Rubik’s Cube and the Group Theory

Math Behind Solving A Rubik’s Cube

By Divvij Chandna

March 2017

A Rubik’s Cube is a puzzle that has stumped many geniuses for decades. First invented in 1974 by Erno Rubik, the puzzle is so difficult that it even took its inventor a month to figure out how to solve it. A Rubik’s Cube has 43 quintillion ways to scramble it. If you started drawing every permutation on a piece of paper and stacked them up, it would take 364 journeys to Pluto and back to complete the stack! In this entire stack there is only one paper with the solved cube drawn on it and it’s your job to find it. A 3×3 Rubik’s Cube is anyway one of the easier cubes to solve, the number of combinations of a 7×7 cube is around the number of atoms in 1080 universes.

If the Rubik’s Cube has so many permutations, how do the world’s fastest cubers solve them in under 5 seconds? A Rubik’s Cube is solved using a mathematical concept known as the Group Theory. According to  the Group Theory, the Rubik’s Cube group consists of  43 quintillion scrambles and all these elements have to follow certain rules or axioms which are closure, associativity, identity and inverse. Closure is the property that states if an operation is performed between two elements, the outcome will be a part of the set. In this case the operation being the movement of the faces of the cube. Associativity means that if elements a and b are operated and the outcome is operated with c, the same result is achieved when a is operated with the outcome of the operation between b and c. Identity is the element which can be operated with any element to give the element itself , which is the solved Rubik’s Cube in this case. Lastly, the inverse of an element is that element which will give the identity when operated with that element.

Cube notations

Using these properties of the Group Theory, a large number of cases can be removed from the set of all permutations by looking at the symmetry of various cases. Now, if a person attempts to solve the cube in a single step, it would be the fastest and most efficient method as it has been calculated that every case of a Rubik’s Cube can be solved in under 20 moves but this would require learning quintillions of cases, a feat that can only be performed by supercomputers at this time. So, we have to solve different parts of the cube in different steps which makes this puzzle easier but increases the number of moves required. The most popular method used by speed cubers today includes four steps but require memorisation of 78 algorithms for different cases. As the number of steps increase, the number of cases decrease, making the puzzle easier to solve. A Rubik’s cube can easily be solved by beginners in 7 steps by learning just 6 algorithms which isn’t a really difficult task.