`

Rubik’s Cube Best Algorithms

Top 5 speedcubing methods

by

Daniel Ross

Table of Contents

Table of Contents

Introduction

Chapter 1 - The Rubik’s Cube: The Greatest Puzzle Invented

Chapter 2 - Mathematics and Algorithms Behind the Cube

Chapter 3 - Rubik’s Cube Basic Notation

Chapter 4 - Beginner’s Method in Solving the Rubik’s Cube

Chapter 5 - Advanced Methods: The Fridrich Method

Chapter 6 - Advanced Methods: The Roux Method

Chapter 7 - Advanced Methods: The ZZ Method

Chapter 8 - Finger Tricks

Chapter 9 - God’s Number and Rubik’s Cube

Conclusion

Introduction

Let me just say that I am really proud of you. Getting this book is a proof that you are among the selected few who decided to step up this game and see beyond the rewards of solving the Rubik’s cube.

You see, not all who tried solving the greatest puzzle toy of all time became successful. Many, in fact millions of them, gave up and convinced themselves that solving the cube is too difficult.

But you are different. I bet you really want to solve the cube; that’s why you have purchased this book. Or probably you have already solved your Rubik’s cube and you want to better understand its mechanism.

As you will learn in this book, the Rubik’s cube is not just a random toy with random colors that you can solve by haphazardly turning and rotating. No, the Rubik’s cube is a great example of what mathematical thinking can do. It follows specific sequences and patterns that you can only understand once you understand and speak its language: algorithms.

As suggested by the title of the book, you will learn the best algorithms that master speedcubers are currently using to solve the Rubik’s cube in as fast as four seconds. You will also learn some tips and tricks on how you can improve your skills in solving the Rubik’s cube.

Once again, thanks for downloading this book. I hope you find it helpful!

Chapter 1 - The Rubik’s Cube: The Greatest Puzzle Invented

The Rubik’s cube is among the quite few puzzles to have survived and became very popular worldwide, even though it has been introduced in the 1970s, before even most of its present patrons even existed.

A Brief History of the Rubik’s Cube

To better appreciate the Rubik’s cube and be proud of its legacy, we will take a short look at its rich history. Before Rubik’s cube was introduced in the public, similar puzzle toys have been released in the market. Several years before Erno Rubik received a patent for the 3 x 3 puzzle, a 2 x 2 puzzle toy was invented and introduced to the public by Larry Nichols. In the 1970s also, Frank Fox patented a 3 x 3 x 3 puzzle toy. But neither Nichols’ nor Fox’s toy took off the market. Rubik’s cube became very popular.

Erno Rubik received the patent for his 3 x 3 Magic Cube in 1975. The device was originally created for him to study an object with parts that can independently move. Rubik’s cube was first introduced in Budapest two years after. Ideal, a toy company, later helped Rubik to bring the magic cube to the world.

World Wide Success

Rubik’s Magic Cube enjoyed positive success around the world. It was released globally in 1980. In the same year, the puzzle toy also won the prestigious Toy of the Year Award. The reason behind the Rubik’s cube success is because children and adults find it entertaining and challenging. In fact, most customers are adults, and after only two years, more than 100 million Rubik’s cube has been sold around the world. The cube was well-praised by critics because of its simplicity yet difficulty to solve.

The Inventor

Erno Rubik is a Hungarian architect and inventor of the Rubik’s cube. He was born in 1944 during the height of the World War 2. He studied at the Technical University in Budapest in 1967 and started postgraduate studies in interior architecture and sculpting. During the time that he invented the Rubik’s cube, he was an architecture professor at the Budapest College of Applied Arts.

As a professor at the academy, Rubik was very fond of constructing geometric figures. Thanks to this hobby, he was able to design a cube, composed of 27 blocks made of wood. It took him over a month to solve the puzzle cube, which he later used as an effective tool to teach permutations in his algebra class.

This cube became very popular in the academy and soon in Budapest, and the rest is history.

Rubik became a tenured professor in 1987, and in 1990 he was elected as the president of the Hungarian Academy of Engineering. During his tenure, he built the International Rubik Foundation who works for supporting talented engineers in Hungary.

Currently, Rubik is primarily working on developing video games and still a main global icon in architectural topics.

Little is known about Rubik as he is an introvert and avoids public appearances. He rarely signs autographs, so if you have one, you are quite fortunate. He is not even known for attending speedcubing events.

What is known is that as a result of his invention, Rubik became one of the wealthiest Hungarians today.

Infringement Issues

After the remarkable success of Rubik’s cube in the world market, Larry Nichols filed an infringement case against Rubik and Ideal Co in 1982. Nichols won the case in 1984, but Ideal Co filed an appeal case submitting new evidence that Rubik’s design was original because it used interlocking plastic parts while Nichol’s design used magnetic blocks. In 1986, the appeals court decided the case for Rubik’s design. It was then formally established that the 3 x 3 Magic Cube invented by Rubik was his original design .

Difficulty in Solving the Puzzle

Rubik’s cube became a very popular toy, but some customers complained that it is impossible to solve. Hence, Ideal Co later releases tutorials and guide books to teach people how to solve the puzzle. The most popular tutorial was “You Can Do the Cube” by Patrick Bossert, which was released in 1981.

Rubik’s 4 x 4 Revenge

In 1981, a new version of the magic cube was released by Ideal Co. The Rubik’s 4 x 4 Revenge was also a massive hit, as it posed an even more challenging puzzle. The 4 x 4 cube was not originally invented by Rubik, even though the creator of the cube authorized the name as a marketing strategy.

Rubik’s Cube World Championships

Only after a few years after the global release of the Rubik’s cube, in spite of the millions of cubes that were remain unsolved by that time, cube fanatics decided to take the puzzle to the next level - solving the Rubik’s cube as fast as they can.

The first Rubik’s Cube World Championship happened in 1982 where 20 contestants from around the world compete to become the fastest Rubik’s cube solver. The first World Champion was Minh Thai of United States who solved a 3 x 3 Rubik’s cube within 22.95 seconds.

The Fridrich Metho d

Although Minh Thai won the World Championship in 1982, the contestant from Czech Republic, Jessica Fridrich became a notable figure in the competition, mainly because of her method in solving the Rubik’s cube.

Fridrich used an easy-to-follow layer method that is a strategy used by most cube fanatics today. She then created easy algorithms in solving the cube, and in 1997 she published her methods on the Internet. The Beginner and Advanced Method discussed in this book were based on Fridrich’s algorithms.

World Cube Association

The World Cube Association (WCA) was established by Tyson Mao and Ron Van Bruchem in 2004, mainly to promote speedcubing contests. WCA also created a website where anyone can access speedcubing results and try to beat the fastest speedcubers in the world.

Because of WCA’s efforts, speedcubing developed into a worldwide hobby and the competitions hosted by WCA became an avenue where world records are made.

The current world record holder as the fastest speedcuber is Feliks Zemdegs of Australia. He solved a 3 x 3 Rubik’s Cube at an amazing 4.737 seconds during the POPS Open Melbourne Australia in 2016 .

Health Benefits of Solving the Rubik’s Cube

Aside from the fact that the Rubik’s cube is an entertaining piece of puzzle, solving it actually has benefits for your health. Below are the top health benefits of solving the Rubik’s cube:

Relieves Stress

Primarily, solving the Rubik’s cube can help you relieve stress.  It takes a lot of focus and brain effort to solve the cube. While you are performing algorithms that you will learn in this book, your brain is working hard to think ahead of your next move.  This is actually a form of meditation or mindfulness, as it can allow you to forget about the stress caused by the daily grind of life. And so, your mind will be forced to enter into a relaxed state. Regular brain workout using the Rubik’s cube has been discovered to lower blood pressure levels.

Enhances Your Capacity to Solve Problems

Puzzles that require a ton of mental effort, such as learning a new method to solve the Rubik’s cube, can be quite beneficial for your mental health. Once your brain becomes accustomed to solving complicated problems such as the Rubik’s cube, your mental capacity to solve problems will be improved. Hence, you will be better equipped in enriching your life. Challenging yourself to solve the Rubik’s cube as fast as you can is a great way to get those brain cells firing and work on areas of your brain that are not usually tapped.

Lowers Risk of Brain Diseases

Some studies reveal that solving puzzles such as the Rubik’s cube can fight mental diseases such as Alzheimer’s and Dementia. We spend a lot of time and money on taking care of our physical health, but we usually overlook our brain, which controls everything we do. Solving puzzles such as the Rubik’s cube will ensure that your brain will stay sharp as you grow old.

Slows Mental Ageing

Everyone will get old; there’s no way to escape ageing. However, working on puzzles such as the Rubik’s cube can help your brain to stay sharp. Just as how doing daily morning jogs can help you stay fit, regular stimulation through puzzles such as the Rubik’s cube can help keep your mind healthy and slow down the effects of ageing.

Solve Your Rubik’s Cube Today

Sure you can try to beat Zemdeg’s record, but before that you need to learn how to solve a Rubik’s cube, and then gradually increase your solving time.

The first step is to learn the mathematical background and algorithms of the Rubik’s cube, which we will discuss in the next chapters.

Chapter 2 - Mathematics and Algorithms Behind the Cube

The Rubik’s Cube , although classified as a puzzle toy, attracts mathematicians because it is regarded as a great symbol of mathematical concept materialized physically. It is crucial that you have at least a minimal background on the mathematics and algorithm behind the Rubik’s cube so you can better appreciate its design and process of solving. In mathematical sense, the Rubik’s cube is the manifestation of a permutation group. The cube has six different colors, with each one repeated specifically nine times. Hence, the Rubik’s cube is regarded as an ordered list with 54 elements with numbers between 1 and 6 with every number signifying that the color is being repeated exactly 9 times.

