How to solve a Rubik's cube in five seconds30 November 2015, by Geoff Smith

Credit: theilr, CC BY-SA

This week, 14-year-old Lucas Etter set a new worldrecord for solving the classic Rubik's cube inClarksville, Maryland, in the US, solving thescrambled cube in an astonishing 4.904 seconds.

The maximum number of face turns needed tosolve the classic Rubik's cube, one that issegmented into squares laid out 3x3 on each face,is 20, and the maximum number of quarter turns is26. It took 30 years to discover these numbers,which were finally proved by Tomas Rokicki andMorley Davidson using a mixture of mathematicsand computer calculation. The puzzle does have43,252,003,274,489,856,000 (43 times 1018, or 43quintillion) possible configurations after all.

So how do the likes of Lucas Etter work out how tosolve Rubik's cube so quickly? They could readinstructions, but that rather spoils the fun. If youwant to work out how to do it yourself, you need todevelop cube-solving tools. In this sense, a tool isa short sequence of turns which results in only afew of the individual squares on the cube's faceschanging position. When you have discovered andmemorised enough tools, you can execute themone after the other in order as required to returnthe 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 tofind tools. The trouble is that most experiments justscramble the cube horribly and you forget what youdid so you cannot undo your moves.

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

The final move of reassembly will be the reverse ofthe screwdriver trick: rotate one face 45 degreesand apply gentle pressure to put the final pieceback in place.

Sequences of moves of a cube form something thatmathematicians call agroup. If A is a sequence ofmoves, then let A-1 (that's "A inverse") be the samesequence of moves performed in reverse. So if youperform A and then A-1, the cube will be in thesame state as was it when you began. The same istrue if you first perform A-1 followed by A.

It’s a common problem. Credit: tangi_bertin, CC BY-SA

Now suppose that B is another sequence of moves.

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Many tools have the form of what mathematicianscall a commutator: do A, then B, then A-1 and finallyB-1. If A and B commute, so that performing A then B is the same as doing B then A, then thecommutator does nothing. From a mathematicalpoint of view, a commutator measures failure tocommute, and is a key notion in group theory.When I had a Rubik's cube in one hand, and ascrewdriver in the other, it was natural to look athow commutators behave.

Think of the overall structure of the differentconfigurations of a Rubik's cube as a labyrinth,which has that many chambers, each of whichcontains a Rubik's cube in the state whichcorresponds to that chamber. From each chamberthere are 12 doors leading to other chambers, eachdoor corresponding to a quarter turn of one of thesix faces of a cube. The type of turn needed topass through each door is written above it, so youknow which door is which. Your job is to navigateyour way from a particular chamber to the onewhere the cube on the table is in perfect condition.

The tools that you have discovered are ways ofgetting nearer to the goal. So you don't need toplan your route in advance, you just execute therotations of each tool so that you get steadily closerto and finally reach the winning chamber. Themathematical result in Rokicki and Davidson'spaper shows that, no matter where you are in thelabyrinth, it's possible to reach the winning chamberby passing through at most 26 doors – although theroute you find using your tools is not likely to bethat efficient.

How to put this to use to solve the cube in fiveseconds? Someone like young Lucas Etta who isinterested in speed solutions will not only havememorised a large number of tools, they'll alsohave practised them until they can perform it veryquickly. This is mostly a matter of dexterity andpractice, but it's also important to have a high-quality cube that can be manipulated smoothly andwith great precision.

Others, rather than going for speed, develop theskill of solving Rubik's cube while blindfolded orwith the cube held behind their back. In thecompetitive version of this variation, the solver is

given a limited amount of time to study thescrambled cube and plan their solution, before theyhave to carry out their solution from memorywithout looking at the cube again.

In terms of our metaphor of a labyrinth, thiscorresponds to all the Rubik's cubes in all thechambers being removed, except for the one onwhich you start. You can't take that cube with you,but you can study it carefully and plan your wholeroute to the winning chamber in advance. Quite afeat of memory, and not for those with just apassing interest in the cube.

This story is published courtesy of TheConversation (under Creative Commons-Attribution/No derivatives).

Source: The Conversation

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APA citation: How to solve a Rubik's cube in five seconds (2015, November 30) retrieved 25 August2018 from https://phys.org/news/2015-11-rubik-cube-seconds.html

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