How can one solve a Rubik's cube without relying on guides/algorithms?

Question

I am wondering who is able to solve Rubik's cube just by thinking without knowing the rules behind it. As far as I know the inventor Ernő Rubik was professor of architecture and he intended to help students to learn 3D imagination. However from my point of view it is really ridiculous and probably to hard for most of us at least in reasonable time. For me personally it is really hard to memorize all implications of either move after 2 or even 3 turns.

What special logic or memorization techniques could be applied to solving a Rubik's cube without relying on guides or taught algorithms? In other words, how would one go about solving a Rubik's cube "blind", from a general mechanical understanding and first principles?

Answer 1

There are three techniques that allow you to come up with useful move sequences for solving a cube.

Conjugation

This is where you already have a move sequence that does one thing and allows you to apply it more generally.

Suppose for example that you know a move sequence X that twists the two adjacent corners at the front of the top layer (i.e. at UFR and UFL). However, maybe you want to twist to corners that are not adjacent. You might for example want to make a pretty pattern where two diametrically opposed corners are twisted (i.e. UFR and DBL). Then all you have to do is some setup move(s) to bring the two corner pieces you want to twist to the top front, apply the sequence X, and then undo the setup move(s).
In this example, X could be R'DR FDF' U' FD'F' R'D'R U.
The setup move to bring DBL to UFL is simply L2.
The resulting move sequence is then L2 X L2, or L2 R'DR FDF' U' FD'F' R'D'R U L2

This process is called conjugation. You conjugate the move sequence X so that you can apply the twists to any two corners of the cube rather than just the two at the top front.

Commutation

This technique allows you to come up with a useful move sequence such as X in the first place.

Suppose you are solving the top layer first, and have only one corner left to do. So the top layer is solved apart from the URF corner, and the corner piece that belongs there is below it at DRF. You can insert it in two ways - using the right face by R'D'R, or with the front face by FDF'. If you choose the wrong one, the corner will be put in its correct place but twisted. If you put it in with R'D'R and it was wrong, you can take it out again by undoing those moves (i.e. R'DR) and then using the front face instead (i.e. FDF').

From this we have a short move sequence R'DR FDF' that twists a single corner in the top face. Lets call it Y. Unfortunately Y is only useful on the first top layer, since it messes up the pieces of the other two layers.
This is where commutation comes in. Consider what happens when you do Y U' Y' U. The first Y twists the top right corner. The U' temporarily replaces that twisted corner by another. The Y' twists that replacement corner in the opposite direction. Finally the U puts the corners back where they came from.
The clever thing is what happens to the bottom two layers. It is not affected by the U' or U so it only "feels" the Y and Y' which cancel each other out. This means that everything is left as it was except for the two twisted corners.

A commutator is a move sequence of the form YZY'Z' where Y and Z are any move sequences you like. The magic happens when there are very few pieces that are affected by both Y and Z. For example in this case Z=U' affected only pieces in the top layer while Y affected mostly the bottom two layers except for one corner piece of the top layer.

So commutation fully explains how the move sequence X works.

There are various tutorials online about conjugation and commutation/commutators and how they apply to the Rubik's Cube and similar puzzles.

Repetition

Another technique that is mentioned less often but which can be very useful on occasion is repetition. Consider the commutation of two adjacent face moves, for example RF'R'F. The faces have 3 pieces that overlap, so the result of this commutator is not quite so simple. It affects 3 edges which are moved around in a 3-cycle, and affects 4 corners which are swapped in pairs.

Nevertheless this move sequence is very useful if you repeat it three times. By doing this the 3-cycle of edges are returned to where the started, so that only the 4 corners are affected.

tldr;

With these three techniques you can build enough useful move sequences to solve almost any twisty puzzle. The move sequences may not always be as short as they could be, but they will be easy to memorise because you know the underlying structure that they were built from and which explains how they do what they do.

Answer 2

Logic

Every method for solving a Rubik's Cube comes down to algorithms at some stage. These algorithms are a set of moves which you memorise and perform when a set of criteria is met. For example, the popular "T-Permutation" will swap two corners and two edges in a T shape and is used to solve the last layer in most methods.

For most people, they are able to complete one face of the cube. This can easily be achieved through trial and error, with a low level of thinking.

To complete a layer, one must place the corners and edges in such a way that three squares in each surrounding face are also complete:

Notice that the middle layer can be rotated so that all four centre squares are correct, too.

