Rubik Cube Almost Solved Assignment

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Written by:Dr. Christine von Renesse

Why the Rubik's cube?

Solving the Rubik’s cube was one of the main themes in my Mathematical Explorations class this semester. It took about one hour of every week for the whole semester for almost all students to finish the cube, so this was a big time commitment. During the semester I regularly worried: would they all be able to solve the cube eventually? How much help do they need right now so they don’t give up? Is it worth spending so much time on a project that my students might consider a “useless skill”? It took a lot of persistence to stick with it, but we did it!

My students believed for most of the semester that they would never ever be able to solve the cube. Watching them overcome this belief was powerful. How often do you get an email in the middle of the night from non-major students in a core class just to tell you that they just solved a mathematical problem? One of the main goals of this course is for students to change their beliefs about their mathematical abilities and to become more persistent, confident and creative in problem solving. And the Rubik's cube does just that. Here is a quote from a student's journal half into the semester:

"The explorations I enjoy most are the ones with the Rubik’s cube. I never would have believed I could solve any part of the cube. At the start of the class I would’ve been happy to get one row filled with the same color. Now, I am getting closer and closer to solving the cube. It is really enjoyable to me, and I am really proud of myself with each week and with the more progress I see".
DAoM Student, Spring 2016

How is this mathematics?

At first students are very surprised to find Rubik’s cubes in a mathematics class, since their perception of mathematics is very narrow. After reading Lockhart’s lament, watching the movie “the proof”, and doing other mathematical activities and proofs (mostly from the music book) they understand that solving the cube is doing mathematics. They also connect that, just as Andrew Wiles made mistakes, they are making mistakes and need to be persistent when trying new strategies or trying to make sense of mathematics.

"This class is definitely changing my views on math. Only 2 weeks in this is easily the most rewarding and enjoyable math class I have ever taken. Between the magic trick and the Rubik’s cube and the clap and count exercise I have been thinking critically about math from day one. [...] I think the way this math explorations class is being taught is exactly how Paul Lockhart would want it to be. There’s no memorizing equations or tests, just a teacher guiding student’s through mathematical thinking. I definitely view math as one of the arts now after the argument made in this paper and hopefully math education does get reformed in the public school systems".
DAoM Student, Spring 2016

"I may never need to solve a Rubik’s cube, or know the Pythagorean Theorem; but I will know how to come up with strategies to solve a real life problem or something similar. Math teaches people dedication and creativity; to never give up looking for new answers."
DAoM Student, Spring 2016

Algebra and reasoning skills

But my students also acquire algebraic thinking skills. Traditionally, students in my Mathematics for liberal arts courses do not enjoy algebra. They are either able to perform the procedures but find them boring or they never succeeded in Algebra and hate it. As an example, most of my students did not know how to “foil” the expression $(x+y)^2$. At the end of the cube project the students make sense of long chains of variables that represent moves on the cube, they invert them, modify them and invent their own. My students will most likely not need the traditional Algebra skills in their lives, but being able to create and make sense of an algebraic system, if they need one, is a handy skill to have.

You can see in the student work to the right how a student grapples with the algebraic notation and makes sense of combining moves. The students were supposed to find a combination of moves that gets the missing white face into the top layer (there are several!), find the Singmaster notation, and explain why their moves make sense. This was an assignment during the first weeks of the semester. The student invents meaningful non-traditional notation, using $D^{-1} 2$ to express doing the move $D^{-1}$ twice, and using $MV$ to denote the middle, vertical slice of the cube.

Is solving the Rubik’s cube a procedural or a conceptual activity?

It certainly has both aspects and I want to make sure that my students learn to think conceptually about mathematics rather than procedurally. What is the point to memorize yet another procedure that allows one to solve the cube? So let’s look at where the conceptual thinking comes in:

My students have to figure out themselves how to solve the first layer of the cube. This is a very difficult conceptual activity for them. They need to make sense of:

  • The difference between edge cubies and corner cubies. Students often believe that a square of some color could move anywhere on the cube and it is a big step for them to realize that for instance corner cubies can only move to other corners.
  • The difference between orientation and position. A cubie can be in the correct position but in the wrong orientation.
  • How to get cubies to the correct place without messing up cubies that are already in the right place in the top layer.
  • How to record a chain of moves on paper and how to record the reasoning of why particular moves make sense.

You can see in the student work to your left how students represent the changes of the cube without tracking that several faces belong to the same cubie. It took many weeks before they realized that it would be much easier to track cubies instead of cubie-faces. Interestingly I had tried to tell them that this is not the most efficient way to represent changes (and I was much less subtle than usually!) but none of them followed my lead. I think they needed to go through their own representations to make sense of what works best for them. If they still hold on to the belief that all cubie-faces can move separately then of course it doesn't make sense to track them in "groups of cubies"!

