Capstone

Solving The Rubik’s Cube
Solving The Rubik’s Cube, Analyzing The Fridrich Method
Hans K. Coburn
Northwest Christian University
April 8, 2016

Abstract
The Rubik’s Cube was made famous in the 1980s when it was distributed to stores nearly
world wide. In 1982 the first speed cubing competition took place. Ever since, there have been
different solving methods and strategies to decrease the number of moves needed to solve and
lower the amount of time needed to solve it. One of the original competitors in that first
completion, Jessica Fridrich, invented her own method for solving the cube. Over the years her
original method has been added to by other speed cubers. That method, which is used to some
degree by most every speed cuber today, is known as the Fridrich Method.
In this capstone, I will be setting out to analyze some of the very basic algebra behind the
Rubik’s Cube. This will allow a better understanding of how to finish the last layer of the cube
using the Fridrich Method. This final step in the method is the permutation of the last layer (also
referred to as PLL). These PLL algorithms that will result are not the only algorithms needed to
solve the cube using the Fridrich method; however, the same type of group theory is behind the
construction of all algorithms for the cube. I will also include an intuitive guide to solving the
cube that anyone can use to solve without the use of the internet to look up algorithms.
Keywords: Speed cubing, Permutation, Groups, Cubie, Cubicle,

A History of the Cube
The Rubik’s Cube was invented by Ernõ Rubik in 1974. Rubik was a professor, at the
time, teaching architecture in Hungary where he was born. Although the cube was invented in
the 70’s, it did not become popular till the 1980s. In response to the cube sensation came the
invention of the 4x4x4 cube called Rubik’s Revenge in the mid 80’s. (Rubik’s, n.d.). Originally
it was believed that the cube would not become popular because of how difficult it was to solve.
Of course, today, with the use of the internet, almost anyone can figure out how to solve the cube
with enough patience. In reality, the cube became so popular that it birthed the competitive sport
known as speed cubing. (Rubik’s, n.d.).
Some, who stumbled their way through the cube, discovered that the cube could be
solved in different ways. Some of these methods were more intuitive than others. Different
methods gave way to the ability to solve the cube in faster times. Because of the competitive
nature of humanity, a sport emerged from the cube as it tested a balance of dexterity and
memory. On the 5th of June, 1982, the first world championship for speed cubing was held in the
birthplace of the cube, Hungary. (Rubik’s, n.d.). Present at that competition was a speed cuber
named Jessica Fridrich and Mirek Goljan. Both were crucial trailblazers of solving methods and
labeling methods which make the generation of algorithms possible. The method that Jessica
Fridrich used became known as the Fridrich Method. Her efforts, and the efforts of many of the
other earliest speed cubers has given us the exceptionally fast solving times that we see today.
(Fridrich, n.d.). The current world record time for solving the traditional 3x3x3 cube is now a
shocking 4.9sec (set by Lucas Etter in the Fall of 2015)! (World Cube Association, n.d.).
Getting Familiar with the Cube

There is a lot of mathematical complexities behind the Rubik’s Cube. Before we can
begin analyzing the cube’s structure, we must first become comfortable with the basic make up
of the cube. Once we understand the make up the structure, we can determine the number of
arrangements the cube can be in. We will get familiar with calling these arrangements of the
colors on the cube, permutations of the cube.
Pieces of a puzzle
There are three different types of pieces that make up the cube. Before we identify these
pieces, it is important to clarify our language. From here on out, we will refer to a particular
piece (either an edge or corner) as a cubie and the particular places that the cubies can live we
will call cubicles. This basic labeling is quite standard when talking about the cube. The first,
and simplest category of cubies, are the center cubies. There are six center cubies which cannot
move relative to each other making them fixed in the correct position at all times. The second
category of cubies are the edge cubies. There are twelve edge cubies each with two sides. This
means that not only can they change positions, they can also be permuted. Think of our edge
cubie that contains the white and blue colors. Now examine how many different spaces it can be
placed in on the white side of the cube. There are four edge cubies on each side so we know that
there are four options for our white and blue cubie to be in. However, that same cubie can be in
those same four cubicles and be turned around so that the blue side is touching the white side.
This is what permutations relative to cubies means; they can exist in a certain location two
different ways. Lastly, our third type of cubie is the corner cubie. There are eight corner cubies
all containing three permutation options. As we saw with the edge cubies, the corner cubies have
one more side to them meaning they can appear in a certain location three different ways.