After this point, most people trying to do it themselves struggle. It is impossible to move an edge piece into its place on the middle layer without rotating one of the sides. This leads to the breaking up of the initial face, causing progress to be reset.

To place these edges an algorithm must be used.

It is possible to complete both the first and middle layer without using algorithms. This method is called F2L (First two layers) because it involves solving both layers simultaneously. I won't go into detail here, however in short it still leaves the last layer, which must be done using algorithms.

How does an individual "work out" how to solve one then, without looking up the algorithms?

Well, how did people find these algorithms in the first place? By playing with the cube and observing the effects that groups of moves have, it is possible for one to discover a set of moves which only moves a small number of parts as the end result. Once a set of algorithms is found, you must find a method which makes use for them.

For example: if you don't have an algorithm that moves corners, you will need to find a method that always results in a final phase which does not need to move corners. The "corners first" method achieves this, where the corners of the whole cube are put into place and the edges are done using edge-swap algorithms.

Summary

In short, it is possible to find a method yourself, however few people have been able to do this. All of those who have have done it through finding and using algorithms.

TL:DR

1. You need algorithms.
2. Most people can do one face.
3. Some people can do a layer, using observation, calculations, and logic.
4. It's possible to solve two layers without using algorithms; the final layer requires them.
5. Algorithms can be found through trial and error, recording results and comparing them.
6. Once a few algorithms are found, you must construct a method which uses them.
7. Very few people have done this, and I think that very few people have the patience.
8. TL:DR: Use algorithms. It takes time and patience.

Answer 3

I haven't solved the 3x3x3 Cube myself. I bought mine in 2010, and used a tutorial to solve it. In 2012 I started a twisty puzzles collection, and I currently have more than 350 puzzles in my collection. So even though I haven't solved the 3x3x3 Cube myself, I have solved quite a few other puzzles myself (after I gained knowledge, understanding and ideas from all the other puzzles I have).
Some of the puzzles I own are custom made, 3D-printed, or made by myself, so they are fairly unique, and only a handful of people in the entire world have these puzzles. Because of this, there also aren't any tutorials or solutions out there for some of these puzzles.

So how do I tackle those puzzles? How do I solve them myself? (Short disclaimer, I'm more of a collector than a solver, so I'm certainly no expert or good solver.)

1. When I have a new puzzle, the first thing to figure out is how it works of course. This may be a pretty obvious step, but sometimes a puzzle may work slightly different than it looks like on first look. I look at the moves that are possible, and how many (and which) pieces move around. I also look at how many times I have to repeat this before it's back where it was (like four times on a 3x3x3 Cube, five times on a Megaminx, three times on a Pyraminx, etc.).
2. Then I usually choose between two basic strategies:
   a. Solve by piece-type: I personally mainly use this method with puzzles I solve myself, because it's easier to grasp imho.
   b. Solve layer-by-layer: This is the method I've learned from the 3x3x3 Cube, and later also for other puzzles like the Megaminx, Pyraminx, etc.

3. Whether I solve by piece-type or layer-by-layer it really depends on the puzzle. When the puzzle is derived from another puzzle I already know, like the 3x3x3 Cube, I use a method as close to that as possible. A simple example: I would first reduce a 5x5x5 Cube to a 3x3x3 by creating 3x3 centers, and 1x3 edges, and then solve it as such. Or a bandaged 3x3x3 I would also solve as a regular 3x3x3 Cube and use algorithms and strategies from multiple different 3x3x3 methods I've learned (Beginner's Method, Cuboid, Triamese, own algorithms, etc.) to bypass the bandaged pieces and still solve layer-by-layer (although sometimes it can't be helped, and by piece-type is a better strategy for a certain bandaged 3x3x3).

   So let's say I solve per piece-type. What does piece-type mean, you say? When we look at a 3x3x3 Cube, there are three different piece-types: Centers, Corners and Edges. When we look at a Tangram Cube Extreme (which I recently figured out how to solve myself), we see six different piece-types. It's a bit too advanced and off-topic to explain what each piece-type is, or how the Tangram Cube Extreme works for now.. The important part is, when I've figured out how to solve the Tangram Cube Extreme, I noticed that it's best to start solving the middle-layer (piece-types F and G). Why? When I try to solve the other pieces, it's very easy to come up with short algorithms which keep the middle-layer (F & G) solved. Kinda like solving the second layer of a 3x3x3 Cube, without destroying the already solved first layer.
   After I had solved the middle layer intuitively, I came up with a three-cycle for the big B and C pieces. When I used this algorithm I came up with, I noticed it also moved the pieces D and E around, so therefore I knew I had to solve B and C first with this algorithm, before moving on to D and E.
   After all B and C pieces were correctly aligned with the corners (A) and middle layer (F & G), I came up with a new three-cycle for the pieces D, which wouldn't disturb/destroy the pieces A, B, C, F nor G (which we had already solved). And finally I used an algorithm I already knew from another puzzles to purely three-cycle the E-pieces to finish the solve.

