Last year, 14-year-old Lucas Etter set a new world record for solving the classic Rubik's cube in Clarksville, Maryland, in the US, solving the scrambled cube in an astonishing 4.904 seconds.

The maximum number of face turns needed to solve the classic Rubik's cube, one that is segmented into squares laid out 3x3 on each face, is 20, and the maximum number of quarter turns is 26. It took 30 years to discover these numbers, which were finally proved by Tomas Rokicki and Morley Davidson using a mixture of mathematics and computer calculation. The puzzle does have 43,252,003,274,489,856,000 (43 times 10^{18}, or 43 quintillion) possible configurations after all.

So how do the likes of Lucas Etter work out how to solve Rubik's cube so quickly? They could read instructions, but that rather spoils the fun. If you want to work out how to do it yourself, you need to develop cube-solving tools. In this sense, a tool is a short sequence of turns which results in only a few of the individual squares on the cube's faces changing position. When you have discovered and memorised enough tools, you can execute them one after the other in order as required to return the cube to its pristine, solved condition.

These tools require experimentation to discover. Here's how I did it myself: go on holiday with a Rubik's cube and a screwdriver. Do experiments to find tools. The trouble is that most experiments just scramble the cube horribly and you forget what you did so you cannot undo your moves.

Now you have a choice, either buy another Rubik's cube, or take out your trusty screwdriver. Turn one face through 45 degrees, and place the screwdriver under a central piece of the rotated face. Using the screwdriver as a lever to gently prise it out, it's then easy to take the cube apart completely and reassemble it in pristine form.

The final move of reassembly will be the reverse of the screwdriver trick: rotate one face 45 degrees and apply gentle pressure to put the final piece back in place.

It's a common problem. (tangi_bertin, CC BY-SA)

Sequences of moves of a cube form something that mathematicians call a
group. If *A* is a sequence of moves, then let *A ^{-1}* (that's "A inverse") be the same sequence of moves performed in reverse. So if you perform

*A*and then

*A*, the cube will be in the same state as was it when you began. The same is true if you first perform

^{-1}*A*followed by

^{-1}*A*.

Now suppose that *B* is another sequence of moves. Many tools have the form of what mathematicians call a commutator: do *A*, then *B*, then *A ^{-1}* and finally

*B*. If

^{-1}*A*and

*B*commute, so that performing

*A*then

*B*is the same as doing

*B*then

*A*, then the commutator does nothing. From a mathematical point of view, a commutator measures failure to commute, and is a key notion in group theory. When I had a Rubik's cube in one hand, and a screwdriver in the other, it was natural to look at how commutators behave.

Think of the overall structure of the different configurations of a Rubik's cube as a labyrinth, which has that many chambers, each of which contains a Rubik's cube in the state which corresponds to that chamber. From each chamber there are 12 doors leading to other chambers, each door corresponding to a quarter turn of one of the six faces of a cube. The type of turn needed to pass through each door is written above it, so you know which door is which. Your job is to navigate your way from a particular chamber to the one where the cube on the table is in perfect condition.

The tools that you have discovered are ways of getting nearer to the goal. So you don't need to plan your route in advance, you just execute the rotations of each tool so that you get steadily closer to and finally reach the winning chamber. The mathematical result in Rokicki and Davidson's paper shows that, no matter where you are in the labyrinth, it's possible to reach the winning chamber by passing through at most 26 doors – although the route you find using your tools is not likely to be that efficient.

How to put this to use to solve the cube in five seconds? Someone like young Lucas Etta who is interested in speed solutions will not only have memorised a large number of tools, they'll also have practised them until they can perform it very quickly. This is mostly a matter of dexterity and practice, but it's also important to have a high-quality cube that can be manipulated smoothly and with great precision.

Others, rather than going for speed, develop the skill of solving Rubik's cube while blindfolded or with the cube held behind their back. In the competitive version of this variation, the solver is given a limited amount of time to study the scrambled cube and plan their solution, before they have to carry out their solution from memory without looking at the cube again.

In terms of our metaphor of a labyrinth, this corresponds to all the Rubik's cubes in all the chambers being removed, except for the one on which you start. You can't take that cube with you, but you can study it carefully and plan your whole route to the winning chamber in advance. Quite a feat of memory, and not for those with just a passing interest in the cube.

*Geoff Smith, Senior Lecturer in Mathematics, University of Bath*

*This article was originally published on The Conversation. *

## Comments

I just recently - 2 weeks ago - learnt how to solve a Rubik's cube. What the article says about "tools" makes a lot of sense. It is not about randomly moving the pieces until you get to a solution. That is near impossible, as one move will impact other positions, and you are likely to muck things up somewhere if you do not know what you are doing.

It's not an easy thing to learn, but YouTube certainly helps. My first solves took over half an hour or more, with a lot of notes and a lot of mistakes. Now with a tonne of practice, at least half an hour a day, I average around the 2 minute mark. Not world record pace, but not bad for someone just starting.

Practice a lot. Try different videos, as some are easier to understand or more complex than others. Take notes and be patient. You are not going to learn 15 point algorithms in just 5 minutes on the first try. A good cube can certainly help to. The "Rubik's cube" tends to be quite hard to turn - it's designed to be hard to dismantle or modify so kids cannot tamper with it, but that makes it very stiff. A site like www.speedcube.com.au sells good stuff at a good price for Australians.