Possible Arrangements of the Cube
To say there are a lot of possible arrangements of the cube is an understatement. In order
to calculate this enormous number, one must first think back to the different cubies that make up
the cube. There are eight corner cubies. Each cubie can move to a different position. One could
take a corner cubie and place it into one of eight cubicles. That gives us eight possible
permutations already. Now with one corner occupied, there are only seven possible cubicles left
for the next corner cubie. Following this logic, one can find that the resulting number of possible
permutations is: 8 ∗ 7 ∗ 6 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 40,320. This is called factorial written as: 8! =
40,320.
Next we analyze the edge cubies. There are 12 edge cubies and 12 respective cubicles
resulting in twelve factorial possibilities: 12! = 479,001,600. We now know the total number of
arrangements of the cubies within the cube are 12! ∗ 8!, which a number that involves thirteen
zeros. There is one thing we are forgetting however, and that is that edge cubies have two sides
and can be oriented two different ways, and similarly, corner cubies have three different
orientations they could appear in. Think about our corner cubies for a moment. Each cubie can
be in three different positions independently of the other corners. This gives us an extra 38
possible positions. Using the same logic we obtain an additional 212
possibilities from the edges.
Our new total is now 8! ∗ 12! ∗ 38
∗ 212
, which is an enormous number. This means that
the cube could potentially appear with about 5.19 ∗ 1020
different arrangements of what appears
to be a simple six colors mixed up among six sides. (Chen, n.d.)
Achievable Permutations of the Cube
We would like to find out how many arrangements of the cube are in fact possible
through legal moves. We call these arrangements of the cube permutations of the cube. We must

first start with a basic definition of what a permutation is. A permutation is the result of some
operation or function 𝜙. This function 𝜙, is a map from a particular set A to the same set A. 𝜙
must also be one-to-one and onto (Fraleigh, 1967). When thinking about the cube, all of our legal
moves can be seen as a permutation of the cube. (By legal moves, I am meaning the intended
turning of the sides of the cube. An illegal move would be dismantling and reassembling the
cube.) We will dive into a more detailed examination of the moves on the cube after we have
found the number of possible permutations of the cube.
This number is all well and good and is achievable if the cube is dismantled and
reassembled; however, not all 5.19 ∗ 1020
solutions are achievable through simply turning the
different sides. The cube is said to be an even permutation group; this cuts our 8!12! in half
(Rosen, 2007). Because of the nature and symmetry of all six sides of the cube, our number is
again decreased by a factor of two for our edge cubies and a factor of three for our corners
cubies. For a more detailed description of why this occurs, I recommend reading Tomas
Rokicki’s article “Twenty-Two Moves Suffice for Rubik’s Cube.” In the end, we find that the
actual number of permutation of the cube that are achievable via legal moves is actually:
8! ∗ 12! ∗ 38
∗ 212
3 ∗ 2 ∗ 2
This new number is
1
12
𝑡ℎ the size of our original number. After all this discussion, we can now
say that there is only one solution out of 4.325 ∗ 1019
possible arrangements of the cube.
Analyzing Moves of the Cube
Most who have played with a Rubik’s Cube at all, know that each colored side can
rotate. The problem is that some sides have shared cubies. When we turn one side and then
another we get this shuffling occurrence. After a short number of moves we find ourselves lost in

