Early on the first Saturday in January, Tomas Rokicki and a few hundred other enthusiasts packed a large lecture hall in the Moscone Center in downtown San Francisco. A major math conference was in progress, and Rokicki, a retired programmer from Palo Alto, California, had helped organize a special two-day session of “serious recreational mathematics” to celebrate the 50th anniversary of the Rubik’s Cube. Erno Rubik, the Cube’s inventor, was the showstopper at 8 a.m., via videoconference from southern Spain.
Rubik, a Hungarian architect, designer, sculptor and retired professor, participated in a question-and-answer session with Rokicki and his co-hosts, Erik Demaine, a computer scientist at the Massachusetts Institute of Technology, and Robert Hearn, a retired computer scientist from Portola Valley, California.
Rokicki asked Rubik about the first time he solved the Cube: “Did you solve the corners first?”
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These days, new cubers learn on YouTube, watching tutorials at 1.5x speed. Rokicki recommends the old-fashioned strategy instead: Set out on a solitary path and figure out a solving method, even if it takes weeks or months. (Computer scientist Donald Knuth did it in less than 12 hours, starting at his dining room table in the evening and working straight through until morning.) Corners-first is a common route, because once the corners are solved, the edges can be slotted in relatively easily. Rubik said he did indeed do it corners-first. Rubik, who is known for his philosophical approach to cubology and life in general, added, “My method was understanding.”
‘Cubitus Magikia’
Rubik dates the cube to the spring of 1974. He was preparing for a course in descriptive geometry and tinkering with the five Platonic solids, and was particularly taken with the cube. But as he wrote in his 2020 memoir, “Cubed, The Puzzle of Us All,” for a long time “it never occurred to me that I was doing a puzzle.”
By his 30th birthday, in July 1974, he had created the structure, realized its puzzling potential, and—after playing with it on and off for a few months—first solved the Cube. In January 1975, he filed a patent application, and in late 1977, the “Magic Cube” debuted in toy stores across Hungary. Travelers carried it “in their luggage, along with other Hungarian delicacies such as sausage and Tokaji wine,” he recalled.
One ardent exporter and ambassador was David Singmaster, a mathematician who wrote the book “Notes on Rubik’s ‘Magic Cube’.” In it, he outlined a notation for the faces — Up (U), Down (D), Right (R), Left (L), Front (V), Back (B) — that would help him orient the cube and refer to its pieces, positions and rotations. He also provided a step-by-step solution guide. And he reported a danger: Dame Kathleen Ollerenshaw, a British politician and amateur mathematician, had developed a case of “‘cubist thumb,’ a form of tendonitis that required a minor but delicate operation to relieve it.”
CubeLovers was one of the first Internet mailing lists — the inaugural message was sent in July 1980 by an MIT student: “I don’t know what we’re going to talk about, but a new mailing list can’t hurt (too much).” In March 1981, by which time the Cube had been renamed Rubik’s and was populating toy stores in the U.S., cognitive scientist Douglas Hofstadter diagnosed the craze as “cubitis magikia” — “a serious mental disorder involving itching of the fingertips, relieved only by prolonged contact with a multicolored cube,” he wrote in his column for Scientific American. “Symptoms often last for months,” he added. “Highly contagious.”
By November 1982, the craze had died down — “Rubik’s Cube: A Craze Ends,” read the headline in The New York Times. But it was revived in the 1990s by the World Wide Web. In 2023, Spin Master, the toy company that now owns the brand, sold 7.4 million units worldwide, including both the classic Cube and the related Twisty puzzles. Ben Varadi, Spin Master’s co-founder, noted that Rubik’s has “95% brand awareness” — virtually everyone has heard of it. According to Rubik’s lore, 1 in 7 people on Earth has played with the Cube. “It gives me hope for the world,” Rubik told the San Francisco audience. “It brings people together.”
Complexity from simplicity
After the Rubik’s session, Rokicki gave a lecture on the mathematical aspects of the cube. He began by noting that the cube breaks down into some 43 billion billion colorful combinations. “A fairly large number,” he said, possibly more than all the grains of sand in the world.
Part of the puzzle’s appeal is its complexity born of its simplicity. The cube is made up 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, allowing the structure to rotate.