You can rotate the cube’s six faces, so you can identify six basic permutation operations that you can rearrange in a specific manner. Combining and repeating these permutations will lead to new permutations that you can rearrange the list in another manner.

The permutation below shows how the elements of the list can be rearranged by turning the D layer of the cube.

Remember that in mathematics, a permutation group is a group whose elements are permuted of the ordered list. The group operation is the permutations that rearrange the set in a particular manner. The cluster of all the permutations of a set is regarded as a symmetric group.

Now, let us further examine the properties of the Rubik’s cube as a mathematical structure:

1. The cube is associative because the permutations in a row could be combined together such as R(B’L) = (RB’)L (Don’t worry if you can’t understand these notations. You will learn them in the next chapter.

2. The cube has a neutral element, because there’s a permutation that doesn’t rearrange the whole set. For example, LL’

3. The cube has an inverse element because each permutation has a corresponding inverse permutation. For example, L - L’

4. The cube is commutative, which is not a required condition for the group permutation, but if you will notice FR is not RF but BF is FB.

5. The cube has a certain degree of permutation. The degree of permutation refers to how many times you have to perform a permutation to go back to the original arrangement of the group. Each permutation in the Rubik’s cube has a definite degree.

This is just a brief discussion on the mathematics of the Rubik’s cube, so we can lay the groundwork of the right mindset in solving the puzzle. The Rubik’s cube is not a random object, but it follows a certain sequence, in this case, permutations, so it can be solved.

Algorithms of the Rubik’s Cube

The algorithms of the Rubik’s cube refer to specific set of operation in the puzzle that reorients the pieces in a particular manner. As we have already discussed, the Rubik’s cube is a permutation group with 54 fields with values equal to 6*9. Hence, you can apply operations (turns, rotations, and mixture of these) which will lead you to solve the puzzle or mathematically restore the pieces in their original arrangement.

Chapter 3 - Rubik’s Cube Basic Notation

Notation is crucial to solving a Rubik’s cube. Speedcubers use notation to describe a rotation on the cube by using letters. You must learn how to read notation so you can easily perform all the algorithms and follow the tutorials in the following chapters.

To solve a Rubik’s cube, without thinking how fast you must do it, you only need to learn six letters. However, if you are aiming to become a speedcuber, you need to learn advanced notation so you can learn Advanced Methods described in this book.

Basic Rubik’s Notation

There are six faces in the Rubik’s cube, which is noted by single letters. These are the following:

R - Right

L - Left

U - Up

D - Down

F - Front

B - Back

The colors that you can see on every face will depend on how you are holding the Rubik’s cube, but it is best to describe the relative moves rather than the colors in the faces.

Remember, you can rotate every face in either clockwise or counter clockwise. You should follow this direction when you are looking directly at the face .

Another easy method to follow a notation is through the right hand rule. You have to point your right thumb away from the specified face, and then curl down your fingers. You will determine the counter clockwise direction by following the curled fingers.

Basic Movements

The most basic move is the 90-degree clockwise rotation described by a single letter. Take note that single letters are always in bold, so you can easily distinguish them. Below are the basic moves in solving the Rubik’s cube:

R - Rotate the right face in a manner that the side on the top will move to the back

L - Rotate the left face in a manner that the side on the top will move to the front

U - Rotate the upper face in a manner that the side in front will move to the left

D - Rotate the down face in a manner that the side on the top will move to the right

F - Rotate the front face in a manner that the side on the top will move to the right

B - Rotate the back face in a manner that the side on the top will move to the lef t

Advanced Notation

It is also crucial to learn advanced notation for solving Rubik’s cube as you will always encounter them while reading algorithms.

Special Notations

Signals can be added to the end of the notations to signify a special move. The following examples use the L face as the basic movement, but you can apply them for any face.

L2 (pronounced as L two) refers to the movement where you need to rotate the left face by 180 degrees.

L’ (pronounced as L prime) refers to the movement where you need to rotate the left face by 90 degrees counter clockwise or opposite the direction of L as described above.

2-Layer Movements

Small letters (r, l, u, d, f, & b) signify that you have to move the specified face and the middle layer adjacent to it. For instance, l means you have to move the L face as well as the layer between the L and R at the same time in the same direction as a regular movement for L. The signals 2 and ’ could also be used for these notations .

Remember, even though L and r will result to a similar change in the Rubik’s cube, these two are still different. These two movements will leave different orientation. L will keep the same color on the center of the F face, while r will move this color to the U face. Hence, r is actually the equivalent of Lx and not L.

So, what is Lx?

The small letter x refers to another Advanced notation, which refers to the movements that could mean rotating the whole cube without turning any of the face. The 2 and ’ can be used for these movements and double the movement or reverse the direction accordingly.

Below are advanced modifiers:

x - signifies movement of the whole cube on the axis connecting L and R faces, which rotates 90 degrees clockwise to the direction of the R movement

y - signifies movement of the whole cube on the axis connecting D and U faces, which rotates 90 degrees clockwise to the direction of the U movement.

z - signifies movement of the whole cube on the axis connecting B and F faces, which rotates 90 degrees clockwise towards the direction of the F movement.

Take note that the whole cube movements are not actually necessary in solving a Rubik’s cube, but you can still use them in algorithms so you can reorient the puzzle. Double turns are also possible, and you can also execute these moves in two directions.

Piece Notatio n

Edge, center, and corner pieces are the three types of pieces found on a Rubik’s cube.

Take note that there is only a single center piece for every face, so these are noted with a large letter of the face where it belongs. For example, U marks the upper center piece.

A corner piece signifies three faces adjacent to it. FRU refers to Front, Right Up and Corner.

An edge piece signifies by the two faces where it belongs: FU refer to Front Up and Edge

Slice Movements

Basic notations refer to the movement of the outer layers. However, there are advanced methods requiring movement of the middle layers. The following notations are used for slice movements.

H signifies movement of the horizontal slice between D and U facing the direction of a U movement

V signifies movement of the vertical slice between L and R facing the direction of an R movement

C signifies movement of the central slice between B and F facing the direction of an F movement.

Take note that V can also mean r R’ or L R’ x, so this notation, although simple, is not strictly followed. The former movements are actually extended descriptions of what you must do with your hands (if you are using your right hand). A left-handed person will usually do a l’ L to get the same result as V. However, there are instances when it is easier to follow the concept than the descriptive hand movements .

Now that you have learned the basic and some advanced notations, let’s start learning how you can solve your Rubik’s cube.

Chapter 4 - Beginner’s Method in Solving the Rubik’s Cube

There are several methods in solving the Rubik’s cube with different difficulty levels such as beginners, speedcubers, and even bizarre levels such as solving the cube while being blindfolded, while underwater, or while being chased by a giraffe.

In this Chapter, we will learn the beginner’s methods in solving the Rubik’s give, in which time doesn’t matter. It may take you hours, weeks or even months to solve your cube, but the important thing is, you will solve a Rubik’s cube, which can bring you a great sense of accomplishment.

More often than not, beginners can intuitively solve one face of the cube, and after that they can be lucky enough to solve two faces. But you may need help if you want to solve all the six faces. So this Chapter will show you one of the best algorithms to solve a Rubik’s cube.

The solution described in this Chapter divides the Rubik’s cube by layers, and it will be fairly easy to solve each layer when you follow the algorithms. I have tried to be very descriptive in each step to help you in every step of the way.

I trust that you have read the previous chapters, especially on the notation, so you can be guided with the algorithms.

Just a review: F refers to front, R for Right, U for Up, B for Back, L for Left and D for Down. Take note that counterclockwise movements are signified by a degree, hence R’ (pronounced as R prime).

Here’s an easy step-by-step algorithm for solving the Rubik’s cube .

Step 1 - Form the White Cross

Let us start by solving the white face. First, you have to form a white cross. Remember, to focus on the color of the center parts. This step is fairly easy, as you can easily form a white cross even without a tutorial. You just need to move the white parts to their places without mixing up the white pieces that is already there.

Have a go and try solving the white face without even referring to the algorithms. This will make you really proud of yourself once you accomplish this on your own. This step is actually not that difficult, because there is no need to mind other faces that are not yet solved.

You could identify where a part comes based on the color of the center pieces that are not interchanging places. Take note that each edge should also fit to the piece in the center.

The key here is to move the first edge to the correct spot properly oriented, then the second part so you will not jumble the parts that you have already placed correctly. Once you properly placed the fourth piece, you should concentrate not to mix up the pieces that you have already solved. Hence, it gets more difficult in every step.

The most challenging situation is if you have already solved all the three ages but the last piece has been placed incorrectly. Below is an example:

You can solve this case by following this algorithm: F R' D' R F' F '

Another challenging case is if the last white piece is placed on the front face, but it can be difficult just to rotate it to the top because of wrong orientation. See image below:

You can solve this case by following this algorithm: F' R' D' R F' F'

There are also cases that the last piece in the second layer will be oriented incorrectly such as the image below:

You can solve the above case by following this algorithm: R' D' R F' F'

Step 2: Solve the Whole White Face

Solving the cube’s first layer is fairly easy. Just as with the previous step, you could know this technique by heart after you spend some time with a Rubik’s cube .

You don’t need to remember complicated algorithms to solve the whole white face of the cube. All you have to follow is some short algorithms, which requires no memorization at all.