mess of colors. Before we can begin our look at solving the cube, we must have a brief look at
how the different moves of the cube interact with one another.
Groups
When looking at solving the cube we need a general understanding of group theory.
Group theory is built off of the definition of a group. A group G is a set closed under a binary
operation. There are three axioms that must be satisfied in order to say that G is a group: First,
the binary operation must be associative. Second, there must be an identity element e such that
𝑒 ∗ 𝑥 = 𝑥 ∗ 𝑒 = 𝑥 for every 𝑥 ∈ 𝐺 (A good example of this is the number 1 when multiplying
integers. Any number multiplied by 1 simply equals that number. So in the group of integers
under multiplication, 1 is the identity element.). Third, there must be and element 𝑥−1
such that
𝑥 ∗ 𝑥−1
= 𝑥−1
∗ 𝑥 = 𝑒 meaning that 𝑥−1
is the inverse of x (Fraleigh, 1967).
Mail Run
The Rubik’s Cube can be expressed as a group. In order to understand what this means in
terms of the cube, let us look at a traffic example. Suppose that there is a set of streets. These
streets are all one-way and they form a square. Now, suppose that these streets each have
mailbox at each corner and a delivery worker can follow the one-way roads to move mail from
one box to the next. In this set up, there can only be one parcel of mail in each box. Here is what
it looks like so far:
a b
d c
Figure 1
The letters a, b, c, and d are what we will call the mail in each box. Now when the delivery
worker’s route we will call M. So when we perform M once, we will find that the mail that was

in a is now in the mailbox b. Similarly, the mail that was in b is now in mailbox c. We can
express this operation in this way: 𝑀 = (𝑎, 𝑏, 𝑐, 𝑑). We can also tell the worker to reverse their
route which we will notate as 𝑀−1
. All 𝑀−1
means is that the worker will run the same transfer
of mail backwards (moving a to d, d to c etc.).
Looking back at our definition of a group we realize that all three criteria have been met.
(One might ask what the identity element is for this group, but it is simply the delivery worker
taking the day off). This mail run is a group; not a very interesting group, but a group none the
less. Let’s take a look at a series of streets adjacent to the mail run we just looked at. It is another
block of one way streets with mailboxes on each corner. Because these sets of streets are
adjacent we can see that these two sets of streets share two mailboxes; particularly mailboxes b
and c. The new picture looks something like this:
a b g
d c h
Figure 2
Suppose that we call our original mail run M and our second mail run N. So now if we
performed N the mail in box b would move to box g, mail in box g would move to box h, h
would move to c, and c would move to b. N can them be defined by the following: 𝑁 =
(𝑏, 𝑔, ℎ, 𝑐). We now have two distinct operations that we are calling M and N. If we were to
perform M and then N we would get something that looked like this:
d b a
c h g
Figure 3

If we were to write out the different paths that the mail takes in terms of operation MN we would
get the following: 𝑀𝑁 = (𝑎, 𝑔, ℎ, 𝑐, 𝑑)(𝑏). Something interesting happens to b and we perform
MN. The mail in b gets moved down the c with M but then when we follow that move with N, b
ends back in its original place. No matter how many times we perform MN, b will always stay in
the same place.
One other observation made about the relationship of M and N is that they do not
commute. Integers under addition commute, for example: 2 + 3 = 3 + 2 = 5. Integers under
subtraction are not commutative, for example: 3 − 2 ≠ 2 − 3. Our two operations do not
commute because 𝑀𝑁 ≠ 𝑁𝑀. We know from above that 𝑀𝑁 = (𝑎, 𝑔, ℎ, 𝑐, 𝑑) (𝑏) where if we
took NM we would get: 𝑁𝑀 = (𝑎, 𝑏, 𝑔,ℎ, 𝑑)(𝑐). If N and M did commute, it would not matter
what order we arranged them in; we would get the same outcome. Also, if we applied the inverse
of both N and M after taking MN we would gain the identity. We can prove that M and N do not
commute by showing that 𝑀𝑁𝑀−1
𝑁−1
≠ 𝑒. Referring back to figure 4 we can say that 𝑒 =
(𝑎)(𝑏)(𝑐)(𝑑)(𝑔)(ℎ). Now, let’s see what 𝑀𝑁𝑀−1
𝑁−1
looks like:
b a g
d h c
Figure 4
We find that 𝑀𝑁𝑀−1
𝑁−1
= (𝑎, 𝑏)(𝑐, ℎ)(𝑑)(𝑔). In other words, a and b trade places as do c and
h, while everything else remains unchanged. Now we can say with certainty that M and N do not
commute because of what is called a commutator. In our example, the commutator of M and N is
𝑀𝑁𝑀−1
𝑁−1
.
Commutators on the Cube

