How hard is it to solve a Rubik’s cube? Can we quantify the answer? To answer these questions, we have to consider the number of rearrangements that can exist in a Rubik’s Cube. In order to understand this, you need to have an idea about permutations! Simply put, permutations are the number of different ways you can arrange a given set of items. Consider 2 letters (Say A and B). How many ways can you arrange them BOTH? It’s 2 (either AB or BA). If we take 3 letters, this becomes 6 ways (ABC, ACB, BAC, BCA, CAB, CBA). If the number of letters is 4, then this goes up to 24 possibilities. Mathematically, it can be proved that if the number of objects is N, then possible permutations is N! (N! = 1*2*3*4….*N.) For example, if there are 5 letters, the number of permutations is 5! = 1*2*3*4*5 = 120

Back to Rubik’s cube then. You can see the 8 corner pieces, which can be arranged in 8! ways. Each corner piece can be arranged in 3 orientations, and it gives us 3

^{8}possibilities for each permutation of the corner piece. In addition, there are 12 edge pieces, which can be arranged in 12! ways and since each edge piece has 2 possible orientations, each permutation of edge pieces has 2

^{12}arrangements. It can be shown that only ⅓ of permutations in corner cubes and ½ of permutations in edge cubes are correct. Additionally, only ½ of the permutations are correct, because we can’t permute only one cube and always have an even number of permutations. (This is called parity). Finally, the number of rearrangements that are possible in a Rubik's cube is:

(8! * 3

^{8 }* 12! * 2^{12}) / (3 * 2 * 1) = 4.3252 * 10^{19}To get an idea of how large this number is, imagine you making one move per second in a Rubik’s cube. It would take more than 1,000,000,000,000 years for you to achieve this number of moves!!

To solve a Rubik’s cube, do we really need to consider all the possible arrangements? No, we do not. Because whatever the arrangement is, it does not change the middle cube of sides. They are relatively fixed. So we know the color that belongs to each side, and when we consider each cube carefully, we can see that each cube is different from the others. That means each cube has a unique place. Now we can start finding the eligible places for cubes and put it with the correct orientation. You don’t have to be a magician to do that.

The easiest way to solve a jumbled Rubik’s cube is to take it apart, and put it back together. But instead, there are plenty of proper methods available online. For amateurs like us, the beginner’s method is recommended. After you get the hang of it, Rubik’s cube can be an endless fun activity, rather than a frustration. Now that you know how, it’s just a matter of when are you solving your first Rubik’s cube.

[1] "Rubik's cube invention: 40 years old and never meant ... - The Telegraph." 19 May. 2014, https://www.telegraph.co.uk/technology/google/10840482/Rubiks-cube-invention-40-years-old-and-never-meant-to-be-a-toy.html. Accessed 13 Aug. 2019.

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This article was sent in by Miraj Chamara Samarakkody. He is a graduate in Mathematics (First Class Honours), and currently working as an Instructor in Mathematics at Faculty of Engineering, University of Peradeniya. You can find him on LinkedIn

ReplyDeleteNice work ayya...🤘🤘

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