1b, a preliminary **R-**move is necessary
to make parallel the two lines of exchange. ** Note that this "R" is
only a quarter-turn, i.e. through 90 degrees.** Afterwards, you need
to reverse the preliminary move - with an anticlockwise quarter-turn of
that right layer, which is written

Similar tactics will sometimes be necessary in order to apply operator 1a.

> Sometimes your preliminary manoeuvre might consist of two or three
moves. Look at Figure 2 below:

**FIGURE 2: **Crosswise exchange of two pairs of edge-pieces.

Here, a useful preliminary would be
**R²L²D** -
thereby lining up the pieces for operator 1a. Afterwards, you *reverse
that manoeuvre *- which involves working it backwards *whilst performing
the inverse of every move *(when clockwise quarter-turns are changed
to anticlockwise ones). In this instance, that requires **D ^{-1}L²R²**.
The complete sequence is therefore:

> Complicated preliminary manoeuvres are best remembered in terms of
the colours of the relevant faces. Beginners may find it helpful to temporarily
note down those colour sequences: this should help you to reverse the manoeuvre
after applying whatever operator is required. With Figure 2, in terms of
colours that manoeuvre is:

Red**²**.Black**²**.Green
[... the bottom face is Green].

*> However, beginners might prefer to tackle Figure 2 by first
applying operator 1b - thus giving a configuration similar to Figure 1.*

----------------------------------

> Often, you will be confronted with a triangular exchange of edge-pieces,
as in Figure 3. You could of course subject it to 1b
- preferably reorientating your cube so as to use **R²F².R²F².R²F²**,
because that will leave you with the "two pairs swap" shown in Figure 1.

Indeed, this is one way of tackling a three-way exchange ... namely
to bring in a couple of 'alien' pieces (which *you* can usually choose
- i.e., it is best to try and make the task as simple as possible) and
then perform a "**two pairs** swap". That will leave two incorrect pieces
remaining in the original triangle. These can be rectified using an additional
"two pairs swap", which at the same time puts the 'alien' couple back where
they belong.

**FIGURE 3: **Triangular exchange of three edge-pieces.

Always examine it carefully to see which piece needs
to go where.

> However, it is better to learn another operator - which restores Figure 3 in just six moves:

__2.__ F²RL^{-1}.U²LR^{-1}

You may find it easier to remember **RL ^{-1} **as two "wheels"
turning together, written

Focussing on the "wheels" moves, notice that **LR ^{-1}**
simply moves them back. That may be shortened to

Figure 4 shows this type of move.

**FIGURE 4: **A "wheels" combination, written **R****w.**

-------------------------------------------------------------

> Solving partially scrambled cubes (containing only two colours on each face) will give plenty of practice with operators 1a, 1b and 2. However, sometimes you will need to look very carefully to see which pieces are "partners" for an exchange.

For example, look at the situation depicted (from different angles)
by Figure 5. You might be tempted to turn **F ^{-1}** as a preliminary
manoeuvre, hoping to align the two pairs of "eyes" for applying
1b, but they will then be paired off incorrectly. Instead,

**FIGURE 5 **(above and below).

Look very carefully before lining up the pieces for operator
1b.

Figure 5 can of course be unscrambled in other ways, e.g. apply operator
1b straightaway without a preliminary manoeuvre - followed by operator
2.

__________________________

> Here is another operator which may be learned and practised with a
partially scrambled cube (and which should also prove useful later for
a *complete* unscramble):

** 3. RB.R^{-1}B^{-1}.RB.R^{-1}B^{-1}.RB.R^{-1}B^{-1}**...
best memorized as

Note that all moves are

**FIGURE 6: **Exchanging two pairs of corners.

> Some people find B-moves difficult. (I manage by looking down from
above, so that the right hand can turn the back layer quite naturally.
*L-moves*
probably require maximum concentration, however). Thus, instead of the
above version of operator 3, you may prefer **(RU.R ^{-1}U^{-1})³**
... producing Figure 7 below.