Now that we have spent a long time talking about these mailboxes and delivery routes it
must be made clear that we could be talking about the relationship between corner cubies
existing on two sides of the cube. Let’s now say that we are holding the cube with the white side
face up, the blue side facing towards oneself, and the the orange side on the right side. Now let’s
define a clockwise turn of the white face as U (for up) and a clockwise turn of the orange face as
R (for right). Notice that these two moves, U and R, both affect all the cubies that have both
white and orange colors attached to them. The assumption is that they do not commute. We will
prove they do not commute by use of the commutator 𝑈𝑅𝑈−1
𝑅−1
. If one performs 𝑈𝑅𝑈−1
𝑅−1
on a solved cube they should notice that the cube is no longer solved (and remember that e in this
case would be a solved cube).
Looking back at what happened to the mail in our delivery example is exactly the same
relationship that the corner cubies have when looking at the operations U and R. One point to
take note of is, this relationship we have examined, only takes into account cubies in cubicles. It
does not examine the permutation of each cubie. One will find some interesting rotation of
corner cubies with the commutator 𝑈𝑅𝑈−1
𝑅−1
. To keep things simple, let’s refer to our corner
pieces with the same letter as the mailboxes from earlier. (For example, if the white face is U and
the orange face as R, the a cubie will have white, red, and green colors on it.) If one performs the
commutator 𝑈𝑅𝑈−1
𝑅−1
twice they will find all corner cubies return to their correct cubicles
(just like in the mailbox example); however, their orientations will have changed slightly. Both a
and b have rotated clockwise, while c and h have rotated counter-clockwise.
One of the amazing things that happens when mapping the movements of the Rubik’s
Cube is that commutators, such as the one seen above, show up in a vast majority of the
algorithms used to solve the cube. This is an example of how proving something seemingly

simply, such as finding whether two sides commute or not, can actually be an essential part of
solving something as complex as the Rubik’s Cube.
Solving the Cube
There will be three methods of solving the cube discussed here. The first method is
known as the beginner’s method because it is the method that most learn first. It is also the
method that is contained in the instructions paper that comes with the cube from the Rubik’s
brand. The second method, that can cut one’s solving time drastically if performed properly, is
the Fridrich method. This method combines certain steps of the beginner’s method and involves
a much larger list of algorithms to be learned. The last method explained is a way in which one
can create their own algorithms to solve the cube. It is an explanation of how one can solve the
cube using their own intuition as opposed to internet videos.
Beginner’s Method
The Beginner’s method involves six steps; solving the cube by layer. A common mistake
is to forget that cubies only have one cubicle that is the solved location. This simply means that,
just because it has a white sticker and is next to the white center, does not mean it is in the
correct cubicle. For example, the cubie that has white and blue on it, has only one correct cubicle
that is edge cubicle between the white and blue faces.
Before we begin to solve, I would like to simply clarify all the moves of the cube. We
will be seeing algorithms fairly soon and it will be important to know what they mean. R stands
for one rotation of the right side clockwise. L stands for the left side, D for the downward face, F
for the forward face, U for the upper face, and B for the backward face. Each move, as we saw
earlier, has an inverse that means a counter-clockwise turn.

