Bright and early on the first Saturday in January, Tomas Rokicki and a few hundred fellow enthusiasts gathered in a vast lecture hall at the Moscone Center in downtown San Francisco, US. A big maths conference was underway and Rokicki, a retired programmer based in Palo Alto, California, US, had helped organise a two-day special session about “serious recreational mathematics” celebrating the 50th anniversary of the Rubik’s Cube. Erno Rubik, the Cube’s inventor, was top of the show at 8am, via video conference from the south of Spain.

Rubik, a Hungarian architect, designer, sculptor and retired professor, took part in a Q&A session with Rokicki and his co-organisers — Erik Demaine, a computer scientist at the Massachusetts Institute of Technology, US and Robert Hearn, a retired computer scientist of Portola Valley, California, US.

Rokicki asked Rubik about the first time he solved the Cube: “Did you solve
corners-first?”

Corners-first is a common route, since once the corners are solved, the edges can be slotted in with relative ease. Rubik said that, yes, he indeed did corners-first. “My method was understanding,” added Rubik, who is known to take a philosophical approach to cubology and life.

Cubitus Magikia

Rubik dates the Cube to the spring of 1974. Preparing a course on descriptive geometry and tinkering with the five Platonic solids, he had become especially taken by the cube. But, as he wrote in his 2020 memoir, Cubed, The Puzzle of Us All, for quite a while it “never once occurred to me that I was creating a puzzle”.

By the time of his 30th birthday, in July 1974, he had created the structure, realised its puzzling potential and — after playing with it intermittently for a few months — solved the Cube for the first time. He submitted a patent application in January 1975, and by the end of 1977 the “Magic Cube” had debuted in toy stores in Hungary. Travellers spirited it out “in their luggage, next to other Hungarian delicacies like sausage and Tokaji wine”, he recalled.

In March 1981, with the Cube having been renamed for Rubik and populating American toy stores, cognitive scientist Douglas Hofstadter diagnosed the craze as “cubitis magikia” — “a severe mental disorder accompanied by itching of the fingertips, which can be relieved only by prolonged contact with a multicolored cube”, he wrote in his column for Scientific American.

Complexity from Simplicity

After the session with Rubik, Rokicki gave a talk on mathematical aspects
of the Rubik’s Cube. He started with the fact that it scrambles into some 43 billion billion colourful combinations. “A reasonably big number,” he said, possibly more than all the grains of sand in the world.

Part of the puzzle’s appeal is the complexity that emerges from its simplicity. The Cube is composed of 20 smaller “cubies” (eight corners and 12 edges centered between the corners) and six face-centre pieces attached to the core. The core mechanism is anchored by a 3D cross, around which tabs on the edge and corner cubies interlock in a geometrically ingenious way that allows the structure to rotate.

The cubies display 54 colourful facets, nine each of white, red, blue, orange, yellow and green. In its solved state, the Cube’s six faces are configured such that all nine facets are the same colour. Turning the puzzle scrambles the colours — in total, there are precisely 4,32,52,00,32,74,48,98,56,000 possible positions into which the many facets can be permuted.

All the while, the puzzle’s essential form — its cubic-ness — remains unchanged. This feature demonstrates group theory, the mathematical study of symmetry: a so-called symmetry group of a geometric object is the collection, or group, of transformations that can be applied to the object but that nonetheless leave the structure preserved. A square has eight symmetries: it can be rotated or reflected four ways each and it’s still a square. A plain cube has 48 symmetries. The Rubik’s Cube has some 43 quintillion.

Cubic Encounters

There are many paths to solving the Cube. During his lecture, Rokicki zeroed in on a specific number: what is the minimum number of moves necessary to solve even the most scrambled positions?

Rokicki set out to calculate this quantity, known as God’s number, in 1999. In 2010 he found the answer: 20. He had the help of many talented people, particularly Herbert Kociemba, a German hobbyist cuber and programmer known for his namesake algorithm. The feat also benefited from a lot of computer time donated by Google, and another algorithm that took advantage of the Cube’s symmetries, reducing the number of necessary calculations by a factor of 48, and in turn reducing the necessary computing power.

Rokicki’s current obsession is identifying all of the God’s number positions — they are “extremely rare, really hard to find”, he said.

During the Q&A session, Rokicki asked Rubik about the hollow Void Cube, by Japanese inventor Katsuhiko Okamoto, who has created dozens of variants of the original. Somehow, the Void is missing the central cubies and the interior mechanics that hold Rubik’s iconic invention together. On this subject, Rubik got philosophical again. “Perfection is an idealistic encounter,” he said. He understood the curiosity-
driven explorations, adding something, taking something away. He preferred the classic combination of cubies and colours. “I love the sound of the Cube as well, the movement,” he said.

Rubik added later that he wasn’t so keen on puzzles that are designed merely to be puzzles. He said, “I love the puzzling content of life and the universe as it is.”

NYTNS