There is a Melbourne tournament held on March 19th and 20th. Check it out to see some good competition. Feliks Zemdegs, one of the fastest cubers and an Australian to boot, will be there.

Curbing can be challenging but addictive. Have fun! Any questions? Feel free to ask. :-)

Here is a link to the Melbourne tournament. Registrations have closed, but people are welcome to visit and spectate.

http://www.speedcubing.com.au/CubingClassic2016/

It might be nice for LH or Kotaku to cover a WCA (world cubing association) official tournament. Don't know if any of them are in Melbourne though. I think Sydney has an official tournament once a quarter, if that helps. :-S

"The maximum number of face turns needed to solve the classic Rubik’s cube, one that is segmented into squares laid out 3×3 on each face, is 20, and the maximum number of quarter turns is 26"

I'm pretty sure I can solve it using more than 20 or 26 turns.... :P

I think they mean minimum, not maximum. In theory, any layout on the Rubik's cube can be reached in that number of turns from a solved (all sides one colour only) position. Getting to that position though from one random layout to another for a human is likely to take more turns than that.

In case you are wondering, the world record for minimum number of turns - that being a person seeing a cube, not turning it but writing down the sequence of moves to solve it, is 23. Pretty amazing stuff. :-)

The point they were trying to make is that it's not as complex or intimidating as people think it is. :-P

Going back to the 80's, you could buy books that would lay out how to solve a rubiks cube. Was very traditional, and would show how to rotate the faces to get the one specific color needed from one spot to another. Very slow, but was a simple way to learn. Using that, I was able to get down to around 90 seconds.

These days, its more pattern recognition than anything, and sort of a 6 degrees of separation ideal. To get from a solved state to any random state, and vice versa, has a limited number of moves before it cross over itself, and thats where the speed solving comes into it.

Dont take the long way around, take the shortcuts. I'm well past being interested in solving the Rubiks Cube again, but its good to see that its still a thing.

It's not the minimum number of moves. Theoretically, even a randomly shuffled cube could be solved in 1 move.

It's a bit clunky, but correct to say the maximum number of moves

needed, even though most people will take many more.Probably the minimax (minimum maximum), or the best possible solution to the worst possible cube state.

Let me just make some further clarifying comments and clear up some common misconceptions.

It’s not about maths. You do not need a PhD in advanced mathematics or an IQ of 200 to solve a Rubik’s cube. That’s like saying you need to be a physicist to drive - it’s nice, but it is not necessary. Solving a cube is more about pattern and spacial recognition, the memorisation of algorithms – which are just set patterns of moves – and muscle memory. Many of the top cubers in the world are under 20 years old. You don’t need to be young either. You may not break a world record in your 50s, for example, but then you would not expect such an age to say break the 100m world sprint record either. Does that stop people from running? No. It’s a fun hobby, it’s not always about the records.Longer scrambles– semi random moves to put the pieces out of alignment –do not make the cube harder to solve. Eventually you will get to a point where if you make another move, you will bring the cube closer to being solved, not further away. Sure, depending on the solving method used, certain configurations are harder to solve than others, but random shuffling for a few seconds is generally as effective as shuffling for half an hour. Think of it like a deck of cards. You can only shuffle for so long before you start to see sequences of cards form again.You don’t need to work out your own algorithms or solutions. There are plenty of tutorials and theories developed already, especially on YouTube. You don’t need to reinvent the wheel. Sure, you can develop your own move sequence or tricks, but it’s not essential. If you are going to go down such a route, I’d recommend looking at tutorials of a basic solve for inspiration, work from a solved cube so you can see the impact of your moves more clearly, and take a lot of notes so you know what you just did. Having a slow but reliable solve method helps as well so you can fix up any random pattern you make which you cannot seem to fix. Again though, all of this is unnecessary to enjoy or solve the puzzle.You don’t need to do things from memory. That’s a good goal to have, and not a hard one if the methods you use are not too complex. Taking notes and re-watching videos is okay to learn. You don’t expect people to be able to have a conversation in a new language without taking notes to help them. It’s the same principle, it gets better with time but it is a new skill you are learning. Be patient with yourself.I mentioned practice. Half an hour a day can seem like a lot. Most of that time is not spent doing timed solves or learning new algorithms or methods every day. A lot of it is just “slow solving”, where one has a cube and just tries out moves, or gives it a random scramble and then attempts to solve it while doing something else, like commuting, or talking to a friend, or watching a show. You don’t need to focus on a cube all the time, at least after you have gotten the hang of a basic method. Think of it like knitting or Sudoku or playing with your phone – you don’t need your brain engaged all the time in those sorts of activities to enjoy them.Most importantly though,

give it a go, challenge yourself and do not be intimidated by it! :-DLast edited 07/03/16 12:00 pmI agree with all of these, i set myself a challenge to solve a rubiks cube a few years ago and eventually muscle memory just takes over memorising algorithms.

I have just learnt how to solve a 4x4 cube, not much harder but it's a bit more involved.

The parities and final edge solve always drive me nuts. A lot of videos I have seen on 4x4 suggest those are "intuitive". It's not. Never assume your audience automatically understands you. >_<;

Do you solve the 4x4 like a regular rubiks cube once the centres and edges are paired? If you do i'll find the couple of videos that i found helpful, the corner algorithm is quite easy but the edge one is a bit more involved but i just practised the moves on a scrambled cube, that way if you bugger it up you don't have to solve majority of the cube again.

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