Layer one of the beginner’s method. Layer one is the most intuitive step in solving the
cube. Nothing is in the correct place yet so there is no need to worry about corrupting any
previous work. The first thing one must do is solve the edge pieces for one side. One might
typically start with the white side because it is one of the easier sides to recognize. Placing the
edge cubies is not too difficult as long as one keeps in mind to line up not only the white side (if
they are starting with the white side) but also the edge cubie’s second color. This step is
completed when there is a white ‘X’ or cross on the starting side.
Next, the corners of the first layer are added. One must be slightly more cautious now
because there are solved cubies that we do not want to disturb. Placing the corner cubies can be
done by simply bringing down a side (with the moves R, L, F, or B and their perspective inverse
turns) then using the D move to pair the desired corner cubie with its matting edge. It is
important to keep in mind again, that getting a corner in place with white facing up does not
mean that one has solved the first layer. This step is complete when the first side is complete and
all four adjacent sides have a 1x3 layer matching the centers.
Layer two of the beginner’s method. Moving on to the second layer, it is typical to now
hold the one fully solved side down. If one solved the white side first, the next task is to identify
the remaining edge cubies that do not contain yellow. We are looking for these second layer
cubies in the top layer so that we can freely position them above where they need to go. These
edge cubies will have two colors that match two of the four second layer sides. To position these
cubies, it is important to move the desired edge cubie so that the face color of the cubie matches

the color of the center facing towards the one operating the cube. Then one will need to either
‘drop it down’ to the right or to the left depending on the second color.
Figure 5
To place the edge to the right the algorithm is: (𝑈𝑅𝑈−1
𝑅−1
)(𝑈−1
𝐹−1
𝑈𝐹). To place an edge to
the left perform the following: (𝑈−1
𝐿−1
𝑈𝐿)(𝑈𝐹𝑈−1
𝐹−1
). It is important to realize that these are
just two commutators put together. Commutators come up in nearly every algorithm on the cube.
Using these two algorithms, one can finish the second layer.
Layer three of the beginner’s method. The first goal of this step is to achieve a yellow
cross on the top face of the cube. In other words, we will be permuting the group of edge cubies
in the final layer. This is done with a very simple conjugate algorithm. (In terms of the cube, a
conjugate algorithm consists of a set up move/moves, an algorithm to manipulate what is needed,
and then the inverse of the set up move/moves.) Performing 𝐹(𝑅𝑈𝑅−1
𝑈−1
)𝐹−1
manipulates our
desired edge cubies. If one encounters a line, holding the cube with the line running from left to
right then performing the algorithm will generate the cross. If one encounters a small L like
shape, orient it into the top-left corner. If one encounters only the yellow center and all edges
permuted wrong, it does not matter which side faces towards the operator. One may have to
repeat this algorithm up to three times depending on the permutation encountered.
Once all the edge cubies have been oriented correctly, it is time to align the edges with
their matching sides. By rotating the top layer (in other words performing U as much as desired),
one can see how what needs to be moved to properly align the edges with the four side centers.

The only algorithm needed to permute these edge pieces is 𝑅𝑈𝑅−1
𝑈𝑅𝑈2
𝑅−1
. It cycles three
edges cubies. Figure 8 depicts the cycle of this algorithm.
a
d b
c
Figure 6 (a view of the top of the cube)
One will have to perform this algorithm at most two times. This step is complete once all edge
pieces are oriented and permuted correctly.
The final step is permuting the group of corner pieces remaining and then permuting each
corner cubie. The algorithm needed to cycle these corners is 𝑈𝑅𝑈−1
𝐿−1
𝑈𝑅−1
𝑈−1
𝐿 and is
depicted in the following figure:
a b
d c
Figure 7
It is important to remember that the corners in this step will not necessarily have the yellow side
facing up. One must check to make sure that the three colors on the corner cubie match the three
sides it touches.
The last step of the beginner’s method is to permute the corner cubies. This is done by
holding a cubie that must be spun in the upper, right, face corner and performing the following
commutator: (𝑅−1
𝐷−1
𝑅𝐷)(𝑅−1
𝐷−1
𝑅𝐷). Performing this algorithm once rotates a corner cubie
counter-clockwise. One must permute one corner at a time. After achieving the correct
permutation of the cubie, one can simply perform U until the next cubie that requires rotation is