There are three common cases encountered by people who are on the second step.

The first step is when the lower right white piece is oriented on the right face. The challenge is to move it properly without messing up the pieces that you have already solved. Refer to the image below.

You can easily solve this by following this short algorithm:

The second case is when the lower right white piece is oriented on the Front face. Again, the challenge is to move this piece on the right face, without messing up the pieces that you have already solved.

You can easily solve this case by following this short algorithm:

The third case is the most difficult one. This refers to the situation where the white corner piece is located on the white layer, but it is on the wrong face or badly oriented. You may need to take this out to the layer bottom and perform the past algorithms. Look at the image below:

You can solve this complicated case by following this algorithm: R' D2 R D R' D' R

The R’ D’ R D Algorithm

The algorithm R’ D’ R D is a common algorithm used in solving the white layer, and it works every time. The only disadvantage is that it is a bit longer compared to the optimal algorithms.

For example, you want to move the white corner piece into the F-R-U position. The key is to move the right corner piece under this area, rotate the right piece you want to the F-R-D spot and perform the R’ D’ R D permutation until you get the right piece in the white face.

According to the initial position, you may need to follow this algorithm:

If the white piece is on the R face such as below:

You can solve this by following this simple algorithm:

But if the white piece is facing down such as this:

You need to perform the algorithm three times. Hence, you should do:

And if the white piece is facing forward such as this:

You need to perform the algorithm five times. Hence, you should perform this:

Step 3 - Solve the Second Layer

You can easily solve the first two steps with your natural intuition without any algorithm guide. However, in this step, when you solve the second layer, most people would find it difficult and even give up as there are too many movements to predict just to solve this layer.

You need to remember two main algorithms so you can move the edge pieces from the yellow face to the second layer. The challenge is not to mix up the white face that you have already completed. Remember that in our examples, we are holding the Rubik’s cube with the white center in the U position. Now, you have to turn it upside down, so you can get a better view of the puzzle.

The Left and Right Algorithms

You have to remember two basic algorithms in this step: The Left and Right Algorithms.

Below is a case that can be solved by a Left Algorithm:

The correct permutation for this is:

Below is a case that can be solved by the Right Algorithm:

You can solve this using the following permutation:

Bad Orientation

If the edge piece is located on the right spot but it is not located in the upper layer or badly oriented, you have to fix this in two stages: 1 - move it on the upper part of the cube.

You can solve a badly oriented white piece by following this permutation:

Step 4 - Form a Rubik’s Cube on the Upper Part of the Cube

At this point, you should already have solved two layers of the cube’s bottom. The remaining face to solve is the yellow one. In the fourth step of the beginner’s method, you need to create a yellow cross on the upper face of the Rubik’s cube. In this stage, it really is not important if the edge pieces are not matching the color of the pieces on the side center. You can easily switch them up in Step 5.

In this step, you have to remember this short algorithm:

Aside from the already solve faces, there are also two common patterns in this stage: A line or an L shape as shown in the image below:

If you encounter L-shape yellow pieces, you have to perform the algorithm F R U R’ U’ F’ two times, which will create the line in stage 1.

Follow the algorithms below to see for yourself how you can change the upper layer of the cube. Remember that the cube’s orientation is crucial. Hence, the L shape must be changed as shown in the images, and the line should be horizontal.

Take a look at the image below:

You can solve this case and form a yellow cross on the top of the cube by following this algorithm:

There is another situation, which is quite similar to the one described above. This is in fact a shortcut and will allow you to create an L shape to finish the yellow cross:

Take a look at the image below:

This case can be solved by following this algorithm:

Step 5 - Exchange the Two Yellow Edges in the Upper Layer

In Step 4, you have formed a yellow cross on the upper face of the cube. However, the yellow edges are often not all matching to the colors of the pieces in the side center. In this Step, you need to reposition the pieces to solve the cube.

In this Step, you need to remember this algorithm: R U R’ U R U U R’ U

Once you get into this case, the first thing that you have to do is to move around the upper layer to find the two edges that you have to swap. There are instances that you may not find the two pieces to exchange next to each other, because you have to exchange the two pieces that are facing one another. In this situation, you have to perform the algorithm two times.

Hence, you should follow this algorithm:

Look at the image below:

This case can be solved by performing this basic algorithm described in this stage:

Meanwhile, the image below can be solved by performing the algorithm two times:

Hence, you should perform this algorithm:

Step 6 - Place the Yellow Corners

You are now a few steps away from finally solving your Rubik’s cube. Now, you just need to solve the yellow corners. At this point, you need to place these pieces on the right spot. It is not important if the orientation is not correct. You can take care of the orientation in Step 7.

You can use the following algorithm to cycle around the three corners in the top layer counter-clockwise to swap the specified pieces:

You have to look for a yellow corner piece that is already placed in the correct spot. Then, you should put this one in the F-R-U spot of the case and then perform the algorithm U R U’ L’ U R’ U’ L

If you are not successful in getting the pieces on their correct spot, you should perform the algorithm again.

There are also cases that you can’t find a piece in the right placement. In this situation, you should use the same permutation to bring another piece to the right spot.

Because of parity, yellow pieces can be on a right position in only three cases. There will be no yellow corner cubelets in the right position, the four pieces are on the right position, or there is only one correct piece.

Step 7 - Solve the Rubik’s Cube

In this final stage, you should have oriented all the pieces on their right spot. All you have to do is to orient the last layer corners that by now should be placed on the upper face of the cube.

Take note that you should hold your cube in a manner that the top piece that you like to orient is currently on the F-R-U corner. The next move is to perform the algorithm R’ D’ R D two times or in some cases four times, until you properly orient that particular piece.

At this point, you might get confused as the cube seems messed up. It’s fine. It will all go back to their right places once you orient all the pieces.

Then, after one or several U’ movements, you should turn another yellow corner cubelet that you like to orient on the F-R-U corner then perform the algorithm R’ D’ R D again .

Just keep on perform the R’ D’ R D algorithm until you solve the cube.

Example Solutions

Below are some cases with their solution algorithms. You may find these permutations complicated, but these are actually series of R’ D’ R D algorithm. Hence, you must memorize this basic algorithm as well as the method on how you can perform it.

Now that you have solved your own Rubik’s cube, do some practice and once you gain mastery, you can try advanced methods that we will discuss in the next Chapter.

Chapter 5 - Advanced Methods: The Fridrich Method

The popularity of the Rubik’s cube among people who are interested in solving puzzles led to different advanced methods. So, the once seemingly impossible-to-solve puzzle can now be solved in as fast as four seconds.

If you have practiced a lot using the Beginner’s methods found in the previous Chapter, the next step is to level up your game. You can start improving your cubing speed by purchasing a high-quality and soft-to-move Rubik’s cube with great tensioning and great corner cuts so the parts will not pop.

You should also learn remarkable finger tricks that you can find in the succeeding Chapter so you can amaze your family and friends in your speedcubing skills. You can also invest more in this worthwhile hobby by buying a Rubik’s Cube digital timer so you can monitor your progress.

In discussing the advanced methods in solving the Rubik’s Cube, it is crucial to mention the Petrus Method and the Fridrich System (also known as CFOP), which are used by many speedcubers nowadays.

As mentioned in Chapter 1, the Fridrich Method was introduced by Jessica Fridrich in 1982 World Cube Championship. This method divides the puzzles into layers and you have to complete the cube by every layer by performing a set of permutations in every step without mixing up the pieces that you have already solved .

The steps in the Fridrich method are as follows:

1. Cross

2. F2L

3. OLL

4. PLL

Using the Fridrich method requires several algorithms, which are crucial so you can understand the logic between every permutation. Be sure to practice a lot and in time you can easily perform the movements without even thinking about the algorithms in order.

Tips In Solving the Rubik’s Cube Using the Fridrich Method

Before trying the Fridrich Method, it is important that you have already mastered the basic notation as well as the Beginner’s Method. If you are not yet adept in these areas, you should go back and practice some more.

It is entirely on your own accord which face you like to begin with. But usually, you can start with the white face. Take note that the goal in this step is to create a cross at the bottom of the Rubik’s cube in a manner that the sides of the white edges will match the center lateral cubelets.

Master speedcubers can easily project the steps when they preview the cube and they are well-practiced in planning the movements. In most instances, you may need to execute six movements to finish the step, and rarely is the need to execute more than eight steps .

A key trick is to hold the Rubik’s cube in your hand in a manner that the white center is facing down. This will greatly improve your speed. As you practice, you will eventually don’t need to see the white cross as you will develop the ability to somehow see what is actually doing down in the bottom based on the color patterns in the cube, your movements and what you see in the upper part.

Through this, there is no need to turn around the Rubik’s cube, which will save you precious seconds. Some cubers choose to solve the cross on the left face. However, you can also choose the bottom so you can have a great viewpoint and you can perform more finger tricks, which you will learn later on. There is a downside, however, as there is no way to notice if you have already mixed up the cross that you are forming

Step By Step Guide To The Fridrich Method

Step 1 - Form the White Cross

You can solve this step on the upper part and then turn the cube upside down once you are done. However, learning through the following method is recommended if you like to hit lower than 20 seconds. As a heads up, the steps in the second stage can greatly help you to understand the movement of the cube even though it is turned upside down.

You will gradually learn the benefits of this perspective, and you will easily realize that this is the fastest way to complete this stage. And if you are really serious in speedcubing, you have to develop the skill of examining the cube and planning all the crucial movements and perform them even without getting a whole view of the cube. This is in fact a great preparation for blindfolded speedcubing.

