Magical Patterns

Roger Antonsen

May 12, 2016

Scientists are constantly finding out more about it.

Rubik's Cube was invented in 1974 by Hungarian professor of architecture Ernő Rubik and is one of the most famous and iconic puzzles there is. Behind the fifty-four small squares of the cube – nine on each side – there is a lot of beautiful mathematics and computer science.

Although Rubik's Cube has been researched for over forty years, there are always new cube results.

Despite the fact that the cube is quite simple, it can be in over 43 billion billion different states. The exact number is 43 252 003 274 489 856 000. This number is so large that if one solves a hundred of these cubes per second, it would take about the whole age of the universe until now to solve all of them!

This great complexity makes it practically impossible to analyze the cube by going through all the different states – one by one – even with superfast computers. To answer basic questions more powerful tools are required: mathematics and computer science.

The British mathematician David Singmaster (1939–) was one the first mathematicians to analyze the cube in his book *Handbook of Cubik Math* from 1982. Singmaster is also known to have made the *notation system* for Rubik's Cube that is still in use today.

The basis for Singmasters notation system is that we use letters to name the six sides: **U**, **D**, **L**, **R**, **F** and **B** for *up*, *down*, *left*, *right*, *front* and *back*, respectively. If we flatten a cube that is white on top and green in front, we can see all six sides as follows:

One cube move usually consists of twisting on one of the six sides ninety degrees clockwiwe, and it is common to use letters to indicate which side gets twisted. Each move is denoted by a letter.

For example, **R** means that the right side of the cube is twisted ninety degrees clockwise, and if we turn it counterclockwise, we write **R '** instead. For example, the sequence **RUR'U'** is familiar to most people who have experimented with the cube. Here is **RUR'U'** step by step:

Whereas many only see a colorful toy, mathematicians recognize that the cube has the same structure as a so-called *group*. In *group theory*, which is part of abstract algebra, one studies the properties of groups. The cube group is a fine example of such a group.

Groups are frequently used in both computer science and physics as tools for representing and analyzing various types of symmetries. Using group theory, many fundamental questions about the cube may be answered.

For example, we can calculate the number of elements in the cube group, which is the same as the number of states the cube can be in, as follows:

In the video above you can see my colleague Vidar Norstein Klungre solve Rubik's Cube using one of the common ways to solve the cube quickly, called the Fridrich method, named after the Czech professor Jessica Fridrich (1963–).

The method is also called the CFOP method, where the letters stand for "Cross" (C), "First Two Layers" (F2L), "Orient Last Layer "(OLL) and "Permute Last Layer "(PLL).

*Read more next week about how quickly the cube can be solved and how many moves it takes to solve any cube!*