in the place of the previous cubie spun. This process is repeated until all corners are permuted
correctly. The cycle of the algorithm should also be completed at the same time resulting in a
fully solved cube.
Fridrich Method
The Fridrich method of solving Rubik’s cube is one of the most popular solving methods
for speed cubers (Rubik’s, n.d.). This method is more advanced because of the number of
algorithms one is required to memorize in order to perform the method properly. When solving
with the beginner’s method, one can approximately achieve times of one minute. When using the
Fridrich method, one can cut that time easily in half; with practice, sub-thirty seconds becomes
much more realistic. The Fridrich method, also referred to as CFOP, involves four steps:
building the white Cross, solving the First two layers, Orienting the last layer, and lastly
Permuting the last layer.
Building the Cross. Solving the first layer using CFOP involves first building the cross.
This step is generally thought to be entirely intuitive and is the only step that is the same as the
beginner’s method. Through practice, this step should be achievable in no more than 2-5
seconds.
F2L. The next step in CFOP is solving the corners of the first layer at the same time as
solving the edges of the second layer. This step is commonly referred to as F2L (First two
layers). F2L is also very intuitive as opposed to the algorithms needed in the beginner’s method.
This involves pairing the edge pieces with the perspective corners before placing them. Because
of its intuitive nature, both building the cross and F2L take up the bulk of the solving time. A
well practice F2L can be performed in anywhere from 5-15 seconds.

OLL. Now that the first two layers are solved the goal is to orient the last layer, also
referred to as OLL. There is a large amount of possible permutations possible for this last layer.
OLL is only concerned with the orientation of this last layer. There are 58 cases that can occur
when performing OLL. One of those possible occurrences is the orientation of the last layer fully
solved so we do not need an algorithm for that occurrence leaving us with 57 algorithms needed
to complete OLL. Each algorithm for OLL completes the orientation in that single algorithm. (To
begin learning OLL, there is a method of set up moves that involves algorithms already learned
in the beginner’s method called a two OLL). OLL takes a well practiced cuber approximately 2-3
seconds.
PLL. The final step when using the Fridrich method is called PLL which stands for
Permuting the Last Layer. This means that all the cubies are in their correct orientation but not
necessarily in the correct cubicle. Much like the OLL algorithms, the PLL algorithms each deal
with a specific case that can be generated from the previous step. There are 22 possible
arrangements of the cubies in their oriented states; one of which is the fully solved permutation
of the layer resulting in 21 algorithms needed to perform PLL in one algorithm. PLL also takes
approximately 2-4 seconds.
These PLL algorithms are an essential piece of the Fridrich method. Because the concern
of this step is simply with cubies trading positions, the algorithms are only concerned with
mapping the movements of the cubies and not the permutation of the cubies. Just as we saw
earlier, commutators appear a large amount in the algorithms for PLL. Let’s take a look at a
famous PLL algorithm known as the T-perm: (𝑅𝑈𝑅−1
𝑈−1
)𝑅−1
𝐹(𝑅2
𝑈−1
𝑅−1
𝑈−1
)(𝑅𝑈𝑅−1
𝐹−1
).
(Keep in mind that the parentheses are just to help us recognize patterns and each movement

within all algorithms of the cube are associative). If the T-perm is performed on a solved cube
with the yellow face on top, looking down at the cube gives the following:
a 1 b
4 2
d 3 c
Figure 8
So we can say that 𝑇: (2,4)(𝑏, 𝑐) order two. We find that the T-perm trades two corner cubies
and two edge cubies. Because the permutation of the individual pieces does not change, we can
say that the T-perm has order two meaning if you perform it again you will get the identity back.
We can say: 𝑇 ∗ 𝑇−1
= 𝑇 ∗ 𝑇 = 𝑒. This means that T is its own inverse.
Intuitive Method
Most think that the cube is too complicated for them to solve without learning it from the
internet. In order to perform the Fridrich method, or even the beginner’s method, this may be
true; however, with some intuition the cube solvable without these methods. The first thing one
must do when attempting to solve the cube is to become familiar with the cube and how it
moves. Solving for a single layer is not difficult. If someone can do that much intuitively, using
the property of inverse functions, one can fully solve the cube.
The key to this method is to be aware of the particular subgroup one is attempting to
manipulate in some way. Think of a single layer of the cube for example. This layer can be
thought of as a subgroup of the entire cube. Any series of moves we can call M. As long as the
codomain of M is the subgroup we started with, there is a series of moves 𝑀−1
that restore the
cube back to the state it was in before M. If all the cubies of the desired layer before M appear in
that same layer after M, we can rotate that layer in between M and 𝑀−1
. A good example of this
a 1 c
2 4
d 3 b