Take a look at the examples below as well as the algorithms used to solve them:

This case is quite easy. All you need to do is to move the edge piece to its target position. The proper algorithm is F2.

This one shows how you can reorient a piece using the algorithm U’ R’ F R

This one shows how you can place two cubelets using a single step. The permutation is F R2 D2

This last example is known as the Super Flip, or also known as the most messed up Rubik’s Cube. The six-step algorithm for this is R F L B R D.

Step 1 in forming the white cross is really an intuitive phase of the solution sequence. It can be very exhaustive to discuss every possible case because at this point, there are so many possible situations.

You can easily get this far even without the guide algorithms. However, you should practice creating the white cross to increase your efficiency. Remember to do this careful and without stopping. Make certain that the cross is upside down. Once you are successful in this stage, you can proceed to the next step.

Step 2 - Solve the First Two Layers (F2L)

The first two layers, also known as F2L, can be simultaneously completed instead of individually, which can save you a lot of time. In the Step 2 of the Fridrich method, you can complete the four white cubelet corners as well as the central layer edges adjacent to them. You can intuitively solve the 41 possible situations in this step, but you can save more time and your movements will be more accurate by working on a set of algorithms.

Remember, you can be more efficient if you don’t turn around your cube while you are solving it. You should also try to familiarize or even better memorize the algorithms so you could still complete the cube even if you are blind folded.

Take note that in the Beginner’s Method described in the previous chapter, the white pieces as well as the edges of the second layer were divided into two steps. But in this step in the Fridrich method, you have to pair them in the upper layer, then move them into where they fit in.

The basic case below shows a good situation where the red-blue cubelet is on the right spot while you have to complete the white corner. If the red-blue corner is on another part, then you have to perform permutations to move it into the back-upper position. Notice that the algorithm used in this case is R U R’

In the next case, the white piece faces upwards. First, you have to put the corner as well as the edge pieces, and place them on the right position. The correct permutation is R U2 R’ - U R U’ R’

There is no way that you could affect the completed blocks. In the next case, you can do the algorithm U’ R U2 R’ instead of performing the algorithm F U2 F. This is crucial so you will not mix up the already blue-orange pieces.

You can save precious seconds by performing double turns. Just take note that the letter u in the permutations signifies that you have to rotate two upper layer clockwise. We have discussed this in Chapter 2.

Using your intuition, you can solve the case below by performing these steps: move the two pieces to the upper layer, combine them to fill in on their right spot. You can do this by using the algorithm R U2 R’ - F U2 F - U’ R U R’

But you can save more time by following the algorithm R U R’ U’ - R U R’ U’ - R U R’ U’ mainly because this is the same series of sequence that you are just doing several times.

Below are more example cases including their algorithms for solutions.

Step 3 - OLL - Orient the Last Layer

In using the Fridrich method to solve the Rubik’s cube, after you solve the F2L step, you have to orient the last layer or OLL, so the top face of the cube will be all yellow.

It is not actually important if the colors on the sides are not matching, as you can work on them later in the PLL step. But this step may take a lot of work as you have to learn all the 57 permutations below so you can perform this step easily.

Step 4 - PLL - Permutate the Last Laye r

The last step of the Fridrich Method is known as the PLL or Permutate the Last Layer. Once you perform the algorithms for the PLL, you can solve the Rubik’s cube.

Remember, the notation X and Y are entire cube notations and if you see a small u, you have to perform a double layer turn.  You have to rotate the upper layer so you can align as much as pieces you can make, then perform one of the 21 permutations below.

21 Permutations of the PLL Step

1. A 1:x [(R’ U R’) D2] - [(R U’ R’) D2] R2

2. A 2:x’ [(R U’ R) D2] -  [(R’ U R) D2] R2

3. U 1:R2 U [R U R’ U’]  - (R’ U’) (R’ U R’)

4. U 2:[R U’] [R U] -  [R U] [R U’] R’ U’ R2

5. H:M2 U M2 U2  - M2 U M2

6. T:[R U R’ U’] [R’ F]  - [R2 U’ R’] U’ [R U R’ F’]

7. J 1:[R’ U L’]  - [U2 R U’ R’ U2] -  [R L U’]

8. J 2:[R U R’ F’]  - {[R U R’ U’]  - [R’ F]  - [R2 U’ R’] U’}

9. R 1:[L U2′ L’ U2′] -  [L F’]  -  [L’ U’ L U]  - [L F] L2′ U

10. R 2:[R’ U2 R U2]  - [R’ F]  - [R U R’ U’]  - [R’ F’] R2 U’

11. V:[R’ U R’ d’]  - [R’ F’]  - [R2 U’ R’ U]  - [R’ F R F]

12. G 1:R2 u  - R’ U R’ U’ R u’  - R2  - [y’ R’ U R]

13. G 2:[R’ U’ R]  - y R2 u  - R’ U R U’ R u’  - R2

14. G 3:R2 u’  - R U’ R U R’ u R2  -  [y R U’ R’]

15. G 4:[R U R’] y’ -  R2 u’  - R U’ R’ U R’ u R2

16. F:[R’ U2 R’ d’]  - [R’ F’] -  [R2 U’ R’ U]  -  [R’ F R U’ F]

17. Z:M2 U M2 U  - M’ U2 M2 U2 M’ U2

18. Y:F R U’ R’ U’  - [R U R’ F’] -  {[R U R’ U’] -  [R’ F R F’]}

19. N 1:{(L U’ R)  - U2 (L’ U R’)}  - {(L U’ R)  - U2 (L’ U R’)} U

20. N 2:{(R’ U L’)  - U2 (R U’ L)}  - {(R’ U L’)  - U2 (R U’ L)} U’

21. E:X (R U’ R’)  - D (R U R’)  - u2 (R’ U R)  - D (R’ U’ R)

Now, just to recap, the following are the steps in solving the Rubik’s cube using the Fridrich Method:

Step 1 - Form the White Cross

Step 2 - Form the First Two Layers (F2L)

Step 3 - Orient the Last Layer (OLL)

Step 4 - Permutate the Last Layer (PLL)

In the next Chapter, you will learn the Roux Method, which is the primary competitor of the Fridrich Method in solving Rubik’s cube.

Chapter 6 - Advanced Methods: The Roux Method

The Roux Method is a primary competitor of the Fridrich Method. The fastest speedcuber is presently ranks 15th in World rankings for a single solve, even though when it was set, it placed 3rd in the World.

The fastest Roux solver also used this method to win numerous speedcubing competitions in the UK and Europe, which only signifies that this method is really a deserving competitor of the Fridrich Method.

While the Fridrich method mainly depends on a look-ahead strategy with a minimal concentration on solving intuitively (the F2L phase, even though you can solve it intuitively, comes with only specific number of possible cases for every pair, which you will eventually memorize, while the cross has several possibilities), two primary stages in the Roux method involves building blocks.

And in the Roux Method, it is crucial to accurately move the cube fast. In the Fridrich Method, the last two stages have permutations, which could be normally performed. In the Roux Method, the last step uses the M slice and U movements only that could lead to more accurate movements and prevents pops.

The Fridrich Methods is more structured though so it is easier to understand for beginners compared to the Roux Method. This method also utilizes advanced cubing techniques like block building. Hence, many speedcubers who are using Roux method switched to this more advanced method after mastering the Fridrich Method.

Here is the step-by-step process of solving the Rubik’s cube using the roux method:

Step 1 - Create a 1 x 2 x 3 block

In general, this step must take between nine to 12 movements for actual speedsolvers. And there is actually a specific approach to create the first 1 x 2 x 3 block. However, it takes time to master this step. But once you master this step, you can save a lot of time. Again, practice is the key.

One trick performed by speedcubers using the Roux Method to create the 1 x 2 x 3 block is to first create a 1 x 2 x 2 and then expand it into a 1 x 2 x 3. More often than not, this is the normal approach by speedsolvers. You need to observe where the pieces are moving and understand how they move to specific spots, so you can easily build the 1 x 2 x 3  block.

Algorithms for Solving the First Bloc k

Unlike with the Fridrich Method, there are no specific algorithms that you can follow in solving the first block. This first step is more about placing the right set of pieces than memorizing a certain sequence of algorithms.

Rather, you need to follow a specific process. You have to place together two corner-edge pairs and one centre-edge pair before placing them together into a 1 x 2 x 3 block. Certainly, you need a lot of practice before you can do this at a fast speed. Try to do this by yourself first, and if you want more detailed instruction, then you should read below.

Take note that it is not important to follow the list in order, or solve the same block each time you solve a cube. Remember, these are only recommendations for those who are not into speedcubing and more about getting things done.

1. Centre-Edge Pair

Select a centre piece and take note of the color. You must identify this color as the left color. Look for an edge, which shares this color. Move the edge so that the left color stickers are near each other. Take hold of the cube so the pair will be on the left face and the edge will be on the down face (not the back, front, or upper face). This second edge color will now be regarded as our down color.

2. Corner-Edge First Pair

At this point, you should not try rotating the cube. You have to use your centre-edge pair as your anchor for the remaining steps. Regardless of what step you are into, that edge must always be in the left-down face. Once you solve the cube, the down face will be the down color, and the left face will be the left color.

Next, search for a corner with both Down and Left colors. Take note that there should be two. You can use the movements F, B, R, U, M so you can form the upper face. Search for the third color on this corner. This will be either your back or front color. After several movements, you can easily notice it, but for now you can just identify it out in a bit. Before looking for this edge, search at the corner and try to check at the color on the upper face. It is a bad face if the color is the Down color. Move the corner to the upper-right side using U turns, then perform R’ or R movement (whichever could lead you to the corner remaining on the upper face. This will maneuver the corner so that the Down color is no longer facing the ceiling.

