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Solving Rubik’s Cube
When Rubik’s cube first came out in my late teens, I toyed with it for a while and then resolutely put it down.1 I knew it was the sort of bug that if it bit, would consume me to the point of affecting my studies. So I set it aside and shut it out of my mind for one day in the future. That day arrived two months ago, a few decades later. I was casually browsing through the Milnerton flea market when I saw a bundle of cubes in a basket at a stall and picked one up for ten rand.
It was really a cheap made-in-China cube that didn’t spin properly and fell apart after a few hours of twisting, so the next day I bought an official cube at a toy shop, and what a difference it made. 2 It pivoted smoothly which lessened frustration. Now solving the cube doesn’t require much intelligence. Rather, one has to persevere and above all, be systematic. The cube is solved through the methodical application of a set of algorithms which gets one through successive stages of its completion. My aim was to solve it in under ten minutes without referencing external aid.
I went about solving the first two layers on my own when I realised that that the marginal hour I would spend on solving the third layer was not worth the sacrifice of my limited leisure time. So I resorted to investigating solutions that others had independently found. I trawled the internet and found many clips on Youtube helping the cause, some of them overly technical. Eventually I settled on a clip by a brash twelve-year (?) old American boy, not short of confidence, but arrogantly good. He is Sergs B.3
Now a 3x3x3 Rubik’s cube has 43 quintillion possible arrangements, that is, combinations and orientations of its 26 cubelets in space. There are many approaches to solve it. It has been proved that any configuration of Rubik’s cube can be solved in maximally twenty moves, the so-called God’s number.4 That does not mean that a human being can solve any configuration in twenty moves; the large number of possible arrangements places the task beyond human mental capacity. A computer is required. The most efficient algorithm a human being can memorise to solve any scrambled state of the cube involves some forty moves.
Perhaps the best place to start was with the classic beginner’s method. This is essentially the layer-by-layer solution, inserting one cubelet in its place at a time for the first two layers followed by re-orientating and configuring the last layer.5 Another beginner’s method is the corners first solution. There are also methods which rely on intuition rather than memory, but these are more difficult, because you have to delve into the secrets of the cube and understand it thoroughly. The most popular speed-solving method is the CFOP method (bottom cross, first two layers solved two cubelets at a time, orientation of last layer, permutation of last layer).6,7
My interest in the Rubik’s cube was far from any thoughts of methodological finesse or even prowess. I simply wanted to solve any arrangement of the cube in less than ten minutes, the simpler the better. So it would be the beginner’s algorithm for me. Clicking on brash young Sergs B’s Youtube clip, I started grappling with the demonic cube.8 Essentially, he follows the beginner’s algorithm, solving the cube layer by layer. The method involves learning around eight or nine algorithms to be carried out in sequence. The key to many of the algorithms is that they make use of so-called commutators, i.e. moves that interchange or re-orientate the pieces you want by leaving the rest of the cube intact. In practice, the pieces of the rest of the cube actually move at some stage during the algorithm, but they are restored to their original slots after being shifted around. You’re left with a cube as it was before the algorithm, excepting that the target pieces have been moved or re-orientated.
I found the going tough at first, and confined myself to slavishly aping bouncy young Sergs’ moves on the screen. He is fast but clear. Pleasingly, the cube started to yield. After a few attempts and tortuous paths I managed to solve my first cube a month ago. Frustratingly, if you foul up on say algorithm 7 of the method, you slip down in your solution all the way to algorithm 2, and have to work your way up through all the algorithms to get back to where you were. So don’t. Rote following of algorithms soon became unsatisfactory, one soon yearns for intuition. So I set about trying to understand what the algorithms actually did.9 To this end I marked some of the cubelets and saw how they shifted around as I tortured the cube. I also made some sketches of the moves. I can now say that the intuition behind some of the algorithms of the beginner’s method has dawned. Over the past two months it’s been my curious pleasure to pick up the cube a few times a week just before bedtime. I am glad to report that I can now solve the cube without recourse to paper. I timed myself yesterday and solved a scrambled configuration in under ten minutes.10 Hurrah!
That is good enough for me and is as far as I’d like to take it, thank you. Yes it would be nice to understand other approaches to the cube’s solution and to see them in action, but the enigma of the cube has been set to rest in my being. I’m suffused with the quiet satisfaction of knowing.
