Have you ever wondered about the number of possible permutations that a 3×3 Rubik’s cube can have? Ever since its invention, people have been absolutely fascinated with the Rubik’s cube, which today has given rise to the culture of speedcubing. Almost everyone has attempted to solve a Rubik’s cube at least once in their lifetime. The attempt often ends in vain with only a jumbled mess of coloured cubies in no coherent order.
Back in the day, solving the Rubik’s cube was quite a challenge, and people weren’t even sure if a human could solve it. It took almost a month to find the first solution of the Magic Cube. Everyone was searching for it at the Budapest College of Applied Arts, where the inventor, Rubik Erno, was a professor back in 1974.
Before we can start calculating the permutations of the 3×3 cube, we need to first understand its structure.
Structure of the Rubik’s Cube
The 3×3 Rubik’s cube has six faces. Each face has a different colour – red, orange, blue, green, yellow and white. These colours are fixed. There are 12 edges and eight corners, which are movable. This is what causes the permutations of the Rubik’s Cube.
Now, we need to focus on and analyze the corners and the edges to find out the number of permutations.
Since there are eight corners, the number of ways to arrange these corners is 8! Or 40,320.
A corner is composed of 3 different colours. For a corner, the position of each colour is fixed relative to the other colours. Therefore, each corner has actually three different possible configurations like white-red-green, red-green-white and green-red-white.
You can also only orient seven corners independently. The orientation of the eighth corner will be done automatically based on the orientations of the other seven corners.
Therefore, the number of permutations arising from the eight corners is 8!x3⁷
Since there are 12 edges in the Rubik’s cube, the number of ways to arrange these 12 edges is 12! Or 479001600.
Each edge is made of two different colours and therefore can also have two different configurations. Just like the case of the corners explained above, only 11 of the edges can be oriented independently. The twelfth edge will be oriented on its own.
Hence, the number of permutations arising from the 12 edges is 12!x12¹¹
Rubik’s Cube permutation
Another important aspect of being considered is that the arrangements of the eight corners or the 12 edges cannot be swapped. In other words, we cannot swap two corners or two edges in isolation without affecting the neighbouring pieces. You will never come across a cube in a solved state with except only two of its edges or corners swapped.
But we have counted these impossible states; we will, in reality, have only half of the permutations calculated.
This essentially means that, the total number of possible permutations of the Rubik’s cube is:
(1/2) x (8!x3⁷) x (12!x2¹¹) = 43,252,003,274,489,856,000
That is 43 quintillion, 252 quadrillion 3 trillion 274 billion 489 million 856 thousand! What’s more interesting is that out of these 43 quintillion permutations, it is possible to achieve a Rubik’s cube solved state in just 20 moves or less! That’s why it is called God’s Number.
God’s Number is the least number of moves needed to solve the 3×3 Rubik’s Cube from any starting position. This number can be defined in many ways. The most common is in terms of the number of face turns required, but it can also be measured as the number of quarter turns. Whereas a quarter turn is either a positive or negative 900 turn, a face turn can be either of these or an 1800 turn.
Google donated 36 CPU-years of idle computer time. It solved every position of the Rubik’s cube in a minimum of 21 moves or less. This was the solution to the ‘superflip’ scramble in 20 moves.
The first estimation of God’s Number was 52 moves in July 1981. This number then gradually decreased to 42 in 1990, 29 in 2000, 22 in 2008, thereby reaching the final number of 20.
It is clearly a challenge to randomly twist and turn the Rubik’s cube and its many adaptations to try and solve them. Since the combinations are endless, it is necessary to know combination strategies, tips and tricks. There are many guides and tutorials that can help you learn to solve the Rubik’s cube in lesser moves. You can also learn from professional cubers and network with other players in the cubing world.
About the author
Writer at cubelelo.com. Her specialties include business research and content strategy. When she’s not working, she enjoys travel adventures and reading literary masterpieces.