Now search for the edge, which has the Left color and that 3rd color. Again, you can use the algorithms F - B - U - R - M. Slide this edge into the U face. If you think that maneuvering the edge into the U face will mess up the corner, you can slide the corner around the U face by performing one or two U turns. Next, move the face turns to move the edge to the U face.

At this point, the corner and the edge must be on the U face. You just need to pair them up. Check the colors on the corner and the edge and make sure that they are on the upper face. You need them to be the same, or else you should perform U movements until the edge turns to the middle slice, which is the vertical slice in the center. Perform M or M’ so that the edge will stay in the upper face and the color will now match the color of the corner.

When the corner and the edge have the same color on the upper face, perform a U movement until the edge reaches the Middle layer. Then, perform an M’ to place the edge to the down layer. Perform U movements until the corner is in the spot near where the edge was and the Down color is on either right or left face.

Then, perform the opposite M’ to bring back the edge. It must be touching the corner and both stickers on the edge must match the non-Down corner of the colors.

Lastly, you can slide this pair adjacent to the center-edge pair to form a 2 x 2 block. Check the colors of the pair on the U face. If these are on the Left, then perform the U turns until the pair is now pointing at the L face.

Take note that the corner and the edge must be on either the Back or Front faces. You can either perform a B or F’ turn to slid this pair down.

If the colors of the pair on the U face are not on the Left, then you should perform a U movement until the whole pair is placed on the R face. Next, perform an L’ movement so that the centre-edge pair color points on the opposing direction of the corner-edge pair. You can perform a U2 to match the pairs up and perform the opposite L’ to lower than the 2 x 2 block back down to the bottom block .

C. Corner-Edge 2nd Pair

I know, the previous step seems very complicated. But once you master it, you can solve the 2nd pair precisely as the first pair. The main difference is that regardless of the face, the first pair will end up on the Back or Front, which will not be possible to solve the second. Hence, never perform the F turns if this would break up your block as well as with B movements.

Step 2 - Create the Second Block

This step is somehow similar to the first, as you also need to create a 1 x 2 x 3 block of solved pieces on the cube’s other side. Focus on the face that is opposing the Left face, and then begin building the block.  Just ignore any piece that will not be part of this block.

Algorithms for the Second Block

There are still no algorithms in the second block. This step is also about placing things on their right places and not memorizing a series of particular permutations. Rather, you have to follow a specific process. You need to place together two corner-edge pairs and one center-edge pair, and then slide them into one large 1 x 2 x 3 block .

This will take practice, and I recommend trying this first intuitively. If you need more details, you can refer to the instructions below. Again, you must take note that you don’t have to follow the order below nor complete the same block each time you solve a cube. These are just instructions for those who are not into speedcubing and more about getting things done.

1. Centre-Edge Pair

This step is similar to the center-edge pair step described in the previous block-building step. The only difference is that for this whole block, the only turn that you can use is the following: U - R - M - r. This will make the process a bit more difficult, but you can make it with practice.

Focus on the color of the R face. You should refer this color as the Right color. Look for the edge with either the Down or Right colors. Move the edge so that the R color will be adjacent. If you find this difficult, you can try moving the edge on the U face, turning it by performing an M movement so that the Down color will also face up, then move it over the right-down slot.

2. Corner-Edge First Pair

You should also not rotate the Rubik’s cube at this point. Find the corner with both the Down and Right colors. Take note that there must be two. By using the turns U - R - M , you can slide the corner so that it will be on the U face. Find the third color on this corner. This will end up as either the back or front color .

Before looking for this edge, find the corner and check at the colors on the U face. It is a bad corner if the color is Down color. Move the corner to the upper-right side using U turns, then perform an R’ or R turn, whichever could lead the corner to remain in the U face. This will move the corner so that the Down color will no longer face the ceiling.

Now, search for one edge, which has the Right color and the 3rd color. Again, you can only use U - R - M turns then slide this edge into the U face. If you think that turning the edge into the U face will turn the corner off, you can slide the corner around the U face by performing one or two U turns. Then you can perform the face turns to maneuver the edge to the U face.

At this point, the corner and edge must be on the U face. The next thing to do is to match them up. Make sure that the colors of the edge and the corner are on the U face as they should be the same. If this is not the case, you should perform U turns until the edge reaches the Middle slice, which refers to the vertical slice in the center. Perform a M’ or M movement so the edge will remain in the U face and the sticker will match the corner color.

When the corner and edge have the same colors on the U face, you can perform a U turn until the edge is placed on the Middle slice. Next, perform an M’ turn to lower the edge to the bottom layer. Perform U turns until the corner is near where the edge was and the Down color is either placed on the Right or Left Face .

You can now perform the opposing M’ movement to bring the edge. It should touch the corner and both colors on the edge must match up with the non-Down colors.

Lastly, you can slot this pair in next to the center-edge pair to form a 2 by 2 block. Check if the colors of the pair are on the U face. If these are on the Right color, then perform U movements until the pair is pointing at the Right face.

Both the corner and edge must be on either Back or Front faces. You can perform the algorithms r - U - r’ or r’ U’ r to slide the pair. Remember, this is quite different compared to the first block as you can’t still use B or F turns.

If the colors of the pair on the U face are not the R color, then you can perform U movements until the whole pair is on the R face. You can now do the R’ turn so that the center-edge pair of the Down color points in the opposite direction of the corner-edge pair. You can perform a U2 to match up the pairs and perform the opposite R’ to drop down the 2 by 2 block back down on the bottom.

3. Corner-Edge Second Pair

I understand this step might seem overwhelming. However, after some practice, you can easily solve the second pair precisely the same as the first pair. Take note that you should not use any B or F turns .

You are now halfway through solving the Rubik’s cube using the Roux method. This part can really take a long time, as this is often the most difficult phase of the entire method. Practice is key.

Step 3 - Work on the Layer Corners

After working on the blocks, the next thing that you have to do is to work on the last four corners then the last six edges. In this step, you will encounter several algorithms, as there are numerous possible moves. However, with the Roux method, you just need to remember two algorithms to complete the four U corners. Take note that you should have not rotated the Rubik’s cube since you have set that first edge for the first block.

In this step, you have to follow a 2-step process. First, you have to orient the corners, which means you have to rotate them the proper way. Second, permute the corners or place them in the right spot. There are several algorithms that you can use to accomplish these steps at the same time, but we can stick with one or both.

Orientation

The algorithm that you have to remember to successfully accomplish this step is the following: F - R - U - R’ - U’ F’.

As it turns out, after placing these several pieces in the correct spot, there are only at least seven probably orientation for your corners. There are also at least seven unique ways for your corners to be turned. All you have to do is use U turns until the patterns match the right form, and then perform the algorithm.

Again, do not rotate the cube. You can at least turn the U face. If these corners are not oriented after performing the algorithm, you can just repeat the process described above until you get the proper orientation.

To determine the right pattern to use on your cube, you can just focus on the positions of the corner colors, which match the top color. If you are not aware of the Top color, you could determine this by checking which color is shared among the four upper corners. Top colors are easy to notice as they are the only colors that are on all fours.

Permutation

Take note of this permutation: R’ U L’ U2 R U’ R’ U2 L

This permutation is the specific algorithm that you have to memorize so you can complete the corners. As it turns out, after placing the corners in the proper orientation, you can only have three probable permutations that they can be placed.

The algorithm above will swap the location of the two Back corners and also leave alone the Front corners.

Again, do not try to rotate the Rubik’s cube.

You can only turn the U face to work on messed up corners into the Back face for switching. If you notice that all the corners are messed up, then you can just do the algorithm above two times. If the corners are on their proper places, then you don’t have to do this step and so you can also skip the algorithm above.

Step 4 - Work on the Last Edges

Now we just have to complete the last six edges, and you can solve the cubes. At this point, you just need to perform several algorithms, and they are very easy. Take note that you must not have rotated your cube ever since you have set that first worked on the first block’s edges.

Again, there are two processes involved in this stage. First, you have to orient the edges, which means you have to flip the edges properly, then permute the edges.

Orientation

Take note of this algorithm: M’ U’ M’

At this point, you just need to use two kinds of moves: M, U turns, which will allow you to complete the remaining six edges.

Remember, this is a 2-part process. First, you need to perform an M turn in order so you can get the bottom or top centers on the actual bottom or top faces. After this, make certain that any M turn you do should be followed up quickly by another so that the Bottom and Top centers will stand on the Top and Bottom face s

Second, you should perform the permutations: M’ U2 M’ until all the colors on the Bottom and Top faces can match either the Bottom or Top centers. It is really not important if there is a bottom color on the Top face. You are all good as long as there is only Bottom and Top colors on the Bottom and Top faces.

You might have realized at this point that there are only very few patterns for the edges that are not still oriented to be in. However, as there is no risk of messing up your completed faces (remember, you should only use M and U turns), you can do this as a bonus exercise. Try to see how many of the sequences you could spot on and try to determine the fastest way to change between these sequences to obtain all the oriented edges.

Permutation

Take note of the following permutations:

U2 M’ U2

M’ U2 M’

There are two steps in this last phase, and you just need to memorize the permutations above.

First, you have to complete the Right and Left edges. One tip I can give you is that you should perform the second algorithm written above and try what you can get. Check if you can use this to get both the Right and Left edges into the D face simultaneously. And another tip: Unless you are using the second algorithm, you must not use a single M’ turn. If you must switch things from the D face to the T face, you can do the M2 instead .

