Mathematics of the Rubik’s Cube
Ernö Rubik invented the Cube in the spring of 1974 in his home town of Budapest, Hungary. He wanted a working model to help explain three-dimensional geometry and ended up creating the world’s best selling toy. More than 350 million cubes had been sold worldwide making it the world’s top-selling puzzle game.
Algorithms exist for solving a cube from any specific position it is currently in, but typically there is no optimal solution; how quick a cube may be solved depends on the individual who is solving the cube and how quickly they are able to deal with the cube. An algorithm is a list of well-defined instructions for completing one’s work from a given situation. During the initial steps of solving the cube, the algorithms effect on other “cubies” are insignificant; however, later on, the chosen algorithm typically destroys the rest of the cube. Many of these algorithms are available to everyone, and memorizing them is beneficial to everyone who participates in Rubik’s Cube competitions. In 1995, with a lot of practice and effort, Michael Reid used a method known with a table of cosets to prove that any cube may be solved in twenty-nine turns. In the past few years, another individual proved the method required fewer steps. The current world record for single time on a 3×3×3 Rubik’s Cube was set by Mats Valk of the Netherlands in March 2013 with a time of 5.55 seconds at the Zonhoven Open in Belgium
Just like a majority of real life situations, the cube also consists of a grand number of combinations and permutations that it can be twisted into. The number of possible positions - permutations - of Rubik’s cube is
That is 43 quintillion ways to possibly organize the cube. To put this into perspective, if one had as many standard sized Rubik’s Cubes as there are permutations, one could cover the Earth’s surface 275 times. If we stacked 43 quintillion pennies, the stack would be tall enough to reach the sun and return to the earth four thousand billion times. If you turned the Rubik’s cube once every second it would take you 1400 trillion years to go through all the configurations. If you had started this project during the Big Bang, you still would not have done it yet, since the universe is just 13.8 billion years old. The puzzle is often advertised as having only “billions” of positions, as the larger numbers are unfamiliar to many.
However, there is a problem; the 43 quintillion ways to organize the cube dramatically increases when you take into consideration the centre cubes. These six centre cubes are more difficult to get to the original orientation. If one marks it the very first time, they will realize that most of the time the centre cubes are positioned differently. That is why, when one distinguishes the beginning orientation, it comes out that there are more ways that the cube may be organized. Thus orientations of centres increases the total number of possible Cube permutations from 43,252,003,274,489,856,000 (4.3×10^{19}) to 88,580,102,706,155,225,088,000 (8.9×10^{22}).
https://www.youtube.com/watch?v=HodNg5d9cxk&list=TLql4bTwS7XPq4WO1W56U3ap70s4lj0hBx