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13 July 2020

The Mathematical Mystery Behind Rubik’s Cube

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Forty-three quintillion. Or more precisely, 43,252,003,274,489,856,000. This impressive figure is the number of tile combinations possible in the Rubik’s Cube (also known as the Magic Cube), the most famous puzzle of all times. Bearing this data point in mind, it is not surprising that the goal of the puzzle — positioning the tiles so that each of the six sides contains only one color — seems to be an impossible mission.

Rubik’s Cube takes its name from its creator, Hungarian sculptor, designer, and architect Ernő Rubik, born on July 13, 1944. In 1974, Rubik patented his creation in Hungary and in 1977 sales in the country began. Just a couple of years later, toy company Ideal negotiated an agreement to sell the cube worldwide. It rapidly became a success, winning recognition in Germany as the Best Game of the Year for the best puzzle category in 1980.

Despite the enormous popularity his cube received, originally Ernő Rubik had not set out to create a puzzle, rather he was researching a structural problem: how to move blocks independently without having to disassemble the cube. As he commented in an article published in 1987 in Discover, the idea of a puzzle based on colors came to him during a test when he was turning the cubes and he saw how they repositioned themselves differently each time.

God’s number

What is considered to be one of the world’s best-selling puzzles also represents a mathematical challenge to which various scientific studies have been dedicated. In 2019, researchers at the University of California successfully created an algorithm that used deep learning to efficiently solve the puzzle, making the least possible number of moves 60 percent of the times attempted.  Another aspect of the puzzle that has been studied is its mechanical engineering design.

El que se considera el juguete más vendido del mundo es también un reto matemático al que se han dedicado varios estudios científicos. Imagen: Zan (Unsplash)
What is considered to be one of the world’s best-selling puzzles also represents a mathematical challenge to which various scientific studies have been dedicated. Picture: Zan (Unsplash)

Above all else, the great unknown that intrigued mathematicians for decades was the minimum number of movements necessary to solve the puzzle from any of the more than 43 quintillion possible combinations, even the most difficult. This figure, known as “God’s number,” has been the subject of research practically since the invention of the cube, with the first attempt dating back to 1981. It was that year when British mathematician Morwen Thistlewaite used a complex algorithm to prove that the cube could always be solved with 52 moves or less. Over the course of three decades, several mathematicians took up the challenge of analyzing the cube and they progressively used new algorithms to reduce the minimum required number of moves: In 1990 it was 42; in 1995, 29; in 2008, 23. Finally, in 2010, a group of researchers proved that there was no initial position that required more than 20 movements to solve the puzzle, thus it was established: God’s number was 20

The odds of starting out making a move to a position that will solve the cube within these 20 moves is one in a billion. Among these unusual positions, there is the well-known “superflip”, in which the corners are correctly positioned relative to the center, but the edges are flipped. This was the first position that proved the puzzled could not be solved in less than 20 moves, back in 1995.

“Speedcubing” and the legacy of Rubik’s Cube

The original Rubik’s Cube soon gave rise to a number of variants: today there are many more challenges beyond the original 3x3x3 cube. From the most obvious developments — such as the “Rubik’s Revenge,” the 4x4x4 version of the cube invented by (the also Hungarian) Péter Sebestény and launched in 1981 — to the “Cuboku”, a hybrid Rubik’s Cube – Sudoku, launched in 2006 with the objective of ensuring each side is is a sudoku, having no repeated numbers. 

Huge player communities have arisen around the original Rubik’s Cube and its knock-offs. “Speedcubing” is the term used to denominate competitions around solving the Rubik’s Cube, or related puzzles, as quickly as possible. Its participants, the “speedcubers” play in official competitions regulated by the World Cube Association. This story has caught Netflix’s attention and will soon debut as a documentary featuring two of the most brilliant speedcubing competitors. 


Sara González for OpenMind

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