**FIGURE 7: **Swapping two pairs of corners on the
R and U-faces.

------------------------------------------

> Figure 8 (below) may be restored using operator 3 after preliminary
manoeuvre **FD ^{-1}**. [Alternatively, you could opt for

This operator 3 is not absolutely essential for restoring a Level Two
configuration. Instead (if confronted with Figure 8, say) you could just
turn
**U²** - then apply the **R²B²R²**
sequence mentioned earlier, and later attend to all the edge-pieces.

**FIGURE 8: **This corner-swap can be required in Level
2.

Operator 3 achieves it without twisting the corners.

Otherwise turn **U² **to
make the exchange-lines parallel and apply **R²B²R².**

--------------------------------------------

> Before it can accept operator 3, Figure 9 below requires preliminary
manoeuvre
**RU** ... (which in terms of colours could be jotted down
here as "Red, Yellow").

**FIGURE 9. I**n Level 2, these corner swaps might

occasionally be encountered (but not by themselves).

> At this early stage, do *not* try and use operator 3 for a crosswise
exchange of corner-pieces *which occupy the same layer -* because
you will then end up with a pair of "twisted" corners. We will deal with
corner-twisting in Level Three, but in the meantime simply turn the offending
layer through 180 degrees - which corrects its corners whilst disrupting
two pairs of edge-pieces, leaving a configuration like Figure
2.

**LEVEL THREE: A COMPLETELY SCRAMBLED CUBE**

> The **RB.R ^{-1}B^{-1}** sequence learned for operator
3 is also part of operator 6. (Operators 4 and 5 will be given later,
in what is perhaps a more logical numbering system).

On a pristine cube, perform
**RB.R ^{-1}B^{-1 }**

**FIGURE 10: **One corner has been twisted anticlockwise
...

(... so it needs a clockwise twist to correct it).

Also, the upper-right edge-piece needs a "flip".

Here then is the complete sequence for the next operator:
** 6. (RBR^{-1}B^{-1})².F.(RBR^{-1}B^{-1})^{4}.F^{-1}**...
It twists two of the front corners - one clockwise and another anticlockwise.
The

Alternatively, try **(RBR ^{-1}B^{-1})².F².(RBR^{-1}B^{-1})^{4}.F²**
... which is essentially the same process, but acting on different corners.
It is always a matter of placing both offending corners on the front face
- (that may require a preliminary manoeuvre) - while remembering that the
piece being corrected is at upper-front-right. Note too that

Also try **(RBR ^{-1}B^{-1})².F.(RBR^{-1}B^{-1})².F.(RBR^{-1}B^{-1})².F²**
... which produces three clockwise twists in the front layer.

> **The Rubik Cube has a "parity law" **by which:

...

Before utilizing this operator 6, it is best to decide in advance whether
you are going to tackle three corners or just two. Remember that they must
all be in the same layer; (it may be necessary to move them there first).
You might have a cube which needs two clockwise and two anticlockwise twists;
if so, correct them as two separate pairs (where a "pair" denotes one of
each type). If, by mistake, you carry out two clockwise twists - say, with
**(RBR ^{-1}B^{-1})².F.(RBR^{-1}B^{-1})².F^{-1}**
then (at that stage) you will notice a lot of other pieces out of position
- because you have used only

> There is also a trick for halving the time spent on producing anticlockwise
twists: work with the left layer instead of the front one, using the lower-left-back
corner as the operating point. **(RBR ^{-1}B^{-1})²**
twists that piece anticlockwise, and

This strategy is therefore useful when confronted with three anticlockwise twists.

Indeed, you can also use it after **(RBR ^{-1}B^{-1})²**
for twisting a second

____________

> One more operator is advisable for tackling a completely scrambled cube. Its function is to "flip" or correct an edge-piece, like the red-yellow one in Figure 10. This next operator involves a new type of turn:

Hold the top and bottom layers still with one hand, and pull the middle
or equatorial layer round with the other hand (Figure 11). An anticlockwise
move is more natural for a right-hander; **E ^{-1}** denotes
a quarter-turn.

**FIGURE 11. **The anticlockwise equator-turn **E ^{-1}**.

The result is shown below. Here, the F-face was originally entirely
white, while the R-face was all red.

