An overview of a method

There are cubes, and then there are cubes that may not always be cubes. One such “Non-Cube” is the Square-1. To learn more about the Square-1, check out the Wikipedia entry. A popular method for solving the Square-1 is the Vandenbergh method (VM). A great resource to understand this method is this page, made by Lars Vandenbergh. If you know the VM , please continue reading. Else, read up on the basic idea of the VM . If you know how to solve cubes other than the 3×3, this should make sense, so please read on.

I was wondering, how do I teach someone to solve the 3×3 if the only cube that they know how to solve is a Square-1? A quick google search came up empty. To answer the question, I created a method for solving the Rubik’s Cube; the method is based on the VM for Square-1.  However, I needed to add a few extra steps to make it a valid method for solving 3×3. On a Square-1, the middle layer is always solved, and the pieces cannot change orientation, but this is not the case for 3×3. I needed to add some extra steps to compensate for these added irregularities on the 3×3.

The first step in executing my method is to solve the middle layer. Solving the middle layer is a mostly intuitive step which is done in relatively few moves, compared to the rest of the method. Compared to the VM , this step is one of the “extra” ones, since not only in the VM but on a Square-1 in general,  the middle layer is always solved.

The second step is to orient the edges and corners on both of the outer layers. This step is also one of the “extra” steps since orientation cannot be changed on a Square-1. To do this step, you need to use OLL algorithms, as in the CFOP method. Along with all 57 OLL algorithms, there is a strange parity case caused by pieces on both the top and bottom layers not being oriented correctly. I won’t explain the cube theory behind this, but I’ll give you a simplification:

On a 3×3, edges can only be flipped in pairs. This means that in the last layer stage of the CFOP solution, you can only have 0, 2, or 4 edges oriented incorrectly. However, in my solution, you have both the top and bottom layers unsolved so that you can have 0, 2, 4, 6, or 8 edges flipped. When you only have one layer unsolved, there is an even number of edges flipped in that particular layer. If two layers are unsolved, there could be four incorrectly oriented edges across both layers and an odd number of incorrectly oriented edges in each layer, such as 1 incorrectly oriented edge in the top layer and 3 in the bottom layer. It is impossible to flip 3 edges or 1 edge, so the edges have to be redistributed across the layers. This is a parity case, as you cannot solve it with the conventional algorithms.

The third step is to get all the pieces onto their respective layer. It sounds like this step should be done before orienting the pieces, but it does not matter. An advantage to doing this step after orienting the pieces is that if you get parity during the last step, you do not need to worry about switching around the pieces from layer to layer. This shortens the parity algorithm considerably.

As to actually doing the third step, it can be completed with only R2, U, and D moves. It is a reasonably intuitive step, with some restrictions, like those I mentioned previously. You do not need individual R moves as an R2 can bring a piece from the top to the bottom layer and vice versa. Another reason why you do not need to make R moves is that all the edges are correctly oriented. Since all the pieces are correctly oriented, you can bring the pieces onto their correct layer with just R2, U, and D moves.

The fourth and final step is to permute both layers. This step also has a parity case, and the cube theory behind this parity case is similar to the theory of the other case. You see, similarly to how edges can only be flipped in pairs, edges can only be swapped in pairs as well. If I explain more of the cube theory behind the parity case, I’ll be repeating what I said about the other parity case, so I’ll move on now.

The fourth step can be done with PLL algorithms since the final steps can be broken down into doing the last layer on both the top and bottom layers. After you do the parity algorithm, if required, you have a standard PLL case on both the top and bottom layers. Just do the PLL case on the top, then turn the cube over and do the PLL case on the bottom.

How does this method compare to other speedsolving methods, such as CFOP or Roux? After some testing, I found that my method wasn’t very efficient. Unlike CFOP or Roux, where you can always look ahead to the next step, my method requires pauses to turn the cube over and look at the case. These pauses are during steps 2 and 4 and add a bit more time to your solves. Having to bring the pieces onto their correct layer after orientation makes it impossible to look ahead to step 4, and this is only one of many inefficient points of this method.

So, this method is not comparable to speedsolving methods regarding efficiency; however I created this method in one afternoon, and this method was the product of a few random musings, my curiosity has been satisfied.

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Cubing Competitions: What are they?

In the field of Rubik’s Cubing, there are competitions held that test your speed at solving the cube. These competitions are an excellent place to meet other cubers, learn new algorithms and such, and to generally have a good time. All things considered, competitions are a good place for cubers of all ages and skill levels.