Coming up with algorithms:

As I told above, I came up with three-cycle algorithms for the Tangram Cube Extreme. But how do you come up with algorithms?

Algorithms usually consist of four parts:
- It can be in the form of A B A' B'. Where both A and B are a single move or a short sequence, and A' and B' is undoing those moves or sequences. These kind of algorithms are usually used to move pieces around (three-cycles, five-cycles, double-swaps, etc.)
- It could also be in the form of A B A' B. These kind of algorithms are usually used to orient pieces without moving them.

Let's look at some algorithms we might know from the 3x3x3 Cube (if you know this Beginner's Method - NOTE that there are A LOT of 3x3x3 Cube Beginner's method. The only thing they have in common is that you solve them layer-by-layer, but apart from that the algorithms and steps can differ a lot):

• D' R' D R D F D' F': This algorithm is used to insert a middle-layer edge on a 3x3x3 Cube, using a layer-by-layer Beginner's Method. As you can see, this consist of D' R' D R and D F D' F', combining two A B A' B' algorithms to achieve what we want: solving a middle-layer edge piece, without destroying the first layer, nor any other middle-layer edge pieces we might already have solved.
• U R U' L' U R' U' L: This algorithm is used to three-cycle three top-layer corners on a 3x3x3 Cube, using a layer-by-layer Beginner's Method. This is again an A B A' B'-algorithm, with the following four parts: U R U'; L'; U R' U'; L.
• ((R' D' R D)2 U)3 U or R' D' R D (2x) used three times: This algorithm is used to orient the top-layer corners of a 3x3x3 Cube, at the end of the layer-by-layer Beginner's Method. This is an A B A' B-algorithm, where B are the U-moves, and A and A' are the R' D' R D (2x) (Note that R' D' R D is a sequence in the form of A B A' B' itself.)

So, that's the theory around algorithm types, but how do we come up with these ourselves? As I mentioned before, I'm no expert, but personally I always try to do some short four or six move sequences and see how it affects the piece-types in the puzzle. If it does something useful, like swapping pieces or creating a five-cycle, I make a commutators (A B A' B') where this sequence I came up with is either A or B, and I use some in-between setup moves to create three-cycles or double-swaps which I can use to solve the puzzle.
My big pitfall when I solve puzzles however, is that I sometimes use this entire 3-cycle I came up with for one part of the solve, as a sequence part (A or B) for another 3-cycle of the next part of the solve. This means I sometimes have algorithms of over 40 moves long to three-cycle or orient some of the final pieces. Pretty inefficient as you can imagine (although I'm still proud to say I've been able to solve some puzzles myself using these long-ass algorithms, which I came up with entirely on my own).

Difficulty of the puzzle

It also all depends on the difficulty of a puzzle, and prior knowledge. A 2x2x2 is obviously easier than a 3x3x3, and a 3x3x3 is easier than a 4x4x4. If you know how to solve a 3x3x3, you are automatically able to solve a 2x2x2 (because it's a corners-only 3x3x3). When you can solve a 4x4x4 (using a reduction method) you can automatically solve a 3x3x3 and 2x2x2.

Your question however, was how people solved a 3x3x3 Cube without any prior knowledge. Not much different than I solve new puzzles tbh. They try to come up with useful sequences and algorithms; analyze the most useful order to solve them in with the algorithms they come up with; and overcome challenges during the solve with cases they hadn't thought of, like parities on even-layered cubes of 4x4x4 or higher; or something as simple a middle-layer 3x3x3 edge that is already at the correct place, but should be oriented correctly / mirrored; or needing a double-swap of top-layer corners instead of a three-cycle. Both 'cases' are easily to overcome (by inserting a random piece in the middle-layer edge position to bump the other out, which you can then insert correctly oriented; or by just using the three-cycle of corners algorithm once, and then one of the four corners should be correct).

Still, I personally think it's impressive when people figured out how to solve a 3x3x3 Cube entirely on their own, without any prior knowledge. Chapeau to them for doing so. And it's always cool to see which method they've developed, and how they came up with it (like the video I linked of Tony Fisher in my comment of the question).