Time for the last step: switch the last four edges on the Middle layer. Take note that the only permutation that you have to remember is the first one, with probably several M’ movements if necessary.

After performing these steps, you should have solved the cube using the Roux Method.

Chapter 7 - Advanced Methods: The ZZ Method

The ZZ Method is another advanced method in solving the Rubik’s cube. It attempts to shorten the Fridrich move count and often used by speed cubers while harnessing their finger tricks. You will learn more about finger tricks in the next Chapter.

This method involves the orientation of edges as part of the first stage to eliminate the need for B, F, or D turns. Meanwhile, the last layer could then be solved in one look as all the edges can be oriented. However, there are at least 493 algorithms, which you have to learn to do that, which is something that only master speedcubers achieved.

As a beginner in this method, you can use the basic OLL and PLL that you have learned in the Fridrich method, which means that you only have to learn at least seven algorithms for OLL rather than 57. But altogether, the ZZ method is easier to execute than the Fridrich method and it saves a lot of precious seconds.

Here are the steps in solving the Rubik’s Cube using the ZZ method:

Step 1 - Start the EO Line

This step requires orientation of all edges on the cube, hence EO while at the same time placing the DB and DF edges. This will reduce the cube to the L - R - U group, so you can save the rest of the cube by turning only the R - L - U faces.

This step takes about six to seven moves, and this is usually regarded as the most difficult phase of the ZZ method. Hence, this is usually divided into two sub-steps:

1. Orienting the Edge

2. Placement of line Edges

How to Detect Edge Orientation

You can say that the edge has been oriented if you can solve it using only L - R - U - D face turns. If you can’t solve the edge through these face turns, then you have a bad or misoriented edge.

The following indicators signify bad edge. Take note that in stating the colors here, we assume that the cube is held with yellow or white on the upper and blue or green on the front.

1. Check the Upper and Down Faces. If you notice:

1. L / R color (red or orange) it is misoriented

2. F / B color ( blue or green ) - you have to look round the edge sides. It is misoriented if the side is yellow or white.

2. Next, check the Front and Back Faces of the Middle Slice. Follow the same rules indicated above.

1. L / R color ( red or orange ) it is misoriented

2. F / B color (blue or green) - you have to look round the edge sides. It is misoriented if the side is yellow or white.

In this method, you have to look at every edge in the Upper layer to identify any bad edge. Then, you should check the Down layer, taking note of the same rules for every edge. Lastly, you have to look at the four remaining middle slice edges. Be sure to check first the Back and Front faces first, taking note of the same rules. You can keep track of the misoriented edges by putting on a finger on every piece you have discovered.

Strategy for Edge Orientation

A quarter turn of the B or F face will cause the oriented edge to turn to bad edge. On the other hand, the same move will turn bad edge to become oriented. Through the F or B quarter turns, you can easily orient at least four edges simultaneously, which is actually the key of this strategy.

Take note that B2 and F2 turns don’t have any effect on the orientation of the edge. The following rules describe the strategy for orienting edges according to the number of bad edges on the cube .

Zero bad edges - All edges are oriented. You are finished with this step

Two bad edges - Turn 1 to F / B and make an F / B quarter turn, switch newly-oriented edge with the second bad edge. Be sure not to mess up other F / B edges, and then perform the final F / B move.

Four bad edges - Move all the bad edges to F / B and perform F / B quarter turn

Six bad edges - Move three bad edges to F / B and perform F / B turn, four remain. Or perform steps as 4 + 2.

Eight bad edges - Perform steps by following 4 + 4.

10 bad edges - Perform steps by treating this case as 4 + 4 + 2. In this case, you have to keep track of two oriented edges, instead of the 10 bad edges. You can use the initial moves to move the two oriented edges out of the F / B, then perform the F and B faces. Lastly, you have to work on the remaining misoriented edges.

12 bad edges - This case is quite rare, but if ever you are fortunate enough to encounter this, you can perform an B and F turn, and you will reduce the bad edges to at least four, then perform the steps for oriented four bed edges .

The following rules can help you in making decision for working on misoriented edges:

1. It is often more efficient to select the Front and Back face with the most misoriented edges the cube currently has for the initial turn for F and B.

2. It is often less efficient if you place a misoriented edge into Front and Back, which eliminates another misoriented edge from Front and Back.

3. If your cube has six or more misoriented edges, it is more efficient to do away with the misoriented edges using both Front and Back faces instead of just one.

4. If your cube has six or more misoriented edges, it is more efficient if the placement of a misoriented edge into a Front or Back face will also place another edge into the opposing Front or Back face.

5. Performing Front or Back moves with only two opposing misoriented edges can be useless in the perspective of an edge orientation, but could be useful in line edge manipulation.

Step 2 - Solve the First Two Layers

This step will solve the first two layers by forming two 1 x 2 x 3 blocks on each side of the line that you have completed in the first step. Because you have already oriented all the edges, you can now solve this step by using R - U - L turns .

In order to complete this step, you have to hone your block building skills. Hence, it is helpful if you have already mastered the Roux method. Through block building, you can simplify this step into the smaller chunks that you can easily recognize and perform easily. Hence, you can do this step into four sub-steps:

1. LH 1 x  2 x 2 block

2. RH 1 x  2 x 2 block

3. LH 1 x 1 x  2 block

4. RH 1 x 1 x  2 block

Take note that the order of execution will greatly depend on the case. Generally, the 1 x 2 x 2 block must always come prior to connecting the 1 x 1 x 2 block, or else you have to perform more moves later on.

Building 1 x 2 x 2 Blocks

In order to form a 1 x 2 x 2 block, a D-layer corner and two edge pieces are required - a mid-layer edge and a D-face edge. Forming the block first involves building a 1 x 1 x 2 block with one edge and corner piece. The other edge will be placed into the proper edge piece. The other edge will then be placed into the proper mid-layer position and will then paired to the block through the U face turns.

You can first build the first 1 x 1 x 2 block with a D-face edge or a mid-layer edge. Because you are free to fully turn the R and U faces without the need to break up anything, these initial 1 x 1 x 2 blocks can be quite easy to make. Below are the four basic cases you will encounter while doing this step:

1. Corner and middle-edge in upper layer

2. Corner and middle-edge in right layer

3. Corner and D-edge in Right Layer

4. Corner and D-edge in Upper Layer

After building the first 1 x 1 x 2 block, you should connect it with the third edge in the middle layer. You can do this in either two ways depending if the block is being paired with a D-layer or middle-slice edge. Below are the two cases you will encounter:

1. Connection with middle-layer edge

2. Connection with Down face edge

Forming 1 x 1 x 2 Block s

You can usually form the 1 x 1 x 2 blocks is often performed by connecting them in the upper layer similar to the process in working with the first two layers following the Fridrich method. But in the ZZ method, you don’t have to rotate the cube, and you can only use L - R - turns. Below are the three basic cases that you will encounter in forming 1 x 1 x 2 blocks. I have also included the permutations that you can use.

R U R’  - R U R’ U R U R’ - R U’ R’ U’ R U R’

R U2 R’ U2 R U’ R’

R U2 R’ U R U’ R’ - R U’ R’ U2 R U R’ - L U R U R’ L’

If the target edge or corner pieces are not placed in the Upper Layer, then you have to bring them up before you can perform these permutations. If the needed pieces are already placed in the Upper layer, but not correctly connected, you have to separate them before you can solve.

Step 3 - Solve the Last Layer

Because you have already solved the edge orientation in the first step, and you have preserved its status in the second step, the edges in the last layer should be now properly oriented. This will open you a lot of options ranging from a basic 2- permutation two-look system or all the way up to a single-look system but you have to learn 493 algorithms altogether.

Let us learn the orientations in the last layer one by one .

OCLL and PLL

In this orientation, the corners of the last layer can be properly oriented in just a single step (OCLL), then will be permuted simultaneously in the final stage (PLL). This is quite similar to the OLL and PLL in the Fridrich Method. However, in the ZZ method, you have quite a few algorithms to memorize, because you have already oriented the last layer.

Take note that OCLL needs at least six algorithms and PLL needs at least 14 algorithms, so you have to perform 20 algorithms for these steps. In average, you may need to move between seven to 12 for PLL.

Meanwhile, by using the partial corner control during the insertion of the last block could do away with the Pi and H OCLL cases. This will allow fewer moves, so you can solve the cube at a faster rate.

COLL / EPLL

In this step, you have to orient and permute the corners of the last layer without messing up the orientation of the edge in a single step. Then, you have to permute the edges of the last layer in the final step. Speedcubers who use the ZZ method prefer this orientation because it has a lower move count compared to the OCLL/PLL and this is even regarded by some speedcubing masters to have cases that are easier to recognize .

Learning COLL/EPLL is also a useful platform before you can learn ZBLL or ZZLL. There are 40 cases in COLL, which can be solved by 25 algorithms. Meanwhile, EPLL is just four cases and only 3 algorithms if you try to mirror the U permutations, which is a total of 28 algorithms for the entire LL.

OCELL / CPLL

This will orient the corners of the last layer while performing the permutations of the edges for the last layer without messing up the orientation of the edges in one single step. In the final step, the corners of the last layers should also be permuted.

The primary benefit of using the OCELL / CPLL is that most of the permutations in the OCELL could be performed in 2-gen or turning only two sides. This provides a major advantage over the COLL if you are into one-handed speedcubing.

