50 years of the Rubik’s Cube: can a tampered puzzle still be solved? | Science

In 1974, architecture professor Ernö Rubik invented a new tool to illustrate geometric concepts to his students at the Budapest School of Commercial Arts. Half a century later, the Rubik’s Cube has not only become one of the best-selling toys in history, but it has also inspired a global culture. Dedicated cube tournaments are held around the world, countless design variations have been created, and it has even raised questions of interest to mathematics researchers. For example, if the cube is tampered with – by peeling off stickers and rearranging them, or by taking apart and reassembling its pieces – can it still be fixed?

The study of the Rubik’s Cube uses a field of mathematics known as group theory. This framework allows us to describe the movements of the cube in abstract terms and, for example, to show that the original cube can still be solved — that is, each face can have a unique, unique color — in 20 movements or less, regardless of the initial color. arrangement. But what happens if the design is slightly changed?

To answer this question, we use the concept of legal configuration, which refers to any solvable state of the Rubik’s Cube. All legal configurations can be obtained from the solved cube by concatenating moves involving rotating one side of the cube by 90 degrees. Essentially, the setup can be reversed by retracing the steps used to solve it. There are a total of 43,252,003,274,489,856,000 legal configurations, and each of these configurations is an element of a mathematical structure that we call a group.

From this point of view, the previous question consists of verifying whether authorizing new movements, such as exchanging the colors of the cube pieces, would create new configurations which would escape the group of legal solutions. And if so, determine if these new configurations could still be resolved. In other words, could modifying the cube in this way create an unsolvable configuration? Or would this create another element of the group, a legal and resolvable configuration?

For example, if you peeled off the 54 stickers from the cube pieces and randomly stuck them back on, could it turn a difficult puzzle into an impossible puzzle? Rubik’s Cube experts will quickly recognize the answer: Legal cube configurations always follow certain rules, and those rules would be easily violated by peeling and re-sticking stickers. In other words, non-legal – intractable – states can be created by this method.

Concretely, in legal configurations, the different types of cube pieces respect strict placement rules. These parts fall into three categories: centers, edges and corners. The edges are made up of pieces that each have exactly two different colors. There are also impossible combinations, since opposite sides of the cube never share pieces.

In a classic Rubik’s Cube, the faces are arranged so that white is opposite yellow, green is opposite blue, and orange is opposite red. For example, if the top side is white, then the yellow should be on the bottom layer. This means that there can never be a white-yellow, green-blue, or orange-red edge, because opposite faces cannot share pieces. Likewise, the corners follow a similar rule and the centers of the cube must always maintain their original position relative to each other, because they do not move with rotations of the faces.

Now, if we modify the cube ensuring that the colors of the pieces still follow these rules of opposite faces – effectively disassembling and reassembling the cube, rather than simply swapping stickers – the question arises: will this configuration be still solvable? However, the answer remains negative. In fact, of all the possible modified configurations — which add up to a total of 519,024,039,293,878,272,000 possibilities — only one in 12 is solvable.

To perform this calculation, a concept from group theory called parity is used. Every move on the Rubik’s Cube, whether typical rotations or swapping pieces, can be thought of as a permutation of the 20 moving parts. Among these permutations, there is a special type called transposition, which involves swapping just two pieces while leaving the rest unchanged.

A permutation is considered even if it requires an even number of transpositions to carry it out, and odd if it requires an odd number. The notion of parity is essential to determine whether a Rubik’s Cube configuration is legal or not. By applying a simple criterion based on permutation parity (and a few other fundamental principles), it is possible to evaluate whether a given configuration is legal or not.

By applying this parity criterion, it is possible to identify all possible modifications of the Rubik’s Cube resulting from an exchange of pieces that are still solvable. This leads to the conclusion that 91.7% of tampered cubes, or cubes whose pieces have been swapped, can never be solved. The parity of permutations plays a crucial role not only in the Rubik’s Cube but also in other puzzles, such as the 15 puzzle, and in deeper mathematical questions, such as solving algebraic equations.

Yago Antolin is a professor at Complutense University of Madrid (UCM) and a member of IInstitute of Mathematical Sciences (ICMAT)

Silvia Centenera holds a degree in mathematics from UCM.

Agata Timon is the coordinator of the Mathematical Culture Unit of ICMAT.

Editing, translation and coordination: Agata Timón García-Longoria. She is the coordinator of the ICMAT program Mathematical culture unit.

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