Solving the cube

There are several different strategies for solving the rest of the cube. The most common strategy, the layer method is pretty procedural and can be learned from youtube or from the instructions that come with the cube. The method we ask our students to use is the “corners first” method (See David Joyner's book). While there are also specific moves that the students receive from us, they need to investigate what exactly these moves do and how to use them. Every time they solve the cube the exact implementation will have to be slightly different which leads to a lot of thinking. There are different strategies students develop to help them figure out the rest of the cube and so there is a lot of conceptual understanding and reasoning necessary as well. But learning the Singmaster notation and correctly using the provided moves also requires students to learn and memorize some procedures.

How do I convince my students that it is a good idea to learn the "harder" method to solve the cube? I keep telling them that the goal is to reason and think hard (brain yoga!) and that there is no point in performing the easy way if we want to learn thinking a.k.a. mathematics. Some students tried both methods but no one pushed back and refused to learn the “corners first” way. And they were even prouder in the end!

An app to solve the cube...

My students found severals apps that give you explicit instructions about how to solve your specific cube, e.g. "Solve your cube" by Felipe B. Valio. At first I wasn't sure how to deal with this. Should I not permit it? (Good luck!) Should I ignore that it exists? Should I incorporate it somehow? I then realized that the app can be very helpful in analyzing the specific moves we were studying. If you have a solved cube you can see much more clearly which cubies have moved where. I asked some students if they would feel less motivated to solve the cube themselves if they had an app that could do it for them, but they didn't think so. And it turns out that they worked just as diligently on solving the cube themselves as before! The only downside was that when I scanned the room and saw a solved cube it didn't necessarily mean that it was time to celebrate.

Group work and a word of caution

Solving the Rubik’s cube seems to be a solitary exercise and it takes the students a while before they reach out to their group members for help. It is difficult for them to have conversations about the cube when they are all “stuck” at different places. Also, students have not developed a language yet to communicate with their peers about the cube. Usually it takes about two weeks for students in my classes to be comfortable with group work and the room tends to buzz with mathematical discussions. The Rubik’s cube leads to more solitary work and so it took almost half the semester before I noticed that same teamwork feeling in the room. I have to think about how I can help my students develop a sense of community earlier while working on the cube.

The Rubik's cube is a big commitment and you can't just decide to stop and switch gears like you can with some other topics. I imagine the students would be very frustrated if the topic suddenly changed or if most of them ended up not being successful in solving the cube. I believe that to be a successful facilitator of the Rubik's cube activity it is really helpful to feel comfortable in teaching inquiry-based and to have taught a mathematics for liberal arts class before. I don't think it is a great activity for "your first teaching IBL experience."

Assessment

I assess my students every class by walking around the room, listening to their conversation, asking them questions to learn abut their thinking, and watching them do mathematics. This participation grade counts for 40% of the total grade.

Almost every week my students have to complete a writing assignment about the cube. For the first few weeks, for instance, they were working on “getting the white face in the correct spot”, see student work above.

After about five weeks I checked during class (like a quiz) that every student was confident in completing the top layer. The time it took them to do this was not important to me at this point.

At the end of the semester, each student had to meet with me one-on-one and show me how he or she solved the cube. They were allowed to bring a sheet with the moves and I asked them to stop every now and then to tell me what they are planning to do next. I gave them a timeframe of 10-15 minutes to solve the cube. The time requirement had the advantage that students tried to solve the cube many times before meeting with me and had seen and solved many of the difficult situations one can face. For example: when you position the edge cubies before the corner cubies, then it can happen that you don’t have a move to deal with the last two cubies. And when you then need to use the corner swap, more of your edges are messed up again!

Open questions (for us)

There were several questions that came up for us as a class that we did not have time to make sense of:

  • How do we know that the “corners first” moves will always be enough to solve the cube?
  • Can we only use corner rotation and never corner swap?
  • Can we only use corner swap and never corner rotation?
  • Why does solving the “corners first” method lead to the fact that the edge rotation move is enough to get all edges in the right spot? Why could there not be two edge cubies left (instead of three)?

It would be great to have the time to dive into some of these questions, but a regular semester is not enough in my experience. It felt important though to acknowledge that there are open questions that we are curious about.

Extension questions

There is always at least one student who either knows how to solve the cube already or is ready to solve it in just a few weeks. This semester I had a student who first wanted to learn the layer methods and then explored the “corners first” strategies. In the end he combined both ideas and managed to solve the cube in under 1.5 minutes. He performed in front of the whole class and is planning to show off his speed cubing at a talent show at his work place.

I also brought to class the 2x2x2, 4x4x4 and 5x5x5 cubes so students who were done with the 3x3x3 cube could explore other challenges. I personally do not know how to solve the other cubes and I told my students often that I was hoping that one of them would develop a method and show me. Why? Because they need to see that we all have limits in our knowledge and that I love to learn more. I model how to be curious and admit not knowing. In doing this, students and professors are all equal learners who are curious about the world of mathematics together.