Similar to COLL, you have to recognize 40 cases in OCELL, which can be solved by 25 algorithms. Take note that CPLL is also just four cases, and most of them are A permutations. The rest of the cases are E permutations and H permutations .

ZBLL

ZBLL is usually regarded as the Holy Grail in speed cubing, because this step solves the last layer by orienting the corners and then permuting the edges and corners all in a single step.

This method involves 494 specific cases, and you need at least 177 algorithms assuming that you apply mirrors and inverses.

Chapter 8 - Finger Trick s

Aside from learning the algorithms, another crucial aspect that you should learn if you are into speedcubing are finger tricks. Sure you can solve the Rubik’s cube without the need for speed, but wouldn’t it be cool to show to your family and friends that you can solve the cube in mere seconds? Very few can do that and that is something that you can be proud of.

When you first develop interest in the Rubik’s cube, you probably have purchased a cheap one online or from a toy store. Usually, these cubes are not designed for speed cubing. Hence, moving a layer may need a whole-handed effort on the layer, and you may need to pull or push some cubes so you can place them at the right position.

Meanwhile, if you are to observe a master speedcuber in a competition, or in popular YouTube videos, you will see that these experts are maneuvering layers using only one finger to place the middle layer across or two fingers in a single movement to perform a double turn. Once you master finger tricks, you can easily use your pinky or ring fingers to execute D turns without the need to rotate the whole cube.

Mastering finger tricks can only be achieved with regular practice. As you familiarize yourself with the algorithms you have learned in this book, executing them while using finger tricks will embed in your muscle memory and help you to become a speedcuber .

The Proper Rubik’s Cube for Speedcubing

Basically, the type of Rubik’s cube that you use will significantly determine your capacity to solve the cube in mere seconds. By chance, you might have been solving the cube for quite some time, or you might have already tried speedcubing or has read speed cubing forums online. So, you will know that the original Rubik’s brand is not great for speed solving.

Seven Towns, the company who owns the Rubik’s brand, produces the cube that are not great for solving the puzzle with finger tricks. This is actually a marketing strategy as the cube is targeted to at least 95% of its market who is not into speed cubing. Hence, there is no actual need for a cube that can be instantly turned in a slight push of a finger.

That is why most speed cubers turn to other producers such as Chinese brands like MoYu and Shengshou. In spite of the fact that these brands are banned for sale in Europe as well as in the US for patent infringement, speedcubers don’t have any option but to buy them so they can practice finger tricks.

Common Finger Tricks on the Regular 3 X 3 Rubik’s Cube

This Chapter is actually quite brief, as we will just simply introduce the most common finger tricks used by speed cubers to maneuver every layer of the cube.

U Turns

To execute U turns, you should use your index finger of your both hands to slightly move the upper layer in the direction that you like (right finger for U and left finger for U’) while keeping all your remaining fingers off the upper layer.

R Turns

To execute R turns, you just need to hold the R face using your right hand, so that your thumb will be on the bottom edge and your first three fingers are placed on the three upper pieces. You should hold the cube using your left hand and then turn your right wrist to perform R2 or R turn. For executing R’ turns, you should hold the Rubik’s cube similarly except with the thumb on the F face and on the back should be your first three fingers.

L Turns

To execute L turns, you should do the same process for R turns, but inverted.

B Turns

There are no tricks to perform B turns, so many speed solvers just resort to turning the cube so that the Bottom layer will be Upper and just do the movements as if it is the U face. Fortunately, rare is the chance that you have to perform B turns for speedcubing, and you will only have to perform them if they are crucial in the algorithm that you have memorized.

D Turns

D Turns are actually the most difficult finger tricks to master. You have to hold the cube from the mid-slice and U layer using your first two fingers and your thumb. Your pinky and ring finger should hover behind the cube. Then, you should use your ring finger to move the layer in the needed direction. D turns are also quite rare for speed cubing algorithms compared to R - L - U turns.

There you go! It will definitely take some time to master these finger tricks. The ideal way to exercise is to try following a scramble as fast as you can so you could easily swap the positions.

Chapter 9 - God’s Number and Rubik’s Cube

God’s number is a term used by a group of researchers, referring to the maximum number of moves it takes to solve any position that a Rubik ’s cube can have. According to researchers, any position can be solved in a maximum of twenty moves using the most efficient algorithm. In this case any twist is considered a move. Note that this is the maximum number and there are many positions where it takes fewer moves to solve.

Most Rubik’s cube enthusiasts know that solving the cube’s puzzle means having to deal with 43,252,003,274,489,856,000 possible positions. While the Rubik’s cube is an extremely simple game, mathematicians have been fascinated by the huge number of possibilities that one can yield by simply twisting the faces of this extremely popular toy. Because of the number of possible scrambles to solve, the Rubik’s cube can turn into an excruciatingly difficult puzzle to solve.

However, it was discovered in 2010 that it is actually possible to solve the most daunting combination on a 3x3x3 cube with merely 20 moves.

Introducing the God’s Number

As you already know, playing the Cube will require you to use an algorithm, or a move sequence that you will use in order to solve the Cube’s puzzle. In order to get all the colors right in a single face, you might need to use an algorithm that will use moves to get the top face right, and then another sequence in order to get the middle, and so on. There are quite a number of algorithms possible that you can use, depending on the complexity of the Cube position that you are trying to solve. Most of the moves that a typical person can memorize will typically take more than 40 moves. Now, thanks to Google’s computers and the minds of the mathematicians that want to solve the mystery behind the Rubik’s Cube, the maximum number of steps to solve any scramble on the cube.

If you are solving the Rubik’s cube puzzle, you will want to perform the least amount of moves, or rather, the most efficient algorithm. For this reason, mathematicians spent a huge amount of time to find the shortest move sequence that can solve the cube. This sequencing is coined the God’s number because of the notion that if God is going to solve the puzzle Himself, he would do it in the most efficient manner.

While 20 moves to solve the Rubik’s cube seem to be low, it is even possible to solve the cube with a lower number of move sequences. This means that there is no possible position in the cube that should take you more than 20 moves to solve. When you think about it, you would be able to even find the optimal, or the least required number of steps to solve the cube, to be less than 20 moves!

A 2x2x2 cube (which has 3,674,160 possible positions) has a different God’s number. You will only need 11 moves if you are going to use the half turn metric, and 14 if you are using the quarter metric. There is no discovered God’s number for the 4x4x4 cube and higher .

How the God’s Number For Solving The Rubik’s Cube was Determined

How was this number determined? Basically the team of researchers created a computer algorithm that solved all 43,252,003,274,489,856,000 possible combinations of the cube. If you are going to use an ordinary computer to perform their computation, it would take you 35 years to complete it, despite having an algorithm and a program that allows them to solve 19.5 billion positions in under 20 seconds. For this reason, John Dethridge from Google made use of Google’s computers in order to speed up the pace of finding all possible solutions for the remaining positions.

Solving all of these combinations in a linear fashion would be an insurmountable task, so the team of researchers undertook a number of steps to make it more manageable.

Since it is impossible to solve all the possible positions on the cube, Tomas Rokicki and his team simplified the process by adapting the group theory. By using this theory, they are able to minimize the number of sets that they need to solve to 56 million, which is done by studying different symmetries of a Rubik’s cube.

For example, if you are going to turn a cube upside down, it has changed positions but it does not require a different solution in order to find the algorithm that solves it. This means that all equivalent positions are disregarded, and all that they needed to solve are the other configurations.

Here are the steps the researchers took:

- Partitioning - Each of the moves was classified into one of 2,217,093,120 sets. Each set was made up of 19,508,428,800. To further reduce the number of sets that had to be solved, they used Symmetry and Set Covering. Using these methods, they had to solve only 55,882,296 sets. Each problem could be solved rapidly by a regular PC.

- Symmetry refers to a position that can be solved the same way even if it is flipped upside down. All you have to do is solve one position, and then flip it around to solve the other. You don’t have to generate a new solution. Since there are only 24 ways that you can orient a Rubik’s cube spatially, and using a mirror add a factor of two to these, they were able to reduce the number of sets that had to be solved by a factor of 48 or 55,882,296.

- These remaining sets were distributed among the computers made available at Google to complete the required computations in a few weeks.

It should be noted that each solution they found was not the optimal one -- that is, the one that actually required the fewest moves. Twenty refers to the maximum number of moves required; there are actually many others that can be solved in as little as eighteen moves.

In fact, the lower bound of eighteen was established as early as 1981 by mathematician Morwen Thistlethwaite or just seven years after the cube was invented. The upper bound was 52, meaning that no matter what the position, it would take no more than 52 moves to solve. Less than ten years later, in 1990, Hans Kloosterman demonstrated that it would take no more than 42 moves.

Move forward to March 2008 and Thomas Rokicki had reduced the upper bound to 25 moves although the lower bound had increased to 20 moves. Michael Reid had established this lower bound in January 1995, and he had also determined the upper bound was 29 moves in the same month .

Rokicki was one of the team of researchers who had definitively established God’s Number. In collaboration with John Welborn, he reduced the upper bound to just 22 moves. Finally, along with three other researchers, he definitively proved that God’s Number was 20 in 2010.

To illustrate an algorithm that can solve the Rubic’s cube in twenty moves, we have provided an illustrative example.

Here is the starting position.

The first step is B’, meaning that you have to rotate the back face of the cube counterclockwise by 90-degrees.

The second step is D’, meaning that you will rotate the down face counterclockwise 90-degrees.

The third step is R, meaning that you have to turn the right face 90 degrees clockwise.

