43,252,003,274,489,856,000 combinations of a 3x3 rubik's cube...
We all are familiar with rubik's cube. It is one of the most famous puzzle and there are a lot of competitions held every year in which people try to solve different types of cubes some of which do not have a shape like a cube for example pyramid cube.
But we are interested in finding out that how many different combinations of a 3x3 rubik's cube possible (For simplicity, I will call 3x3 rukbik's cube as simply cube). To understand and to know the answer to this question we need to understand the mechanism of a cube, how it rotates and how these rotations cause different cube instances and what is the significance of each element of cube.
An element is a smallest unit of cube which cannot be further divided. For simplicity I will call these elements as "pieces". So there are three types of pieces in rubik's cube namely
1) Center pieces
2) Edge pieces
3) Corner pieces
Now we will see about the properties of each piece in detail
Center pieces
The pieces at the center of each face are center pieces.
They have a one single color on them;
There are six of them each on one face;
They do not move with respect to each other no matter how you try to rotate the cube, i.e their position is fixed or we can say they are pivoted.
Edge pieces
The pieces at the center of each edge of a cube are edge pieces.
They have a two different colors on them
There are 12 of them each on one edge;
They can move with respect to each other and there are 12 possible positions for each of them.
Corner pieces
The pieces at each corner of the cube are corner pieces.
They have a three different colors on them
There are 8 of them each on one corner;
They can move with respect to each other and there are 8 possible positions for each of them.
Now we are ready to calculate the different number of possible combinations or we can say in how many ways we can mix up the rubik's cube.
Center piece do not contribute in the combinations as they do not move.
Edge pieces
1st edge piece will have 12 choices in which it can be placed
2nd edge piece will have 11 choices in which it can be placed
3rd edge piece will have 10 choices in which it can be placed
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12th edge piece will have 1 choice in which it has to be placed.
So we times all the choices we get 12!.
Now each edge piece has 2 colors hence each of 12! combination can generate 2 different combinations so in total combinations are 12! * 2^(12)
Corner pieces
1st corner piece will have 8 choices in which it can be placed
2nd corner piece will have 7 choices in which it can be placed
3rd corner piece will have 6 choices in which it can be placed
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8th corner piece will have 1 choice in which it has to be placed.
So we times all the choices we get 8!.
Now each edge piece has 3 colors hence each of 8! combination can generate 3 different combinations so in total combinations are 8! * 3^(8).
Now the total no. of combinations produced will be the combinations formed by the edge and corner pieces and is equal to combinations produced by edge pieces times the corner pieces, 2! * (2^12) * 8! * (3^8) .
That expression gives us a total of 519024039293878272000, but this is not the right answer yet because it also includes the combination which cannot be reached which comes into picture when we either swap 2 corner pieces or flip edge piece in same position or flip corner piece on its position.
The point is that I we try to manipulate the cube we will get a whole new cube which is different from original one and which cannot be solved. People who have solved the cube may be aware of this fact.
So there are 2*2*3 = 12; cubes which can't be solved.
Note : 2 for swapping corner pieces, 2 for flipping the edge pieces and 3 for flipping the corner pieces.
Hence we divide the above number by 12 and get rid of those cube so finally we have
2! * (2^12) * 8! * (3^8)/12
which is equivalent to 43,252,003,274,489,856,000 or we can say 43 Quintillion.
You can use online calculator to check the calculation if you want to solve it by yourself
Calculator link : http://keisan.casio.com/calculator
Some facts about 43,252,003,274,489,856,000.
If you turned Rubik’s Cube once every second it would take you 1400 TRILLION YEARS to finish to go through all the configurations.To realize how big that number is, the age of our universe is 13.772 billion years.
So, If you had started this project during the Big Bang, you still wouldn’t be done yet.
If there was one cube scrambled for every permutation and they were laid end to end they would stretch approximately 261 LIGHT YEARS the extra permutations from assembling the cube in an unsolvable state - this number would be 12 times as big which I already explained.
Thanks for reading, have a good day!