Here is a student quote from the final journal assignment:

"I can honestly say that I went from a mathematician who was worried that he wouldn’t be able to figure out the math problems given to him this semester, to a mathematician who was excited to find a new problem on the board to figure out every day and is saddened to know that this class has come to an end. However I am encouraged, because I know that from now on math will no longer worry me, and I can now say that I am pretty good at math, and feel confident in mathematics. I will continue to search for math problems to overcome, and now I have the skills to solve a Rubik's cube as well, which is something my whole life that I have always wanted to do!"
DAoM Student, Spring 2016

References:

Joyner, David. Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys. Baltimore: Johns Hopkins University Press, 2002.




On January 1, 2017, I asked myself the question: Within one month of practice, can I solve a Rubik’s Cube in under 20 seconds?

On January 24, 2017, after 25 hours of practice, I found out that the answer was yes.

During the month of January, I documented my entire learning process in a series of 31 daily blog posts, which are compiled here into a single narrative. In this article, you can relive my month of insights, frustrations, learning hacks, and triumphs, as I strive towards monthly mastery.

It’s January 1st today, which means it’s time to start a new challenge. This month, my goal is to solve a Rubik’s Cube in under 20 seconds.

Why 20 seconds?

Two months ago, at San Francisco’s Ferry Building, I went to a book event and signing by Ian Scheffler, the author of Cracking the Cube, the predominant book on speed cubing (the competitive sport of solving the Rubik’s Cube).

At the event, I spoke with the author and the dozens of Bay Area speed cubers that attended. I wanted to learn about the major benchmarks in the sport.

Repeatedly, what I heard was “The most significant benchmark in speed cubing is the 20-second mark. Breaking 20 seconds is like breaking the four minute mile”.

At first, I just thought that the equivalence was symbolic, but, that day, I also heard that “If you look at the proportion of cubers who can solve “sub-20” to all cubers (i.e. people who own a Rubik’s Cube), it’s about equivalent to the proportion of runners who can run sub-4 miles out of all competitive runners”.

I’m not exactly sure how this was computed, or exactly how accurate it is. But, nevertheless, it’s clear that 20 seconds is the most notable benchmark in speed cubing.

Thus, I’ve set my sights on a sub-20 time.

My starting point

I actually have a long history with the Rubik’s Cube: I originally learned how to solve it when I was in Middle School, and, ever since, I’ve been casually solving it for fun.

However, I’ve never really tried to go fast. Instead, I prefer the more meditative, tactile approach of smoothly solving the cube.

In fact, I haven’t actually timed myself in eight or nine years. So, to properly kick off this month, I set up a camera and a computer-based timer, and completed five solves at top speed.

The average of the five solves was 48.29. (44.46, 51.83,46.15, 53.62, 45.41).

This isn’t horrible nor is it great. However, if you have no Rubik’s Cube experience, it’s possible to think that I’m not too far away from a sub-20 time.

But, to continue the running analogy, solving a Rubik’s Cube in 48.29 is symbolically equivalent to running a 7-minute mile. In other words, it’s a reasonably respectable time for an amateur, but it’s far from a competitive time for a serious athlete.

A month of deliberate practice

Over the past nine years, I’ve solved the Rubik’s Cube hundreds of times, but never with the goal of improving. As a result, my time has only improved by around 10 seconds from 60-ish seconds to 50-ish seconds.

This month, I’m taking a more deliberate approach with my practice, explicitly looking to shave another 30 seconds off my time.

My hope is that my effort this month reveals the overwhelming difference between nine years of mindless practice and 30 days of deliberate practice.

Time to formulate my plan of attack…

Over the next few days, I’ll be experimenting with a number of training techniques and exercises, to determine the best 30-day Rubik’s Cube training program. In particular, the program needs to get my Rubik’s Cube solving speed consistently under 20 seconds.

To understand my training program and the rest of this month’s posts, it’s important that you first understand the method I use to solve a Rubik’s Cube, which is called CFOP.

Intro to CFOP

CFOP is the most common speed cubing method and is the one that I originally learned.

CFOP stands for 1. Cross, 2. F2L, 3. OLL, 4. PLL, which are the four steps used, in this method, to solve the cube. These acronyms probably don’t mean much, so let me go through each one.

1. Cross

With a fully scrambled cube, I start by finding the white center, which identifies the white side of a solved cube.

I then find the white edge pieces, and move them into place, so that a white cross is formed on the white side.

It’s important that the edge pieces not only match up with the white center, but that they also match up with the correct color of the adjacent centers.

For example, the white and green edge piece should be placed between the white center and the green center. The white and red edge piece should be placed between the white center and the red center. And so on.

The cross is complete.