In a competition, the events that are held are not just 2×2 and 3×3 Rubik’s Cubes. There are a total of 18 events. However, not every event is held at every competition.  The way in which you compete is by doing 5 timed solves in whichever event you are competing in. Your best and worst times are taken out, and the mean of the remaining 3 solves is your result. However, in more extended events such as 7×7 and 6×6, only three solves are done and the mean of those three solves is taken as the result. The method of scoring is a bit more complicated than this, and for the full scoring method, you can click here. 

Where are the competitions held? The answer is almost everywhere! Competitions are held in virtually every major city across the world, from Perth, Australia; to Toledo, Spain. In fact, a competition could be happening near you at this very moment! Furthermore, by going to competitions, you can learn things from other cubers, such as how to execute a particular algorithm, which kind of cube is best for you, and further your knowledge of cubing.

In conclusion, competitions for twisty puzzles are held, and they attract quite a lot of attention. Competitions test your skill in various twisty puzzles by comparing your times against other cubers, and these competitions are great places to go for meeting the cubing community, and just having fun. To find a competition near you, click here.  The link will take you to the World Cube Association’s Competitions page, and I hope you go to compete this summer. Good luck and have fun!

Some more efficient methods.

The methods I described in my previous posts can solve a cube, however they are not close to being very efficient. Therefore, I am outlining a few more efficient methods for each cube in this post, starting with the 2×2.

2X2

A good method for 2×2 is the Ortega method. It is a very useful method and can easily take your times down to less than 5 seconds. This method is done by first solving a face, then solving the opposite face, and finally permuting both layers. There are 13 algorithms to learn in total, some of which you will already know, to solve using Ortega.

A good place to learn this method is Cyotheking’s website.

Pros- Easy to learn, can get quite fast.

Cons-To get world class, this method isn’t sufficient.

Another good method for 2×2 is called CLL. It is a very simple method used by world class solvers. It is done by solving a layer, and then using one algorithm to solve the last layer. There are many extensions of this method which can be used along with CLL to get better times.

A good place to learn this is  Cyotheking’s website, which also has the extensions I mentioned earlier.

Pros- World class times are possible

Cons- Lots of algorithms are needed, particularly if you want to learn the extensions.

3X3

Moving on to 3×3, there are lots of methods available for use in advanced solving. The most popular is the CFOP method, also known as the Fridrich method as it was proposed in the 1980’s by Jessica Fridrich. This method can be called a more efficient form of the beginners method. The steps to it are (1) Solving the cross, (2) Solving the first two layers, (3) Orienting the last layer and (4) Permuting the last layer to solve the cube.

A good place to learn this method is Bob Burton’s website 

Pros- Simple, somewhat intuitive

Cons- Lots of algorithms to learn

Another method for 3×3 is the Roux method, proposed by Frenchman Gilles Roux. His method is where you build two 1x1x2 blocks on opposite sides, solve the last layer corners, orient the last 6 edges, and finally solve the rest with one algorithm.

A good place to learn this is Waffo’s cube Thngy

Pros- Very intuitive, few algorithms

Cons- Hard to understand at first

Pyraminx

For Pyraminx, the method I have described earlier is quite good, if the top 3 edges can be done efficiently. Another method is LBL, or Layer by Layer. It is a very fast method and can be used to achieve times of 5-6 seconds. It is done by building the bottom layer, then finishing the top with one algorithm. There are 5 algorithms to learn.

A good place to learn this is Legoboyz3 ‘s video

Pros- Simple, few algorithms

Cons- Not sufficient for world class times

A bottom first method for solving Pyraminx

Pyraminx is a pyramid-shaped twisty puzzle that is a World Cube Association event. I have developed a method for Pyraminx. A Pyraminx looks like this.

IMG_20170727_131920

I do not know if this method has already been developed or if it is original, so I am just saying what I discovered.  The breakdown of this method is that you first solve the bottom part of the Pyraminx, (shown below left), and then you solve the rest (shown below right).

IMG_20170727_132147IMG_20170727_132328

I have developed my notation to go with this method. It consists of the letters R, R’, L, L’, B, B’, D and D’, which are explained below. The cube has to be held with one vertex pointing down, as shown in the pictures.

The letter R represents a clockwise turn around the right-most vertex, as shown below.

IMG_20170727_132621

The letter R’ represents a counterclockwise turn around the right-most vertex, as shown below.

IMG_20170727_132859

The letter L represents a clockwise turn around the left-most vertex, as shown below.