is what we saw in the beginner’s method in the last step. The commutator (𝑅−1
𝐷−1
𝑅𝐷), when
performed twice, keeps all the the cubies of top layer in the top layer. Even though the bottom
two layers seem to be corrupted, they are restored by simply performing the algorithm in reverse.
All one really needs to complete the cube is their intuition and homemade algorithms through the
process of finding inverses.
Final remarks
After all this discussion on the cube, one might ask how these slick algorithms that
generate small subgroups of a layer of the cube are created; the answer is in the mapping of
generators. Of course to be able to map so many different combinations of moves would take a
long time and would involve a large amount of trial and observation. A deeper study into
computer science would be necessary if one wanted to analyze the creation of these algorithms.
Not to be discouraged by that, one can still map simple systems of moves to develop their
own solving method. Knowing what specific cubies are effected by these systems of moves (or
what subgroups are generated by the function) can be modified by set-up moves to manipulate
the desired cubies. In the abstract algebra world, these “set-up moves” paired with another
algorithm can be referred to as conjugates (Singmaster, 1981). Many experienced cubers
intuitively create algorithms using conjugates and maps of simple systems of moves.
As mentioned above, a vast majority of the cube can be solved intuitively. In order for
this to happen, one must have the patients to observe the movements of the cube. There are
patterns and puzzles all around us in the world. Ernõ Rubik put it poetically when he said: “If
you are curious, you’ll find the puzzles around you. If you are determined, you will solve them.”
(Rubik’s, n.d.). Do not be afraid to analyze the puzzles that can be found, because in order to
solve them we must first observe them.

References
Chen, J. (n.d.). Group Theory and the Rubik's Cube. Harvard University.
Demaine, E. D., Demaine, M. L., Eisenstat, S., Lubiw, A., & Winslow, A. (2011). Algorithms
for Solving Rubik’s Cubes. Algorithms – ESA 2011 Lecture Notes in Computer Science.
doi:10.1007/978-3-642-23719-5_58
Fraleigh, J. B. (1967). A first course in abstract algebra (5th ed.). Reading, MA: Addison-Wesley
Pub.
Fridrich, J. (n.d.). 20 years of speedcubing. Binghamton University. Retrieved February 27,
2016, from
http://www.ws.binghamton.edu/fridrich/history.html
Rokicki, T. (2010). Twenty-Two Moves Suffice for Rubik’s Cube®. Mathematical Intelligencer,
32(1), 33-40.
Rosen, K. H. (2007). Discrete mathematics and its applications (6th ed.). Boston: McGraw-Hill
Higher Education.
Rubik’s. (n.d.). The History of the Rubik’s Cube. Retrieved February 27, 2016, from
https://www.rubiks.com/about/the-history-of-the-rubiks-cube
Ruwix (n.d.). Rubik's Cube advanced Friedrich - Orient last layer OLL. (n.d.). Retrieved April
20, 2016, from http://ruwix.com/the-rubiks-cube/rubiks-cube-solution-with-advanced-
friedrich-method-tutorial/orient-the-last-layer-oll/
Singmaster, D. (1981). Notes on Rubik's magic cube. Hillside, NJ: Enslow.
World Cube Association. (n.d.). Official Results. Retrieved February 27, 2016, from
https://www.worldcubeassociation.org/results/events.php?eventId=333

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