2. F2L (First 2 Layers)

Once the cross is formed, I put the white side of the cube on the bottom, which looks like this.

In this next step, F2L, or ‘First 2 Layers’, I’m focused on solving the first two layers of the cube, starting from the bottom.

Since the center pieces and ‘white + color’ edge pieces are already solved (from the last step), I only need to focus on solving the highlighted areas below.

After placing the correct pieces in these slots, the cube looks like this.

Here, in the solved state, it’s easier to visualize which pieces I moved during F2L: I positioned the four edge pieces (like the blue and red edge piece) between the corresponding centers (like the blue center and the red center). I also positioned the four corner pieces that have a white side and two other colored sides (like the white, blue, and red corner piece) in the correct slot on the bottom layer.

As a result, when I finish F2L, the white side is completely solved.

F2L is complete.

3. OLL (Orient the Last Layer)

At this point, the white side is completely solved and the first two layers of the cube (from the bottom) are solved. Thus, I only need to solve the last layer of the cube.

To do this, I break down the last layer into two parts. First, in this step, OLL, I solve the yellow side, which is also known as orienting the last layer.

At this point, on the yellow face, it’s possible for the yellow pieces to be configured in 57 different ways.

The fastest speed cubers have memorized the 57 different algorithms necessary to most efficiently solve the 57 different configurations (one algorithm for each configuration).

However, if you’re willing to give up efficiency, it’s possible to solve the yellow side with far fewer algorithms. I currently only know 9 of the algorithms, which means I’m highly inefficient at this step.

Upon completing the necessary algorithm, the yellow side is solved.

OLL is complete.

4. PLL (Permute the Last Layer)

After finishing OLL, the cube looks like this.

The first two layers and the yellow side are solved. The only thing left to solve, during this step, PLL, is the rim of the last layer, which is also know as permuting the last layer.

There are 21 different configurations for PLL, which can be solved most efficiently with the corresponding 21 algorithms.

I currently know 7 of the PLL algorithms, which is enough to complete PLL in twice as many moves as a competitive speed cuber.

After executing the appropriate PLL algorithm, the rim of the last layer is solved.

And as a result, the entire cube is solved.

Going into each month, I have a reasonably good idea how I plan to approach that month’s challenge.

This month, my plan was simple: Learn the 62 Rubik’s Cube algorithms I don’t know. Obtain a sub-20 time.

I figured that, since I know only 21% of the algorithms, learning the remaining 79% will drop my time low enough to meet my goal.

It turns out that it’s not this simple.

Right now, algorithms don’t matter

As a reminder, there are four steps to solving the Rubik’s Cube (The cross, F2L, OLL, and PLL), which I explain in yesterday’s post. As a simplification, there are really two parts to the solve: The intuition-based part, which includes the cross and F2L, and the algorithm-based part, which includes OLL and PLL.

My original plan was just to focus on the algorithm-based part, since this approach seemed contained, very measurable, and well-defined. I would learn three algorithms per day, leaving me one week at the end to practice and perfect all the algorithms together.

But there’s a problem: it takes me, on average, 30–35 seconds just to reach the algorithm-based part of the solve. In other words, even if I learned and could perform all 78 algorithms perfectly, at full speed, I wouldn’t even reach that part of the solve until much after 20 seconds elapsed.

Basically, the intuition-based part (the cross and F2L) currently takes way too long, making it unignorable. In fact, since it makes up the majority of my solve time, I will likely need to put much of my focus on these steps.

The 20-second breakdown

According to Rubik’s Cube forums, if I want to solve the cube in 20 seconds, my time should be allocated, more or less, in the following way.

  1. Cross: 2 seconds
  2. F2L: 10 seconds
  3. OLL: 2 seconds
  4. PLL: 6 seconds

Thus, I need to throw out my algorithm-based plan, and instead, create unique plans for each stage to achieve the necessary times.

Of course, I can give and take seconds from step to step, depending on where my strengths lie, but this is a good framework to start with.

Tomorrow, I need to start experimenting with F2L training techniques (since these past three days I’ve spent just learning new algorithms).

I have a hunch that I will be reprising November’s use of a metronome…

As mentioned yesterday, I need to decrease my speed on the intuition-based parts of my solve (the cross and F2L) in order to achieve a sub-20 time.

After analyzing a few of my solves, there’s clearly one major thing currently slowing me down: Cube Rotations.

What is a Cube Rotation?

If I turn the entire cube clockwise or counterclockwise around its vertical axis, I’ve executed a cube rotation.

Executing a cube rotation doesn’t get the cube any closer to the solved state, but still costs valuable time. In fact, not only am I losing time executing the rotation, but I’m also losing time getting reoriented to the rotated cube. (For those who read about productivity, this is a bit like context switching, which has time implications beyond just the act of switching tasks).

As a result, when solving, speed cubers attempt to minimize their number of executed cube rotations, aiming for one or two at the most.