IMG_20170727_132927

The letter L’ represents a counterclockwise turn around the left-most vertex, as shown below.

IMG_20170727_132943

The letter B represents a clockwise turn around the back-most vertex, as shown below.

IMG_20170727_133111

The letter B’ represents a counterclockwise turn around the back-most vertex, as shown below.

IMG_20170727_133136

The letter D represents a clockwise turn around the bottom-most vertex, as shown below.

IMG_20170727_133231

The letter D’ represents a counterclockwise turn around the bottom-most vertex, as shown below.

IMG_20170727_133338

Now for the breakdown of Pyraminx. It is made up of three main types of pieces: Center Pieces, (shown below)

IMG_20170727_133538

Edge pieces,(shown below)

IMG_20170727_133522

And tips,(shown below)

IMG_20170727_133445

The first thing you do to solve is find the  red-green-blue tip and position it on the bottom, as shown below.

IMG_20170727_133708

Once you do that, you need to solve the three edge pieces around the red, blue, and green centers. This step is intuitive, and there’s not much I can explain here. Once you’re done, your cube should look like this, with the bottom part solved.:

IMG_20170727_133755

You can practice the above step to get a better understanding of the Pyraminx. The next step, solving the yellow layer, is broken into two parts: getting the yellow centers on the top face and solving the last three edges.

For the first part,  the first thing you have to do is check if there is a yellow center on the front face. If there is, it should look something like this:

IMG_20170727_133835

Or:

IMG_20170727_133948

If you don’t have any yellow centers on the front face, you should rotate the cube so that at least one yellow center is on the front face. if you can’t get any yellow centers on the front face, then you can move on to the next step.

However, if you do have at least one yellow center on the front face, identify if it is on the left or right. If it is on the left, your cube should look like this:

IMG_20170727_134207

If it is on the right, your cube should look like this:

IMG_20170727_134052

If the center piece is on the left, do the algorithm L B L’ B’ L’.

If the center piece is on the right, do the algorithm R’ B’ R B R .

Now for the final step: Solving the last three edges.

For this step there are a few cases you need to memorize algorithms for.

The first case is where two edges need to be flipped in place, as shown below:

IMG_20170727_134331

The algorithm for this case is L R’ L’ R B’ R B R’

The next case is where three edges need to cycle around, but they are all oriented correctly. There are two variants to this case:

1.  The three edges need to cycle counterclockwise-

IMG_20170727_134411

The algorithm is B’ R B R’ R’ L R L’

2. The edges need to cycle clockwise-

IMG_20170727_134551

The algorithm is B L’ B’ L L R’ L’ R

The third case is where two edges are flipped incorrectly and they need to cycle around counterclockwise-

IMG_20170727_134621

The algorithm is L R’ L’ R

The fourth case is where two edges are flipped incorrectly and they need to cycle around clockwise-

IMG_20170727_134639

The algorithm is R’ L R L’

After you are done with all of this, align the remaining three tips with their respective centers to solve the Pyraminx.

Now that we’re done solving, we should take a look at the move efficiency of this method.  I did fifteen solves using this method from computer generated scrambles, and I calculated the move average of this method to be about 25 moves, which if executed at 2-3 moves per second will take 8- 12 seconds. Of the 25 moves, 4 moves are required to build the bottom, 3 moves are needed to fix the tips, 12 are needed to solve the centers, and 6 are needed for the last 3 edges.

Based on the numbers that I got, one way to make this method faster is to solve the top face centers with one algorithm rather than 2-3 algorithms, more on that later…

This method isn’t as fast as Oka, LBL or Intuitive L4E, but it is a good starter method and you can get decent times with it.

 

 

A space racing video game in Scratch

Now that I have made a 3×3 scrambler in Scratch, I decided to do something a little more complicated. I created a video game, and you can see this video to get the basic idea. The game is about getting as many coins as possible before the countdown runs out, not a race to the finish line. Then, I will talk about how I made it.

game video

What I did was create a “Game Loop”, which is just one big loop that sends all the messages that tell every character (called sprite in Scratch) to do whatever they need to do. For instance, the scrolling scenery that repeats over and over again receives a message from the game loop when the game starts that moves it downward. Each sprite is programmed to react differently to the messages from the game loop, so that I can control every sprite with only a few messages.

This is an image of the game loop

Screen Shot 2016-09-02 at 8.15.50 PM.png

Where it says ” broadcast —- and wait ” or “broadcast —- “, it means that it is broadcasting a message to the other sprites to do whatever they need to do. When setup is broadcasted the scrolling scenery returns to its original position, the spaceships return to the centre, the screen is cleared of the asteroids in the middle and the screen is cleared of the coins.