Now onto the next thing. There’s so much wonder out there.
- “Rubik’s Cube was invented in 1974 by Hungarian professor of architecture, Ernő Rubik. A widespread international interest in the cube began in 1980, which soon developed into a global craze. On June 5, 1982, the first world championship was held in Budapest, Hungary” – Wikipedia. The standard cube has a 3x3x3 segmentation and consists of 26 cubelets. Its edges are 5.7cm long. The challenge is to resolve a scrambled state of the cube such that each side is of only one colour. The official website of Rubik’s cube is at http://rubiks.com/blog/how-to-solve-the-rubiks-cube.
- Rubik’s Cube is a trademark. Apart from standard cubes, there are special cubes for speedcubers which allow one to detach the pieces and assemble them in any configuration for practice. However, one has to take care when assembling such a cube. A fact about Rubik’s Cube is that if you take it apart and reassemble the cubelets randomly on the axes, there will be only a 1 in 12 chance of it actually being solvable by legal moves (that is, without taking the cube apart again). See http://www.ryanheise.com/cube/cube_laws.html.
- He is Sergs B and his website is The Sergs B Tutorials. See https://www.youtube.com/user/TheSergsB
- See http://www.cube20.org/. This is according to the face metric, where each twist of any face of the cube counts as one move. Mathematicians had steadily brought down the maximum proved number of moves to solve the cube from any configuration down from 51 in 1981 to 22 in August 2008. In June 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson and John Dethridge, using 35 CPU-years of idle supercomputer time donated by Google, brute forced every possible combination and proved that the maximum number of moves to solve Rubik’s cube from any configuration is 20. On the quarter turn metric, where each 90 degree twist is counted as one move, God’s algorithm is 26. [Note that these are existence proofs – they prove that a solution exists without saying what the solution is. Existence proofs are intuitive. For example, I can safely state that there are two people on earth with exactly the same number of hairs on their head. How? The average number of hairs on a person’s head is 100000 and there are over seven billion people on earth. There are therefore far more people than the number of hairs per person, so that there are bound to be many people having the same number of hairs on their heads. Now I don’t know what the exact number of total hairs it is that two or more people might share, and there might be a few people on earth that have a unique number of hairs on their scalps. But I know that some people will have the same number of hairs on their head by virtue of the surfeit of people over number of hairs per head. Musical chairs anyone?]
- The small components of Rubik s cube are called cubelets or alternatively cubies. There are six fixed cubes each of a different colour, eight moveable corner cubes each of three colours and twelve moveable edge cubes of two colours each.
- The CFOP method was popularized and improved by Jessica Fridrich of the Czech Republic who published it on her website in 1995. It involves solving the bottom layer cross, the subsequent insertion of two cubelets at a time to complete the second layer before solving the third layer in two steps. It is a more efficient method that the beginner’s algorithm but requires much more memorising. More efficient, but more taxing, therefore more advanced. See http://ruwix.com/the-rubiks-cube/rubiks-cube-solution-with-advanced-friedrich-method-tutorial/. CFOP has been the dominant 3x3x3 speed cubing method since around 2000. It and its variants are used by the vast majority of the top speedcubers.
- Speed cubing competitions have been around since 1982, when the first winner solved the 3x3x3 cube in 23 seconds. The world record for solving the 3x3x3 cube with a fifteen second inspection period beforehand stands at 6.5 seconds today. Speedcubers use specials lubricants on their cubes and must be extremely dexterous as well as algorithm savvy to be competitive. Many variants of the basic competition exist, e.g. blindfold solving, solving with the feet, one-hand solving and solving in the minimum number of moves etc.
- In particular, Sergs B’s solving video is at https://www.youtube.com/watch?v=MaltgJGz-dU
- In fact, there is a website (Ruwix) where you can electronically mirror your cube’s configuration by clicking on an on-screen cube, and the computer works out the moves in Rubik cube notation for you. You do so by entering the combination of your physical cube on the screen by clicking on a 2-D six paneled cube. See http://ruwix.com/online-rubiks-cube-solver-program/.
- I love getting people to scramble the cube for me and many oblige! Although not being able to solve the cube themselves, some have a rare talent for scrambling the cube to ready it for solution by others. It’s an art form in itself!