The black squares came from the left face.

----------------------------------------------------------------------
** 7. (RE^{-1})^{4}**
produces Figure 12 below. Four edge-pieces are flipped, three of which
are on the equator.

**FIGURE 12 **(above and below):

The four flips produced by operator 7 ...

... viewed from the left and from the right.

____________________________________________________________

__Additional Operators__

The six listed above are enough to restore a cube. (In other words, numbers 4 and 5 below are not absolutely essential).

However, you will sometimes find yourself having to correct just *one
edge-pair plus one corner-pair.* If you have most or all of those incorrect
pieces contained in the same layer, then you could simply give that layer
a quarter-turn, enabling you to tackle the edges and corners separately.
(In order to get all the pieces into the same layer, you may need to do
some preliminary work using operators 1a, 1b and 3). But eventually you
will probably decide that it is indeed worth learning operator 5, because
it will make the task somewhat easier.

Many hundreds of operators have been discovered, so these extra ones given below represent just a very small selection. Some may suit you, others may not:

** 2b.** Triangular exchange of edge-pieces:

** 4. **Triangular exchange of corner-pieces:

and

That was deduced using the versatile sequence **RBR ^{-1}B^{-1}**

- here transferred to other faces as

It is really a four-stage process: **UBU ^{-1}B^{-1}**
followed by

then reverse the sequence giving

As explained when operator 2 was introduced,
any triangular exchange can also be contrived using a double application
of a "two pairs" swap.

_ _ _ _ _ _ _ _

> ** Here is a third parity law: **(two have
already been mentioned - involving edge-flips and corner-twists)...

We can only exchange: *[The numbers here below, correspond
to those of the operators]*

(1) two pairs of edge-pieces,

(2) or three round a triangle,

(3) or two pairs of corner-pieces,

(4) or three round a triangle,

(5) or *one edge-pair together with one corner-pair.*

To tackle this last situation we have this other operator -

** 5. U²BU²B^{-1}U².L²B^{-1}L².FU²F^{-1}**
- which swaps two corners along with two edge-pieces. Others are available
for carrying out an exchange like that, but this one is my favourite because
the result is symmetrical.

Furthermore, the above sequence can also help unscramble a "superior"
Rubik Cube *which contains symbols or emblems on all its faces, *(i.e.,
requiring its centre-pieces need to be orientated correctly !) ** Operator
5 turns the back centre-piece 90 degrees anticlockwise. **An

** 6b. (F²UR^{-1}.B²RU^{-1})²**
twists two corners.

** 7b. RF^{-1}UR^{-1}F.E.F^{-1}RU^{-1}FR^{-1}.E^{-1}**flips
the two edge-pieces on the right equator. You could also use

____________

__My personal strategy__

Whereas some people like to restore the top layer completely before moving on to the middle one, others (including myself) prefer to leave one column free, as "parking space".

Thus, I correct the four edge-pieces and three of the top corners, then three or four edges at the base - sometimes together with some on the equator, if convenient. All this can be accomplished just by "common sense". Preliminary manoeuvres for lining up edge-pieces tend to be harder than for corners, so edges are given priority at this early stage. Try and leave at least some of the pieces correctly twisted and flipped, if you can.

After that, operators are indispensable.

_______________________________________________________________________________________________________

__APPENDIX__

**R **= Right; **L **= Left;
**F **= Front; **B **= Back;
**U **= Up (i.e. Top); **D **= Down (i.e. Bottom, but the initial
"B" has already been taken).

**R **then denotes a clockwise quarter turn (i.e. through 90 degrees)
*of
the Right Layer,* as in Figure 13. **R ^{-1}** is an anticlockwise
turn. (Some texts write

"Squared" moves represent 180-degree turns - e.g. **F²**
for the front layer.

**FIGURE 13:**

An **R**-move turns the right-hand layer 90º
clockwise.

------------------------------------------------------------

The *"two-wheels"* move **RL ^{-1}** may be abbreviated
as

**E **and **E ^{-1}** (Figure 11)
both turn the middle layer or "equator"

E-mail: __DLMcN@yahoo.com__

Link to Mathematics

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