The messages inside the “repeat until” block only repeat until a certain condition is met, in this case, until the “time left” variable is less than one. The “game over” message is sent when the “time left” variable is less than 1 to end the game.

Now I’ll go over the basics of each sprite.

The scrolling scenery is two sprites that fit seamlessly on top of each other and take turns filling the screen. The sprites are just the part with the asteroids. The stars are simply a backdrop.

The spaceships are exactly the same but programmed to move along the screen at a certain speed  when specific keys are pressed. They are also programmed to spin when they hit the asteroids or the other spaceship.

The asteroids are programmed to appear at the top of the screen, move down at the same speed as the scrolling scenery, and disappear when they hit the bottom of the screen. The original asteroid sprite is hidden and it creates clones of itself that move down the screen. This is a tedious long piece of code but routine.

This is the script used for making clones of the asteroids and the coins. Sprite 4 is the asteroid and sprite 3 is the coin.

Screen Shot 2016-09-03 at 1.53.01 PM.png

The coins are like the asteroids; they have clones that appear at the top of the screen and move down. However, they fall at a slower speed than the scenery, and they are more common than asteroids. Also, they are programmed to disappear when they touch spaceships and add one to that spaceship’s score. The code for the coins is below.

Screen Shot 2016-09-03 at 1.57.07 PM.png

And that’s all there is to my video game!

If you have any questions, leave a comment and I will reply. Happy Scratching!!!

A 3×3 scrambler in Scratch

According to WCA ( World Cube Association) regulations, your cube must be scrambled by a computer generated algorithm as to avoid some people getting easy scrambles because the person whose scrambling is biased.

So, I decided to make a scrambler using Scratch. Scratch, if you didn’t know, is a programming language that you can use to make simple video games and other programs. I used it for making a cube scrambler. Now, I’ll tell you about it.  My scrambler is for a 3×3 cube.

Screen Shot 2016-08-27 at 4.29.46 PM.png

The picture on top shows the scramble you get after you run my program. The picture on the bottom shows the script I used to run the scrambling algorithm. List is a way to store items in a Scramble program. I used two lists calles moves and scramble. Moves contains all the 18 possible moves, F, Fi, F2, B, …. . The idea is to pick a move at random from the list of moves and add it to the list called scramble. Scratch lets you pick an element at random.

Next, I explain the individual instructions.

 

Screen Shot 2016-08-27 at 4.30.33 PM.png

The top part of the picture, where it reads “when flag clicked” simply indicates that when the flag on the first picture is clicked, the program will start running.

The next part down, where it reads “hide list moves” means that the list of all possible moves is hidden so that you can’t see it.

The next part down, where it reads “delete all of scramble” means that the previous scramble from the program’s last use is deleted.

The yellow, C shaped part, that reads”repeat 25″ means the part within the C will be repeated 25 times. In this case, it means that the “add item random from moves to scramble” will be repeated 25 times.

The next part down, where it reads “add item random of moves to scramble” means that a random item of the list of possible moves will be added to the scramble.

The part at the bottom, which reads “show list scramble” means that the scramble will be displayed on the screen.

And thats an overview of my scramble generator!

Instructions for solving a 3×3 Rubik’s Cube.

Basics

A 3×3 Rubik’s cube is a three-dimensional object with 6 faces, 12 edges, and 8 corners. The cube has three types of pieces on each face.

33 pic 1.png

Pieces labeled 1- These are corner pieces. They have three colours on them. They move during the course of a solve.

Pieces labeled 2- These are edge pieces. They have two colours on them. They move during the course of a solve.

Piece labeled 3- This is a centre piece. This has only one colour on it. This does not move during the course of a solve.

IMPORTANT- An edge piece can only go to the spot of an edge piece and not a corner or centre. The same applies for corners and centres.

(in the illustrations, gray squares don’t matter)

Notation

 

In algorithms, we use letters to denote faces and which way they should turn. Each face can be turned clockwise or counterclockwise.

Front face= F/Fi

Back face=B/Bi

Top face=U/Ui

Bottom face=D/Di

Left face=L/Li

Right face=R/Ri

If there is an i next to the letter, the face is turned counterclockwise, and if there is no i, the face is turned clockwise.

Colour scheme

 

We are assuming the standard colour scheme, which means that white is opposite yellow, blue is opposite green, and red is opposite orange. If white is on the front face, then yellow is on the back face, green is on the top face, blue is on the bottom face, red is on the left face and orange is on the right face

Now let’s hurry up and get solving!

