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The Mathematics of Rubik's Cube



Have you ever wondered why this Rubik's Cube which is often considered as a children's toy fascinates the mathematicians too? Surely because there is too much mathematics involved in it.

What if you take a Rubik's cube and separate every piece of it and then you close your eyes and start assembling it again in any preferred configuration. When you take the first corner piece you'll have 8 possible positions of putting it as there are a total of 8 corner pieces in the cube. Then for the second piece, you'll have 7 possible positions as one position has already been occupied by the first piece you have put. Now, for the third piece, you'll have 6 possible positions and so on until for the last piece you'll have only one position left. So when you multiply them it becomes 8x7x6x5x4x3x2x1 or rather simply 8!. And each piece can be arranged in 3 possible orientations, it gives us 3^8 possibilities for each permutation of the corner pieces.
Now coming to the edge pieces. There are in all 12 of them, giving us 12! ways of arranging them and in addition to it the two possible orientations of edge pieces give 2^12 possibilities for each permutation. Calculating all of the above expression gives us a total of 

 or simply 519 quintillion. Out of this 519 quintillion possible arrangements, not all permutations are possible, only 1 in every 12 of these are actually solvable. This is because there is no sequence of moves that will swap a single pair of pieces or rotate a single corner or edge piece. These 12 possible sets of reachable configurations are often called as 'universes' or 'orbits'.
Dividing this number by 12 it comes down to this number

which is approximately equal to 43 quintillion. This number is the maximum number of possible patterns in the cube that can be solved using the regular algorithms.
Mathematically a Rubik's cube is a Permutation Group because, in math, a permutation group is a finite group G whose elements are permutations of a given set and whose group operation is the composition of permutations in G.
There are certain simple properties of this mathematical structure but for understanding this you must first know a certain set of moves on the Rubik's cube.
When you move the rightmost layer of your cube 90 degrees in the clockwise direction, it's termed as the move R. The same move in anticlockwise direction is R'. Similarly, when you move the leftmost layer 90 degrees clockwise it's L and when anticlockwise it's L'. A 90 degrees turn of the uppermost layer is U and 90 degrees anticlockwise turn is U'. The same is for the bottom layer. 90 degrees clockwise turn is D and anticlockwise is D'. The front layer facing toward you when makes 90 degrees clockwise turn if F and when it turns anticlockwise it's F'. And the back layer facing away from you makes a move B by turning 90 degrees in the clockwise direction and a move B' in the anticlockwise direction.

The properties are-
  • Associative Property: This property states that the permutations in the row can be grouped in any way, for example, the move (RU)R' is exactly equal to the move R(UR').
  • Neutral Element:  which means that there is a permutation which doesn't rearrange the set, for example, when you perform a move R followed by the move R', you restore your initial configuration.
  • Inverse Element: which is quite intuitive this states that for every permutation there exists an inverse permutation, for instance, the move R has its inverse as R'
  • Commutative Property: which is rather not a property here because it's sometimes true and sometimes not like BF is exactly the same as FB but FR is not the same as RF.
Now the next important part the God's Number, don't worry it's not something related to god it's just a name given to it. This number is the maximum of the minimum number of mover required to solve any of the 43 quintillion combinations of the cube, and since July of 2010, we know that this number is 20. So every configuration can be solved in 20 moves or less. This has been given the name God's Number because it was initially thought that only a superhuman could do it so somehow related to god.
So that is all I had to tell you. If you want to dig deeper into the topic then do check the links I have provided below.



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