The cubes display 54 colorful facets, nine each of white, red, blue, orange, yellow, and green. In the solved state, the six faces of the cube are configured so that all nine facets are the same color. Rotating the puzzle confuses the colors — in total, there are exactly 43,252,003,274,489,856,000 possible positions in which the facets can be permuted.
Meanwhile, the essential form of the puzzle—its cubism—remains unchanged. This property is demonstrated by group theory, the mathematical study of symmetry: a so-called symmetry group of a geometric object is the set, or group, of transformations that can be applied to the object but that still preserve its structure. A square has eight symmetries: it can be rotated or mirrored in four ways and it is still a square. A regular cube has 48 symmetries. The Rubik’s Cube has about 43 quintillion.
These symmetries are a “fantastic feature,” Rokicki says, which “really gives the Cube its elegance.”
In a similar vein, the recreational math meeting included discussions on building an origami computer; the controlled art of juggling (as opposed to “joggling,” the uncontrollable chasing of balls); and list problems in knitting.
Barry Cipra, a mathematician and math writer, shared a wooden drawer puzzle he invented and 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 overlap adjacent horizontal rows.
As Cipra spoke, several spectators rushed to the stage (at his invitation) and got to work on finding one of the puzzle’s 2,184 solutions. Among the enthusiasts were Bram Cohen, a computer programmer (and the inventor of BitTorrent, a file-sharing protocol) who designs Rubik’s-like puzzles like the Maltese Gear Cube (in collaboration with Oskar van Deventer); and Rivka Lipkovitz, a high school senior and speedcuber (official personal best in competition, 14.71 seconds; personal at home, 10.75).
Cubic Encounters
There are many ways to solve the Cube. During his talk, Rokicki focused on a specific number: What is the minimum number of moves needed to solve even the most chaotic 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, most notably Herbert Kociemba, a German hobby cuber and programmer known for his eponymous algorithm. The achievement also benefited from large amounts of computer time donated by Google, and another algorithm that took advantage of the Cube’s symmetries, reducing the number of calculations required by a factor of 48, in turn reducing the required computing power.
Rokicki’s current obsession is identifying all the positions of God’s numbers — they’re “extremely rare, really hard to find,” he told the audience. As he spoke, three computers in his home were running at full tilt — their combined 336 gigabytes digging out about 100,000 distance-20 positions a day. So far, Rokicki has a database of about 100 million. “They’re mathematical gems,” he said.
The Cube is also a nice challenge for machine learning systems and robots.
And Maria Mannone, an Italian theoretical physicist and composer, invented the “CubeHarmonic,” a musical instrument developed with collaborators in Japan. “It’s a Rubik’s Cube with musical chords on each facet, a note on each facet,” she explained in an email. “By shuffling the cube, we shuffle musical chords.”
Parisian street artist Invader creates “Rubikcubist” works, figurative canvases composed of hundreds of cubes like a mosaic. Invader’s version of “Les Demoiselles d’Avignon,” Picasso’s first cubist painting, used 1,848 cubes to create a reproduction that was 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 algebra) and non-STEM students (who learn to solve the puzzle, explore patterns, count configurations, and design and build mosaics). “There’s so much depth to this puzzle,” Rose said at the conference in San Francisco. She believes part of the reason the Cube has endured is because it’s “so accessible and fun.”
“It’s a good way to get people excited about learning math,” she added.
By now, all the Platonic solids have been transformed into squiggly puzzle variants. And to riff on the original, there’s the 4-by-4-by-4 Rubik’s Revenge, the 5-by-5-by-5 Professor’s Cube, and on up to the 7-by-7-by-7, the largest cube used in World Cube Association competitions. The 21-by-21-by-21 is the largest cube generally available on the mass market ($1,499.99). The 256-by-256-by-256 exists only in the virtual realm, where it was solved by a team of six in 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 variations on the original. Somehow, the Void lacks the central cubes and internal mechanics that hold Rubik’s iconic invention together. On this subject, Rubik waxed 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 cubes and colors. “I also like the sound of the Cube, the movement,” he said.
Rubik later added that he was not so fond of puzzles that are just meant to be puzzles. He said, “I like the puzzling content of life and the universe as it is.”
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