Answer 4

I can speak as someone who solved the Rubik's cube "without knowing the rules behind it". I solved it by myself, when I was in high school, with no instruction beyond watching some guy solve it one time. He explained I should do one side first, paying attention to the sides of the top layer of cubes, then move down to the middle layer and finally the bottom layer. He also pointed out that the centers don't move (I hadn't noticed yet.) That was all the information I had, and I did not know any of the principles behind the cube or the solutions. My approach was to look for sequences that restored most of the cube to its earlier state, and see if the changes could be useful. I was on summer vacation, and it took all my time over four or five days to figure out how to do it. You could do it too, if you had the patience and took the time.

Who can solve the cube on their own? It's hard, but you don't need to be a genius or an autistic savant to do it. Mainly it takes patience, and motivation. After doing some things over and over, they become automatic and you just start seeing the bigger picture. And with so much repetition, memorization is not really an issue; by the time you work out a way to do something, you'll know the sequence of moves it requires. In fact I am terrible at memorizing move sequences, or much of anything really. When a year or two later I got my hands on a cube tutorial, the steps were different to my own and I saw a couple of interesting shortcuts. Well, I never managed to memorize those new sets of moves. The ones I developed myself I still remember.

I'll admit I didn't work very hard at memorizing the sequences in the tutorial; where's the fun in that? To this day I am mystified that so many people are willing to just memorize instructions from a Rubik's cube tutorial, basically wasting a brilliant puzzle by looking up the answer.

Answer 5

I think there is almost a fundamental contradiction / false assumption in your question:

What special logic or memorization techniques could be applied to solving a Rubik's cube without relying on guides or taught algorithms? In other words, how would one go about solving a Rubik's cube "blind", from a general mechanical understanding and first principles?

namely that one solved the cube from first principles without using guides or taught algorithms.

I was a 17-year-old at school in the UK when the Rubik's Cube first came out, in an (old-style) A-Level class of 10 students, all of us doing "double maths" (Pure and Applied mathematics) plus another subject (i.e. all nerds who instantly loved the Rubik's Cube!)

At that time, there was certainly no internet, initially no published guides, and very little in the way of external resources for how to solve the cube. So, although there were few if any guides to help, it certainly wasn't true that we didn't use taught algorithms. The key, though, was that we taught ourselves the algorithms.

While there may be a very rare, very exceptional people who can solve the cube "holistically", it was almost immediately obvious that any methodical solving of the cube would need a "layered" approach -- you solve part of the puzzle (typically starting with one face) and then find combinations of moves that put further pieces in the correct place without (long-term) disruption of what had already been solved.

The key, then, was to find these "combinations of moves" that could put chosen pieces in chosen places. While performing these combination moves, the "already solved" portions would, naturally, get disrupted, but the point was that at the end of a successful / useful combination you would be left with everything you'd previously solved plus one or more new pieces in their correct positions.

Initially, this was very much trial-and-error and involved an awful lot of disassembly and manually returning the cube to its starting position -- it was much easier to see what a sequence of moves did when starting from a "clean" cube.

Eventually -- and being of a mathematical bent -- we discovered concepts such as conjugation, commutation and repetition that Jaap Scherphuis talks of in their answer.

More complex "sequences" could be created by combining two or more "simple" sets of moves -- sometimes noticing that the end of one simple sequence would "cancel out" the beginning of the next simple sequence (e.g. you might combine FDF' with FRF' -- the F' and F cancel-out leaving FDRF').

Of course, there was a trade-off: longer sequences of moves tended to be more "specialised" -- would only apply to particular states of the cube -- and the time taken to "recognise" when one could be used (not to mention the effort in remembering them all) began to take longer than using multiple "simple", more universal combinations.

Answer 6

By using group theory. Take a look at "Group Theory via Rubik’s Cube" (PDF) by Tom Davis on solving the Rubik's Cube using Group Theory. 59 pages and very technical but it outlines how to approach the cube without guides.

Why is group theory useful to solving a rubiks cube? Because it turns out that the Rubik's cube along with rotating it forms a group. Loosely put: Every rotation has an inverse and every permutation of the cube can be realized from every other permutation (i.e. no matter how many times you rotate the cube in arbitrary ways, as long as you do not take the cube apart and reassemble it there is always at least one way to rotate the faces so that the cube is solved again which should be somewhat obvious from the assumption that every move has an inverse since you can perform n moves and then just perform the inverses of those n moves in reverse order and be back at a solved rubik's cube) and thus we can study the rotations and how they effect the various colors of the cube using group theory. A crucial point of understanding how the Rubik's cube works is understanding that there are subgroups inside the Rubik's cube group and how the rotations and permutations of the different subgroups affect each other.