The fourth step is B’, meaning that you have to rotate the back face by 90-degrees counterclockwise.

The fifth step is D2, meaning that you have to rotate the down face by 180 degrees.

The sixth step is F’, meaning that the front face of the cube should be rotated 90-degrees counterclockwise.

The seventh step is L2, meaning that the left face has to be turned 180 degrees .

The eighth step is B2, meaning the back face has to be turned 180-degrees.

The ninth step is D, meaning the down face has to be turned 90-degrees clockwise.

The tenth step is L, meaning that the left face should be turned 90-degrees clockwise.

Step 11 is D2, meaning that the down face should be turned 180-degrees.

Step 12 is L2, meaning the left face should be rotated 180-degrees.

Step 13 is D, meaning the down face has to be turned 90-degrees clockwise.

Step 14 is R2, meaning the right face has to be rotated 180-degrees.

Step 15 is U, meaning the upper face has to be rotated 90-degrees clockwise.

Step 16 is R2, meaning the right face has to be rotated 180-degrees.

Step 17 is B2, meaning the back face has to be rotated 180-degrees.

Step 18 is L2, meaning the left face has to be rotated 180-degrees.

Step 19 is F2, meaning the front face has to be rotated 180-degrees.

Step 20, the last step, is U’, means the upper face has to be turned 90-degrees clockwise.

Fewest Move Count (FMC) Even t

So why does solving for the God’s number matter? If you are a Rubik’s cube enthusiast, you probably have heard about the Fewest Move Count or FMC. This is an event of the World Cube Association (WCA) that is dedicated to solvers that have spent time in figuring out the most efficient way to solve the Cube.

While the average spectator in the WCA is more concerned about speedcubing and seeing finger tricks in solving a Cube puzzle in less than 10 seconds, many are still interested in the mathematics behind the Rubik’s cube and finding out the shortest way to solve the puzzle. While the competition would only provide you 60 minutes to provide your solution to a given scramble, it would help that you practice solving random scramble by taking only 20 moves or less. When you attend the FMC, you need to take note that any kind of rotation that you make is already considered a move, but if you do a slice move or a wide turn, they may be considered as 2 or 3 moves already.

If you are going to join the FMC, your primary goal is to find a solution that will require you 20 moves or less, which means that you will want to get as close as possible to the God’s number, or take less moves to solve the puzzle. While some people will attempt to solve scrambles during the FWC by doing speedcubing, take note that you will need to also provide the algorithm that you used. Compared to most competitions that you can join, the FMC is only concerned about the solution that you provide and not the time that you took in order to solve the scramble.

The world record for FMC is 20 moves, which is held by Rami Sbahi and Tomoaki Okayama. While this record matched the God’s number, the solutions they have provided is not proven to be the optimal solution for the scrambles that they are provided during the competition .

Discerning the Toughest Positions

The search for God’s number started in 1981 when Mowen Thistlewaite claimed that it will only take him 52 moves is sufficient to solve any of the 43 quintillion possible scrambles in the Rubik’s Cube. Of course, this number was greatly reduced as more efficient moves are discovered.

For 15 years, experts know that there are positions that will need only 20 moves; however, the discovery of the God’s number proves that there is no possible position in the Cube that will require more than 20 steps.

But how many possible positions can be solvable by 20 moves? These positions, known as the Distance 20 positions because of the number of steps they are away from the original face where all the colors are the same in every face. Out of the 43 quintillion possible positions, there are only 490,000,000 positions that are actually solvable by the God’s number.

What’s the toughest position to solve, according to Rubik’s Cube experts? According to cube20.org, this is toughest position that their computers have found so far:

Images from: cube20.org

This is solved by following the algorithm F U' F2 D' B U R' F' L D' R' U' L U B' D2 R' F U2 D2. If you want to follow how this position is achieved, here is a step-by-step illustration of how this is done, starting from the following starting position:

1. F – Flip right face clockwise

2. U' – Flip top face counterclockwise

3. F2 – Flip right face clockwise twice

4. D' – Flip lower face counterclockwise

5. B – Flip back face clockwise

6. U – Flip top face clockwise

7. R' – Flip right face counterclockwise

8. F' – Flip right face counterclockwise

9. L  – Flip left face clockwise

10. D' – Flip lower face counterclockwise

11. R' – Flip right face counterclockwise

12. U' – Flip top face counterclockwise

13. L – Flip left face clockwise

14. U – Flip top face clockwise

15. B' – Flip back face counterclockwise

16. D2 – Flip lower face clockwise twice

17. R' – Flip right face counterclockwise

18. F – Flip right face clockwise

19. U2 – Flip top face clockwise twice

20. D2 – Flip lower face clockwise twice

Doing the Super Flip

The super flip is the first position or combination that has been proven to be solvable by 20 moves. This position was discovered by Michael Reid in 1995. For his solution, he used the algorithm R L U2 F U' D F2 R2 B2 L U2 F' B' U R2 D F2 U R2 U. He used this algorithm to solve the Superflip position, which looks like this:

Image from: https://ruwix.com/the-rubiks-cube/gods-number/

This position is known to most Cube enthusiasts to have all of its corners already solved, while all the edges are flipped in their current position. To get into this position, you will need to perform the following moves.

Original position:

1. R – Flip right face clockwise

2. L – Flip left face clockwise

3. U2 – Flip the top face clockwise twice

4. F – Flip front face clockwise

5. U’ – Flip top face counterclockwise

6. D – Flip lower face clockwise

7. F2 – Flip front face clockwise twice

8. R2 – Flip right face clockwise twice

9. B2 – Flip back face clockwise twice

10. L – Flip left face clockwise

11. U2 – flip upper face clockwise twice

12. F’ – Flip front face counterclockwise

13. B’ – Flip back face counterclockwise

14. U – Flip upper face clockwise

15. R2 – Flip right face clockwise twice

16. D – Flip lower face clockwise

17. F2 – Flip front face clockwise

18. U – Flip upper face clockwise

19. R2 – Flip right face clockwise twice

20. U – Flip upper face clockwise

The Quarter Turn Metri c

There are some people that might tell you that God’s number for solving the Cube is actually 26. This is also true, if you are following the quarter-turn metric, which means that a move is considered to be a single quarter turn on any face of the Cube.

When you look at solving the Cube in this manner, you will need to make more moves in order to get all the faces of the Cube right. It also means that the mathematics behind the algorithm that you need to use is going to be a lot different. Interestingly, there is only one position that requires a maximum of 26 quarter-turn moves.

If you are going to follow the quarter-turn metric and solve the hardest position that requires the most number of moves, you will get to a position that is similar to a Superflip, but has a four spot. This is the position that you will get to:

To do this, follow the algorithm U U F U U R- L F F U F- B- R L U U R U D- R L- D R- L- D D. Here is the step-by-step instruction on how this position is solved:

Original position:

1. U – Flip upper face clockwise

2. U – Flip upper face clockwise

3. F– Flip front face clockwise

4. U – Flip upper face clockwise

5. U– Flip upper face clockwise

6. R’– Flip right face counterclockwise

7. L– Flip left face clockwise

8. F– Flip front face clockwise

9. F– Flip front face clockwise

10. U – Flip upper face clockwise

11. F’– Flip front face counterclockwise

12. B’– Flip back face counterclockwise

13. R– Flip right face clockwise

14. L – Flip left face clockwise

15. U – Flip upper face clockwise

16. U – Flip upper face clockwise

17. R – Flip right face clockwise

18. U – Flip upper face clockwise

19. D’– Flip lower face counterclockwise

20. R– Flip right face clockwise

21. L’– Flip left face counterclockwise

22. D – Flip lower face clockwise

23. R’– Flip right face counterclockwise

24. L’– Flip left face counterclockwise

25. D – Flip lower face clockwise

26. D – Flip lower face clockwise

Software for Solving Using God’s Number

If you want to find a solution that has 20 moves for a given scramble, you might want to use the Cube Explorer program that was developed by the mathematician Herbert Kociemba. This program is developed to find a solution using only 20 steps. However, it is possible that it might not find a 20-move solution for all scrambles – if it doesn’t find a solution with 20 steps, it searches for a solution that has 24 steps, which is still close to the God’s number.

Conclusion

Thanks again for taking the time to buy this book!

You should now have a good understanding of the best algorithms and be able to solve the Rubik’s cube.

Solving the Rubik’s cube is among the many achievements you can truly be proud of - one that you can pass on to your kids and grandkids. Who knows, you can even become one of the world champions and set a world record!

You must understand that nothing is impossible. You can achieve great things as long as y ou work hard for it and you are armed with the right knowledge and experience.

Remember, completing this book will not turn you into a master speedcuber. You can only achieve that rank if you practice every day and commit to yourself that you will never give up.

If you enjoyed this book, please take the time to leave me a review on Amazon. I appreciate your honest feedback, and it really helps me to continue producing high quality books.

You can find my book here:

https://goo.gl/GbSgfN

Table of Contents

Table of Contents

Introduction

Chapter 1 - The Rubik’s Cube: The Greatest Puzzle Invented

Chapter 2 - Mathematics and Algorithms Behind the Cube

Chapter 3 - Rubik’s Cube Basic Notation

Chapter 4 - Beginner’s Method in Solving the Rubik’s Cube

Chapter 5 - Advanced Methods: The Fridrich Method

Chapter 6 - Advanced Methods: The Roux Method

Chapter 7 - Advanced Methods: The ZZ Method

Chapter 8 - Finger Tricks

Chapter 9 - God’s Number and Rubik’s Cube

Conclusion