There is no denying that the Rubik’s Cube is one of the world’s favourite puzzles and has been for a surprising 40 years and more. Many different methods have been created for solving it, and they can be found across books, tutorials and other resources. Professional SpeedCubers can solve a brain Cube or, at times, multiple Rubik’s Cubes in a matter of seconds.
There are many fascinating questions based on mathematics related to a Rubik’s Cube. And that’s what makes it a Brain’s Cube. A move is made on the Cube by rotating one of the six faces by either 90,180 or 270 degrees. There are a staggering 45 quintillion possible states. That can be obtained by applying sequences of moves to the solved state. This is what makes the Rubik’s Cube a challenge to solve, a source of great brain exercise.
The legacy of Rubik’s Cube
The original Rubik’s Cube gave rise to a number of variants. There are many more challenging cubes today that go beyond the original 3x3x3 Cube. From the most obvious developments like the ‘Rubik’s Revenge’ (the 4x4x4 version of the Cube invented by Péter Sebestény launched in 1981) to the ‘Cuboku‘ (a hybrid cube Sudoku launched in 2006), there has been an upsurge in the challenging solves of the variety of Rubik’s Cube.
Solving the Rubik’s Cube
The question of if a scrambled Rubik’s cube of any size can be solved in a certain given number of moves was recently stated as NP-Complete in research. For those who aren’t mathematical lovers, NP-complete is a math lingo for a problem that even mathematicians find hard to solve.
In fact, a few professors at MIT (Massachusetts Institute of Technology) took it on themselves to prove that the problem is NP-complete. The question to understand here was that if there is a path that crosses each vertex exactly once. This would happen in a graph that contains a collection of vertices that are connected by edges.
This is quite similar to a salesperson who aims at finding the shortest route, which visits several cities only once. The NP-complete problems tend to be easy to check if there is a proposed solution. However, the time taken to solve the problem also leads to an increase in the number of inputs. Which ultimately lead to a shooting increase in the number of algorithms.
Researchers are still unsure if there is an algorithm that exists which can solve this problem faster and indirectly help solve a brain Cube in no time. This definitely is amongst the most important unsolved questions. The Clay Mathematics Institute of Cambridge is offering the solver of this problem a reward of $1,000,000.
It is said that every standard 3x3x3 Rubik’s Cube can be solved in a maximum of 20 moves from any starting position. It does not matter how the Cube has been scrambled. In 2010, it was found that 20 is the so-called ‘God’s Number‘ by programmers. They chose this name to suggest that even a deity couldn’t solve a brain cube any faster.
However, a year later, in 2011, a formula was devised to solve a Rubik’s Cube with sides of any measure of length. It was found that the total number of moves required for a cube of side n is proportional to n2/log n.
Finding God’s number for a cube with n=3 took several years of computing time, and it is estimated that the n=4 case would take billions of times longer. God’s number is the upper limit for the most scrambled cubes, but many cubes will not take that long.
Wrapping it up!
Thinking about any problem too much and for too long can be daunting. And when it comes to solving the Rubik’s Cube, the endless algorithms and possibilities definitely make it a brain’s Cube. Solving a Rubik’s Cube requires innovative solutions at a grand scale, in addition to close attention to detail. At the same time, you should have an appreciation for the complexity and resolve to carry on.
No matter how complex the approaches to a Rubik’s Cube might seem, the successful approach will be based on a mind that is open to mathematical possibilities. The challenge is about learning a new skill and making the most of your resourceful mind. If you can already solve a Rubik’s Cube, there are many ways to make it harder for your brain. So, be bold, take chances and keep practising!