Mathematicians and hobbyists have had a half-century of fun exploring the 43 billion billion permutations of Erno Rubik’s creation.
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. A big math conference was underway and Rokicki, a retired programmer based in Palo Alto, California, had helped organize 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 videoconference 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 Massachusetts Institute of Technology, and Robert Hearn, a retired computer scientist, of Portola Valley, California.
Rokicki asked Rubik about the first time he solved the Cube: “Did you solve corners-first?”
These days, new cubers learn on YouTube, watching tutorials at 1.5x speed. Rokicki instead recommends the old-fashioned strategy: Set out on a lone path and discover a solving method, even if it takes weeks or months. (It took computer scientist Donald Knuth less than 12 hours, starting at his dining table in the evening and working straight through to the morning.) 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. Rubik, who is known to take a philosophical approach to cubology and to life in general, added: “My method was understanding.”
‘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 about the time of his 30th birthday, in July 1974, he had created the structure, realized 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.
One avid exporter and ambassador was David Singmaster, a mathematician who wrote the book Notes on Rubik’s ‘Magic Cube.’ Therein he outlined a notation for the faces — Up (U), Down (D), Right (R), Left (L), Front (F), Back (B) — providing a way to orient the Cube and refer to its pieces, positions and turns. He also gave a step-by-step solution guide. And he reported a hazard: Dame Kathleen Ollerenshaw, a British politician and recreational mathematician, had developed a case of “‘cubist’s thumb,’ a form of tendinitis requiring minor but delicate surgery for its relief.”
CubeLovers was among the first internet mailing lists — the inaugural message was sent by an MIT student in July 1980: “I don’t know what we will be talking about, but another mailing list cannot hurt (too much).” 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 multicoloured cube,” he wrote in his column for Scientific American. He added: “Symptoms often last for months. Highly contagious.”
By November 1982, the mania had subsided — Rubik’s Cube: A Craze Ends, declared a headline in the The New York Times. But it was resurrected in the 1990s by the World Wide Web. In 2023, Spin Master, the toy company that now owns the brand, globally sold 7.4 million units, including both the classic Cube and related twisty puzzles. Ben Varadi, a Spin Master co-founder, noted that Rubik’s has “95% brand awareness” — virtually everyone has heard of it. Rubik’s lore also holds that 1 in 7 people on Earth have played with the Cube. “It gives me hope about the world,” Rubik told his audience in San Francisco. “It brings people together.”
Complexity from simplicity
After the session with Rubik, Rokicki gave a talk on mathematical aspects of the 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 the 20 smaller “cubies” (eight corners and 12 edges centered between the corners) and six face-center 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 colorful 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 color. Turning the puzzle scrambles the colors — in total, there are precisely 43,252,003,274,489,856,000 possible positions into which the 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.
These symmetries are a “fantastic property,” Rokicki said, that “really gives the Cube its elegance.”
In much the same spirit, the recreational math gathering included talks about how to build an origami computer; the controlled art of juggling (versus “joggling,” uncontrollably chasing after balls); and enumerative problems in knitting.
Barry Cipra, a mathematician and mathematics writer, shared a wooden tray puzzle that he invented called the bricklayer’s challenge. The setup: four rows of six brick-like blocks of varying lengths. The goal: Arrange the bricks so that none of their vertical joints align between adjacent horizontal rows.
As Cipra spoke, several audience members rushed to the stage (upon his invitation) and got to work trying to find one of the puzzle’s 2,184 solutions. Among this subset of keeners were Bram Cohen, a computer programmer (and the inventor of BitTorrent, a file-sharing protocol) who devises Rubik’s-like puzzles, such as the Maltese Gear Cube (in collaboration with Oskar van Deventer); and Rivka Lipkovitz, a rising high school senior and speedcuber (official personal record in competition, 14.71 seconds; personal at home, 10.75).
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 told the audience. As he spoke, three computers in his home spun away on the task — their combined 336 gigabytes excavate about 100,000 distance-20 positions per day. So far, Rokicki has a database of about 100 million. “They are mathematical gems,” he said.
The Cube is also a good challenge for machine learning systems and robots.
And Maria Mannone, an Italian theoretical physicist and composer, invented the “CubeHarmonic,” a musical instrument, developed with Japanese collaborators. “It is a Rubik’s Cube where, on each face, there are musical chords, a note on each facet,” she explained in an email. “Scrambling the cube, we scramble musical chords.”
Parisian street artist Invader creates “Rubikcubist” works, figurative canvases configured like a mosaic with hundreds of cubes. Invader’s version of “Les Demoiselles d’Avignon,” Picasso’s first cubist painting, used 1,848 cubes in order to make a reproduction the same size as the original.
Lauren Rose, a mathematician at Bard College in New York, uses the Cube as a teaching tool in courses for both math majors (who delve into the algebra) and non-STEM majors (they learn to solve the puzzle, explore patterns, count its configurations, and design and build mosaics). “There’s so much depth to this puzzle,” Rose said at the conference in San Francisco. She believes that part of the reason that the Cube has endured is that it is “so accessible and fun.”
“It’s a good way to get people to want to learn math,” she added.
By now, all the Platonic solids have been transformed into twisty puzzle variants. And riffing on the original, there’s the 4-by-4-by-4 Rubik’s Revenge, the 5-by-5-by-5 Professor’s Cube and going on upward to the 7-by-7-by-7, the largest cube used in World Cube Association competitions. The 21-by-21-by-21 is the biggest cube generally available on the mass market. The 256-by-256-by-256 exists only in the virtual realm, where it was solved by a team of six with 633,494 moves in a cumulative time of about 96 hours.
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.”
This article originally appeared in The New York Times.
Written by: Siobhan Roberts
Photographs by: Akos Stiller
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