Today, before analyzing some video, I imagined that my cube rotating was minimal (perhaps 4 or 5 rotations per normal solve), but after filming myself, I realize I’m an aggressive cube rotator.

Watch as I execute 13 cube rotations just while solving the cross and F2L…

I execute Cube Rotations for two reasons

The first is because I’m trying to find the pieces I need to complete the cross and F2L. I’ll address this problem, which I call Inspection Pauses, tomorrow.

The other reason I execute Cube Rotations is to gain positional leverage. In other words, it’s easier for me to execute certain Rubik’s Cube maneuvers when holding the cube in certain, better-practiced positions. Simply, I’m rotating the cube into more preferable orientations.

While I’m nicely compensating for my weaknesses, this isn’t a healthy practice (if I want to get universally faster). It’s as if I learned how to play a guitar solo only in the key of A major, and every time the band plays anything different, I need to stop the band and ask them to transpose the song into the key I know.

Like a good guitarist, if I want to solve the Rubik’s Cube fast, I need to be comfortable with the full range of possibilities and all the ways they can be presented to me — whether that’s all the possible musical keys of a song or all the possible ways a certain maneuver is oriented on the cube.

How I plan to get better (introducing FRS)

In order to practice cube maneuvers from all possible orientations, and also in order to reduce Cube Rotations, I’ve created a simple exercise called Forced Rotationless Solving (FRS).

During FRS, Cube Rotations aren’t allowed. Instead, I must determine new ways to solve the cross and F2L, from new vantage points, without ever rotating the cube.

I’ve completed about 25 solves today using FRS and am already developing a knack for these new maneuvers. In particular, I’ve gotten good at inserting pairs in the back. I’ll explain specifically what this means in a future post, but generally it means that I can now (more comfortably) solve pieces on the back face of the cube, rather than having to rotate the cube 180 degrees to solve them in the front.

During the rest of the month, I’ll continue using FRS as a warmup exercise before I practice my normal speed solves.

Hopefully, I can drop the number of Cube Rotations during my solves into the low single digits.

Yesterday, I realized that Cube Rotations were majorly slowing me down during the intuition-based parts of my Rubik’s Cube solve. In an ideal world, I’d have only one or two rotations per solve, but instead, I have about 10–15.

As a reminder, these are Cube Rotations…

Yesterday, I discussed the positional reasons for Cube Rotations (and introduced Forced Rotationless Solving, FRS, as a potential solution). Today, I’ll address the other reason for rotations: Inspection Pauses.

Inspection Pauses

During F2L (solving the first 2 layers), I must correctly place eight pieces: four pairs of corresponding corners and edges.

In other words, during F2L, I must create four corner-edge pairs, and then place each pair in the correct position (called a “slot”) on the cube.

If you watch the video above, you’ll notice that I execute a fast series of moves. Then, I pause, rotate/inspect the cube until I find the next pair of pieces, execute another series of fast moves, and so on.

Basically, I’m not able to flow fluidly from pair to pair during F2L. Instead, my F2L is broken up by these inspection pauses.

Looking into the future

If I want to eliminate Inspection Pauses, I need to search for the next pair while I simultaneously execute the necessary moves for the current pair. In other words, I need to look into the future.

This is tough.

So, to practice, I’ve returned to using the Pulse metronome app, which I used in November to become a pseudo grand master of memory.

Here’s the idea: I set the metronome to something modest, like 90 BPM. Then, I attempt to solve the cube by making a turn every time the metronome clicks. As a result, by the nature of the exercise, all Inspection Pauses are (theoretically) eliminated, and the solve is consistent and fluid.

Even solving at 90 BPM, in this way, is surprisingly challenging for me. My brain really needs to work hard to both execute the moves and search for the next pieces simultaneously. But, as a result of this brain strain, I can feel my cube vision improving.

Better cube vision = faster solves, so nightmare-inducing metronome clicks are back in my life for at least the next week.

When I first explained how I solve a Rubik’s Cube, I noted that I always start by creating a cross on the white side. This consistency helps with speedy pattern recognition, but may not be the most efficient method.

So, to address this potential inefficiency, I experimented today with Color Neutrality.

What is Color Neutrality?

Under color neutrality, rather than always starting a solve with the white side of the cube, I must assess each unique scramble and identify which color seems easiest to start with (where easiest is some combination of fewest number of moves and ease of execution). The easiest color could be white, but it could also be yellow, blue, green, red, or orange.

Thus, to be a color neutral cuber, I must be able to 1) Quickly identify the easiest starting color for a particular scramble, and 2) Execute my solve with equal competency and pattern recognition from any starting color.

To determine if Color Neutrality is something worth pursuing, I needed to answer a couple of questions…

How much time would Color Neutrality save me?

To answer this question, I timed myself solving 20 crosses on the white side of the cube. I wanted to see how much time I was saving between an easy cross and a hard cross.