Step 0: scramble your cube

 

Step 1: solve the white cross

Hold the cube with the white centre piece on the top face.

Find an edge piece with white and some other colour on it.  I’ll take the white/red piece. Align the piece directly under its desired spot. Once you have done that, you can have one of two cases.

33 pic 2.png

 

 

If you have the case above, execute the algorithm Di, Li, F, L to put the edge piece into its desired spot. If you  have the case below, where the red side of the edge piece lines up with the red centre, do F, F.

33 pic 3

After solving the edge piece, make sure that both colours of the edge piece match up with the adjacent centres.

Repeat with all 4 edges of the white cross.

 

Step 2 : solve the white corners

After solving the cross, hold the cube with the cross on the top face and look for a white corner on the bottom layer. Once you find it, align it under its spot. Your cube should look like this  –

33 pic 4.png

Now that you are all set, execute the algorithm the algorithm Ri, Di, R, D as many times as needed to solve the corner.

If your corner is in the top face but in the wrong spot, execute the algorithm Ri, Di, R, D to get it into the bottom layer.

Repeat with all the white corners.

Step 3: solve the middle layer

Turn the cube over so that white is on the bottom face and then look for an edge piece that belongs in the middle layer that is in the top layer. Once you have found it, turn the top layer until that edge piece forms an upside down T shape. Once you have that, you can have one of two cases. I’ll use the red/green edge piece. In the figures below the green line on the top indicates that the other side of the edge piece is green. The green line on the side indicates that thet side has the green centre.  33 pic 5.png

If you have the case above, execute the algorithm U, R, Ui, Ri, Ui, Fi, U, F

If you have the case below, execute the algorithm Ui, Li, U, L, U, F, Ui, F

33 pic 6.png

 

If you have an edge piece that is in the middle layer and you want to take it out put it on the left side then execute the second algorithm.

Step 4 : Orient the yellow layer

Part one: get the yellow cross

At this point, you should have the white layer on the bottom, and the yellow face should be in one of these four states –

33 pic 7.png1

33 pic 8.png2

33 pic 9.png3

33 pic 10.png4

 

If you have case 1 or case 2, execute the algorithm F, U, R, Ui, Ri, Fi in the position shown with white on the bottom face. For case 1 you have to execute the algorithm twice.

For case 3 execute the algorithm F, R, U, Ri, Ui, Fi in the position shown with white on the bottom face.

For case 4, move on to part 2.

 

Part 2: get all the yellow on top

Now you can have one of these cases 33 pic 11.png

 

 

1 :one corner

 

 

 

33 pic 12.png

 

 

 

2 :2 corners

 

 

 

33 pic 13.png3  :no corners33 pic 14.png4 :all corners

 

For cases 1 and 2 execute the algorithm R, U, Ri, U, R, U, U, Ri in the position shown with white on the bottom.

For case 3 execute the algorithm mentioned above with red on the front face, yellow on the top face, and white on the bottom face.

For case 4, move on to step 5.

 

Step 5: permute the yellow layer

Now, you can either have the yellow layer solved, or it can be unsolved. If it is solved, turn the yellow layer until the whole cube is solved. If it is not solved, then you need to permute the last layer, or put all the pieces in their designated spot.

Part one: permute the corners

For this part, look at the corners of the last layer. You can have one of three cases.

33 pic 15.png

 

33 pic 16.png

 

33 pic 17.png

For case 1, hold the solved corners in the back and yellow on the top and execute the algorithm Ri, F, Ri, B, B, R, Fi, Ri, B, B, R, R, Ui.

For case 2, hold it with yellow on the top and do the algorithm above twice.

For case 3, move on to part two.

 

Part 2: permute the edges

Now you can have one of two cases –

33 pic 18.png33 pic 19.png

For case 1, execute the algorithm F, F, U, Ri, L, F, F,  Li, R, Ui, F, F to get case 2.

If you have case 2, there two possibilities you can have.

 

33 pic 20.png33 pic 21.png

If you have case 1, execute the algorithm F, F, U, Ri, L, F, F,  Li, R, Ui, F, F with the solved edge in the back and yellow on top.

If you have case 2, execute the algorithm F, F, Ui, Ri, L, F, F,  Li, R, U, F, F with the solved edge in the back and yellow on top.

If your cube is not solved then turn the top layer until the cube is solved.

If you followed my instructions correctly, CONGRATULATIONS!!!