Answer 7

I firstly place the top corners, then bottom corners so I have all the crosses. Then I solve the top and bottom layers middle pieces simultaneously. Then I am left with the 4 middle layer peices which I solve with my longest algorithm.

The theory I went with was that every action was always going to have an equal and opposite reaction so my algorithms generally go in one direction to isolate a piece, then I make one move that changes that piece (and usually it will affect others) then I reverse the moves to put everything else back.

Answer 8

As myself who is still working on it to find my own ways, would like to put some of the ways that didn't work for me. To be precise it worked ONCE :)

1. Use 2x2x2 cube inorder to reduce the complexity. I believe you could step up once you are master with this.
2. Start with reverse engineer. This helps you to learn how cubes position themselves after 'n' moves (preferably on all 3 planes)
3. Next, solve your self with fixed moves. Say do 3 moves and solve it.
4. Number the faces so that it would 'aid' to get the directional perspective

Answer 9

1. Look at what is changing when I move a certain side or sides (just that it changes, not how it changes).
2. Notice that you can return everything to how it was if you just do the steps backward.
3. Notice that the first 2 steps are connected. I know this doesn't sound like helpful advice but trust me... it's a big one. A bit more help: undoing doesn't come right after mixing and the solved side can be used as a test zone for step 1.

Answer 10

It doesn't take too much effort to learn how to solve first two layers, and if you are lucky then the 3rd one will be solved as well. As a young teenager I used this method, and if 3rd layer had errors I would first fix those, and then "repair" the damage I did to the first two layers. With sufficient iterations and some luck you can get it solved. I was left with having everything in place but just two of the center-edge pieces would have to be flipped in-place, which I didn't know how to do before studying tutorials.

Answer 11

My method, valid for any kind of puzzle as goes as follow:

• Try to find what moves do what
• Always use moves like [(ABAB)*N][(ABAB)*N] so only rotate two faces, two corner or two edges, depending on how puzzle rotations are)

I talk about rotate face X, then face Y but not any other face on that four moves. Rotation direction can be different, for example ABA'B' or AB'A'B etc.

Then comes the long part... and a lot patience needed.

I use a lot of papers and I write each 'sticker' move per each of my moves.

For example, if you rotate 90º a normal 3x Rubik Cube (you a rotating a face) then you are moving 9+3*4 stickers, that is 21 stickers, but you are only moving 9 pieces (8 if you ignore the center of the face, warning some ones has multicolor on center so orientation comes important).

When I found a sequence (ABAB)*N that does something interesting, I copy it to solutions book as a 'tool'.

This method makes (with lot of time) you have a book filled with 'tools', some better than others.

Most times you get how to move pieces, so solve the puzzle by kind of pieces, not by layers, etc.

On 3x3 Rubik for example, you solve first all edges (ignoring all corners), then move to correct position each corner till all are in place, then set the correct orientation.

Just as an example of using only ABAB to circle three corners, using common notation will be: (R'FRF')*3 (RTR'T')*3. This one was the one I found yesterday and was the last 'tool' in ABAB challenge I was still needing to end auto-learn Rubik 3x3. Note: that one moves only three corners without affecting the rest, circle is: TFL->TBL->TBR->TFL

Yes, it is very long 24 moves, but each () is ABAB. It is my own challenge of always solve any puzzle by doing only ABAB kind of moves and find them by myself.

And yes, I know there are much faster solutions, but I use the challenge of solving doing only (ABAB) moves.

For some puzzles I also add (ABC) moves, not just two faces or two corner or two edges, I go for three in cascade.

And for *morphix (jumbling) and Flowercopter (it is a mix of corner and edge) I also use mid moves to solve 'jumbling'.

Minimal steps:

• Take note of what does what
• Try and error
• Copy out the 'tools' you find

As more 'tools' you find, you will get how to solve it, which kind of piece first, which next, etc.

For example, for the 3x3 Rubik's, if you have a 'tool' to circle 3 edges, but that moves corners, do not use the 'tool' that moves only corners prior to the 'tool' for edges, etc. The list of 'tools' will tell you what order of piece type.

I talk of 'tool's, a sequence of moves, not an algorithm, because it is fixed sequence, not a "do this if that".