I found that my cross time was between 2 and 4 seconds, meaning perfect Color Neutrality, at best, will save me only two seconds per solve. Of course, two seconds is significant if I want to beat the world record (or if I’m consistently stuck at 22-second solves), but it’s only a tiny boost in absolute efficiency.

How much effort would it take?

Since I’d only be gaining two-ish seconds from Color Neutral solving, I needed to assess the investment level.

To do so, I completed ten solves starting on a color other than white.

Surprisingly, I’ve never actually tried this, so I didn’t know what to expect. The result was worse than I expected: On average, my Color Neutral solves were about 30% slower than my solves starting with white.

This is a big hill to climb.

Of course, with practice, I can overcome this difference, but I’m just not sure it’s worth my attention during this month’s challenge. Especially since I’ll only be shaving 1–2 seconds off my time for doing so.

I’m glad I went through the exploration, but I feel my efforts are better spent elsewhere.

What should YOU do?

If you are just learning to solve the Rubik’s Cube now, or if you ever plan to learn in the future, I would encourage you to start practicing Color Neutral solves as early as possible (assuming you care about speed).

It seems that the better and better you get with one color, the harder it is to justify going backwards to gain Color Neutrality.

In fact, if I was just learning how to solve the cube now, I would probably force myself to start each new solve with a different color than the previous solve. This is an easy mechanism to understand and follow, and will reap nice Color Neutral rewards.

Today, I solved the Rubik’s Cube 100 times, which took about 2 hours (with a few breaks).

I started with 20 Forced Rotationless Solves, which helped improve my muscle memory for maneuvers in less common orientations.

Then, I executed 30 solves to a metronome at 100 BPM, which helped improve my look-ahead.

Finally, I completed 50 regular speed solves to put the pieces together (literally, I guess).

Adding algorithms to my training

Over the past few days, I’ve only been training the first part of my Rubik’s Cube solve, ignoring the second, algorithms-based part.

Now that the exercises to train the “first part” are well-defined, it’s time to start addressing new algorithms as part of my training regime.

I’m not sure if I can learn all the remaining algorithms in the next three weeks, so tomorrow, I’ll figure out which ones to focus on.

When I first described how I solve a Rubik’s Cube, I alluded to the fact that some parts of the cube can be solved using a set of 78 algorithms, but I didn’t say much further.

Today, I’ll explain the fundamentals.

What is a Rubik’s Cube algorithm?

Simply, an algorithm is a set of pre-determined moves that, when properly executed, accomplish a specific task (i.e. “move these particular pieces on the cube into this particular configuration without moving/messing up these other pieces”).

The most common way to express a Rubik’s Cube algorithm is using Basic Notation, which is depicted below.

Under the Basic Notation scheme, R means “turn the right face of the cube clockwise”. R’ (R prime), means “turn the right face of the cube counterclockwise”. And so on, with F = front; B = back; L = Left; R = Right; U = Up; D = Down.

A full algorithm may look like this: F R U R’ U’ F’

When can you use Rubik’s Cube algorithms?

Once you solve the first two layers of the cube, you enter the algorithm-based portion of the solve, which is focused on solving the last layer.

Here’s what the cube looks like at this point.

To solve the remaining part of the cube, you must execute two classes of algorithms.

The first class of algorithms is called Orient Last Layer (OLL), which solves the top face of the last layer.

There are 57 different algorithms to solve each of the 57 possible patterns.

The second class of algorithms is called Permute Last Layer (PLL), which solves the rim of the last layer (and, as a result, the entire cube).

There are 21 PLL algorithms.

Which algorithms should I learn?

To solve the cube, you only need to know four OLL algorithms and four PLL algorithms. But, to solve the cube fast, you need to know as many as possible.

Since I don’t think I can learn all 78 this month, I need to determine which subset to study. Tomorrow, I will finish figuring this out and will write about it.

How should I optimally practice algorithms?

This another question I haven’t yet answered.

Once I determine my target algorithms, I’ll experiment with some training ideas, and share out what I find.

Today, I was trying to select which set of last layer algorithms to learn, when I stumbled on something amazing…

Two-Look OLL algorithms

As explained yesterday, 57 of the 78 Rubik’s Cube algorithms are used for the Orient Last Layer (OLL) step of the solve. Thus, I suspected that OLL would make up most of my algorithm learning effort.

However, surprisingly, after today, that’s no longer true:

This morning, I learned about something called Two-Look OLL, which basically means “If I’m willing to string together two algorithms back to back (instead of using just one algorithm), I only need to learn 10 OLL algorithms to cover all 57 OLL patterns”.

Here are those algorithms:

At the beginning of this month’s challenge (and as of this morning), I only knew a random assortment of 9 OLL algorithms that I picked up over the past many years of casual cubing.

I have no idea why I know the particular set of algorithms I do. I just do.

When I learned about “Two-Look OLL” this morning, the first thought I had was “I wonder how many of the 9 algs I know match the 10 Two-Look algs”. If I just linearly extrapolate the probability (which is perhaps combinatoriclynaive), I would expect something like a 15% match.

However, unexpectedly, I already knew 70% of the two-look algs. Somehow, 7 out of the 9 algs I already knew perfectly matched 7 out of the 10 two-look algs.

And, on top of that, the three algs I didn’t know were only minor variations on others that I did. In fact, because these three algs were so similar to others (i.e. mirror images, one move apart…), I was able to learn all three new algs in ten minutes.

Normally, it would probably take a day to fully incorporate a new alg.

Thus, excitingly, as of today, I know all the OII algorithms I need for this month’s challenge (I’ll prove why two-look is sufficient below).

It feels oddly lucky how much overlap there was between the set of algorithms I knew and the set of algorithms I wanted to know, but I suspect there is some underlying reason that propelled me to learn these particular algorithms in the first place. I’m just not sure what that underlying reason is.

If you’re a speed cuber and have a reasonable guess, leave a comment. I’d love to know.

PLL algorithms

The remaining 21 algorithms (out of the 78) are used for the Permute Last Layer (PLL) step.

Unlike OLL, I’ve decided to learn all 21 this month.

I already know 7, so, over the next two weeks, I will learn one of the remaining 14 algs per day. This will afford me one full week, at the end of the month, where I can execute solves with full knowledge of all the relevant algorithms.

Will this work?

If I just learn Two-Look OLL and complete PLL, will I be able to execute sub-20-second solves? It looks like the answer is yes:

I reviewed the video of solves I made on Day 1 of this challenge, and extracted a few relevant statistics.

On average, I was using 3.2 OLL algorithms and 2 PLL algorithms to complete the five solves.

With two-look OLL and complete PLL, I’ll reduce the number of algs to 2 and 1 respectively. In other words, assuming time maps linearly to ‘number of algorithms’, I’ll be cutting my OLL time by 1/3 and my PLL time by 1/2.

In Solve 1 from the video, which is the most average solve (since I used 3 OLL and 2 PLL algs), it took me 6 seconds to solve OLL and 8 seconds to solve PLL.

Therefore, if I can reduce my times by the amounts above, I will theoretically be able to execute a 4-second OLL and a 4-second PLL, or, in total, an 8-second last layer.

In my post “I already need a new plan”, 8 seconds is exactly how much time I allocated myself for my 20-second solve:

  • Cross: 2 seconds
  • F2L: 10 seconds
  • OLL: 2 seconds
  • PLL: 6 seconds

I’m shifting some of my PLL time to OLL to compensate for two-look, but it seems like it will work out.

Tomorrow, I’ll start practicing the new algs and determine which training methods are optimal.

Yesterday, I determined that I need to learn 14 more Rubik’s Cube algorithms this month. Today, I learned one of the 14.

Here was my approach :

  1. On the train ride to work, I practiced the movements of the algorithm over and over again. While doing so, I wasn’t trying to use the algorithm in context or solve the cube. Instead, I simply wanted to learn the feeling of executing the algorithm. I repeated the algorithm about 200 times until I felt I could execute it without consciously considering each of the individual moves.
  2. On the train ride home, I executed the algorithm again, but this time, I watched what happened to the patterns on the cube. In particular, I was trying to learn how I could quickly identify the pattern and associated algorithm (based usually on only seeing two or three sides of the cube), so that I could execute the new alg in flow during a solve.
  3. Once I got home, I solved the cube completely 40 times. During these solves, the new algorithm came up three times, giving me the chance to assess how smoothly (and correctly) I could execute it. The first time didn’t go so well, but the other two times flowed perfectly.

While this approach isn’t too profound, it got the job done.

Over the past 11 days, I’ve solved the Rubik’s Cube probably 400–500 times, which is a lot. In addition, I’ve also focused on learning new algorithms, and training my pattern recognition abilities with a few different exercises.

Ever after all of this practice so far, I still don’t seem to be getting much faster. In fact, some of the speed gains I saw in the first few days seem to be reverting (i.e. I’m going slower than I did ~4 days ago).

I have four theories why this is happening:

1. My cube is getting worse at the same rate I’m getting better

As I practice more and more with my Rubik’s Cube, it’s out-of-the-box buttery smoothness is breaking down. I think this is from a combination of dust and general wear and tear. In other words, the cube is getting harder to turn and sometimes locks up when I’m going really fast.

It’s possible that, as my cube gets slower, I’m getting generally faster, so the effects cancel out and I’m left with an unchanging time.

The potential solution: Lube.

Yes, Rubik’s Cube lube is a real thing and very popular amongst speed cubers. I’ve ordered some on Amazon and will put it to use once it shows up.

2. My brain is getting overloaded

Two months ago, when I was trying to learn how to memorize a deck of cards in less than two minutes, between days 7–10, I dipped to my worst level of performance. My brain had been subjected to so much new information over the previous week, and hadn’t yet caught up. Once it did, after a few days, I made a big leap forward in my progress.

It’s possible I’m experiencing a similar phenomenon now.

The potential solution: Don’t get discouraged and push through. If I don’t see positive changes in a few days, I should reassess this theory.

3. I’m practicing poorly

I’ve noticed that, during real solves, I ignore a lot of the techniques I’ve been practicing. For example, I’ve been working hard to reduce my cube rotationsand improve my look-ahead via two main directed exercises. But, when it’s time to apply these principles to an actual solve, I freak out, forget everything, and spin the cube frantically. This is clearly not good.

The potential solution: Combine my training exercises with actual solves, forcing myself to actively look-ahead and keep the cube positionally controlled.

4. Poor pre-solve planning

Before every solve, speed cubers are allowed 15 seconds to inspect the scrambled cube. During this time, it’s advised to make some sort of plan.

Typically, most speed cubers can plan out the entire cross and one F2L pair during inspection. On the other hand, I seem to only plan, on average, two of the four edge pieces necessary to form the cross.

As a result, my cross is still too slow, and my transition to F2L is usually terrible. After I finish the cross, I pause for maybe 1–2 seconds just to assess the situation.

This is not good.

The potential solution: Practice planning out the full cross during inspection. Also, practice tracking an F2L pair while executing the cross, so I can smoothly transition from the cross to F2L.

(Not sure what “the cross” or “F2L” are? Read my post about how I solve the Rubik’s Cube)

Validating these theories

When learning new skills, especially for speed, it’s always a good practice to form hypotheses about where the major inefficiencies lie, and then attempt to validate/invalidate these theories.

That’s what I plan to do over the next few days, which will educate my training moving forward.

Yesterday, I took some time to reflect on the Rubik’s Cube progress I’ve made so far this month. I mentioned that I wasn’t progressing at the pace I was hoping, and I presented four theories as to why I thought this might be.

Interestingly, when I started writing the article, I only had three theories in mind. The fourth came to me while I was proofreading.

I tested this fourth theory today, and shaved two seconds off my average solve time. It also reminded me something important about life.

The Fourth Theory: Poor pre-solve planning

Yesterday, here’s the theory I put forward:

Before every solve, speed cubers are allowed 15 seconds to inspect the scrambled cube. During this time, it’s advised to make some sort of plan.
Typically, most speed cubers can plan out the entire cross and one F2L pair during inspection. On the other hand, I seem to only plan, on average, two of the four edge pieces necessary to form the cross.
As a result, my cross is still too slow, and my transition to F2L is usually terrible. After I finish the cross, I pause for maybe 1–2 seconds just to assess the situation.

Today, as a follow-up, I attempted to better use my inspection time, which I expected I could improve over the next week with deliberate practice.

However, as soon as I started actively using my prep time, I was able to plan for the entire cross during inspection without a problem. I guess this is something I was already able to do (and was previously just being mentally lazy about).

The result: My solves were two seconds faster, on average, which is a nice bonus and a great help in my quest towards a 20-second solve.

It’s interesting that I didn’t think about this inefficiency earlier in the month.

A hiccup in my approach

Typically, the way I approach learning a new skills is as follow:

  1. Breakdown the entire process end-to-end
  2. Identify which parts of the process I’m worse at / are causing the most inefficiencies
  3. Identify which parts of the process offer the most upside growth potential
  4. Overlay these two lists to determine how to prioritize my practice
  5. Practice
  6. Repeat, as appropriate

However, when I was originally working through this approach, I was completely blind to “the inspection phase” as part of the end-to-end process. In fact, in the post where I deconstruct the Rubik’s Cube process, I never even mention inspection.

This was a big miss.

Why did I originally miss this?

I’m not actually surprised I missed inspection as part of my deconstruction.

I fell into a very common learning trap. Or maybe it’s just a life trap in general:

In life, we are often so driven to succeed at our goals, that we jump right to execution without taking a moment to plan for and setup our success. As a result, even though it feels like we’re moving faster, we’re actually going slower.
Usually, this is the result of mental laziness. It’s easier to distract our brains with “progress”, than it is to thoughtfully plan and reflect on our goals.

My eagerness to solve the cube forced me into this not-usually-so-optimal behavior.

Anyway, I plan to use all 15 seconds of my inspection time moving forward.

On Day 1 of this month’s challenge, I filmed myself solving the cube. My average solve time was 48.29 seconds.

Today, I filmed myself again (solving the cube 20 times). During today’s session, my average speed was 32.88 seconds, which represents solid progress.

Here are three of my best solves from today:

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