On the first Saturday in January, bright and early, Tomas Rokicki and a few hundred enthusiasts gathered in a huge conference room at the Moscone Center in downtown San Francisco. A big math conference was underway. Rokicki, a retired programmer who lives in Palo Alto, California, had helped organize a special two-day session on “serious recreational mathematics” to celebrate the 50th anniversary of the Rubik’s Cube. Erno Rubik, the cube’s inventor, headlined the show at 8 a.m. by video link from southern Spain.
Rubik, a Hungarian architect, designer, sculptor and retired professor, participated in a question-and-answer session with Rokicki and the other organizers: Erik Demaine, a computer scientist at MIT, 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?”
Today, those who join the Rubik clan learn on YouTube, watching tutorials at 1.5 times the speed. But Rokicki recommends the traditional strategy: setting out on a solitary path to discover a method for solving it, no matter if it takes weeks or months. (Computer scientist Donald Knuth took less than 12 hours, starting at night and working nonstop until morning at his dining room table.) Many start at the corners, so the edges can fall into place relatively easily. Rubik revealed that he had, indeed, made the corners first. Rubik, who is known for his philosophical approach to the cube (and life), added: “My method was to understand it.”
‘Cubitis magic’
Rubik dates the idea for the cube to the spring of 1974. He was preparing a course in descriptive geometry and, playing around with the five Platonic solids, became particularly taken with the cube. Still, as he wrote in his 2020 memoir, Cubed, The Puzzle of Us All, for quite some time “it never once occurred to me that I was creating a puzzle.”
By the time he turned 30 in July 1974, he had created the structure. He had realized its potential as a puzzle, and after playing with it on and off for a few months, he solved the cube for the first time. In January 1975, he filed a patent application, and by the end of 1977, the “Magic Cube” had debuted in some toy stores in Hungary. Travelers carried it “in their luggage, along with Hungarian delicacies like sausages and Tokaji wine,” he recalls.
An enthusiastic exporter and ambassador of the puzzle was David Singmaster, a mathematician who wrote the book Notes on the Rubik’s Cube. In it he laid out a notation for the faces – Up (U), Down (D), Right (R), Left (L), Front (F), Back (B) – which offered a way of orienting the cube and referring to its pieces, positions and turns. He also gave a step-by-step guide to solving it. And he denounced a danger: Dame Kathleen Ollerenshaw, a British politician and recreational mathematician, had presented a case of “‘cubist’s thumb’, a form of tendonitis which requires minor but delicate surgery for its relief.”
CubeLovers was one of the first mailing lists on the Internet: The inaugural message was sent by an MIT student in July 1980: “I don’t know what we’ll be talking about, but another mailing list can’t do (too much) harm.” In March 1981, after the cube had officially acquired the Rubik’s name and filled American toy stores, cognitive scientist Douglas Hofstadter diagnosed the frenzy as “magikia cubitis”: “a severe mental disorder accompanied by itchy fingertips, which can only be relieved by prolonged contact with a multi-colored cube,” he wrote in his column for Scientific American, adding: “Symptoms often last for months. Highly contagious.”
By November 1982, the craze had subsided: “Rubik’s Cube: The Furor Is Over,” declared a New York Times headline. But the Internet brought it back in the 1990s. By 2023, Spin Master, the toy company that now owns the brand, had sold 7.4 million units worldwide, including the classic cube and other versions. Ben Varadi, co-founder of Spin Master, notes that the cube has “95 percent brand awareness”—virtually everyone has heard of it. One in seven people on Earth is said to have played with it at some point. “It makes me feel hopeful for the world,” Rubik told his audience in San Francisco. “It brings people together.”
Complexity from simplicity
Following the session with Rubik, Rokicki gave a talk on the mathematical aspects of the cube. He began by explaining that it can create around 43 trillion colour combinations. “A pretty 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 arises from its simplicity. The Cube is made up of 20 “cubies” (eight corners and 12 edges centered between the corners) and six center pieces attached to the core. The core mechanism is anchored by a three-dimensional cross, around which the edge and corner cubes interlock in a geometrically ingenious way that allows the structure to rotate.
The cubes feature 54 coloured facets, nine of each colour: white, red, blue, orange, yellow and green. In their solved state, the six faces of the cube are configured so that all nine facets are the same colour. When the puzzle is rotated, the colours are mixed: in total, there are exactly 43,252,003,274,489,856,000 possible positions in which the facets can be permuted.
However, the puzzle’s essential shape—its cubicity—remains unchanged. This characteristic is demonstrated by group theory, the mathematical study of symmetry: the symmetry group of a geometric object is the set or group of transformations that can be applied to that object without altering its structure. A square has eight symmetries: it can be rotated in four ways or reflected in four ways and still be a square. A simple cube has 48 symmetries. The Rubik’s cube has about 43 trillion.
These symmetries are a “fantastic property,” Rokicki said, that “bring a great elegance to the cube.”
In a similar spirit, the recreational mathematics meeting included talks on how to build an origami computer; the controlled art of juggling (as opposed to joggling, which involves chasing balls around in an uncontrolled manner); and enumeration problems in weaving.
Barry Cipra, a mathematician and math writer, shared a wooden puzzle he created 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 are aligned between adjacent horizontal rows.
As Cipra spoke, several members of the audience rushed up on stage (at his invitation) and set to work trying to find one of the puzzle’s 2,184 solutions. Among them were computer programmer Bram Cohen (the inventor of BitTorrent, a file-sharing protocol), who designs Rubik’s-like puzzles such as the Gear Cube (in collaboration with Oskar van Deventer); and Rivka Lipkovitz, a high school senior and speedcuber (official personal record in competition: 14.71 seconds; personal record at home: 10.75).
There are many ways to solve the cube. During his lecture, Rokicki focused on one particular number: what is the minimum number of moves needed to solve even the most scrambled positions?
Rokicki set out to calculate this amount (known as “God’s number”) in 1999. In 2010, he found the answer: 20. He enlisted the help of many talented people, notably Herbert Kociemba, a German Rubik’s Cube aficionado and programmer best known for his eponymous algorithm. The feat was also made possible by a lot of computing 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, thereby reducing the computing power needed.
Rokicki’s current obsession is identifying all the positions of God’s number, because they are “extremely rare, really hard to find,” he told the audience. As he spoke, three computers in his home were busy at the task: Each day, their combined 336 gigabytes excavate about 100,000 positions within 20 moves. So far, Rokicki has collected about 100 million in a database. “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 a musical instrument called the “CubeHarmonic,” developed with Japanese collaborators. “It’s a Rubik’s cube where on each face there are musical chords, one note on each facet,” she explained in an email. “By shoving the cube, we are shoving the musical chords.”
Parisian street artist Invader creates “Rubikcubist” works, figurative canvases configured like a mosaic of hundreds of cubes. For his version of Les Demoiselles d’Avignon, Picasso’s first cubist painting, he used 1,848 cubes to make a full-size reproduction of 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 nonmath majors (who learn to solve the puzzle, explore patterns, count its configurations, and design and build mosaics). “This puzzle is very deep,” Rose said at the San Francisco conference. Part of the reason the cube has endured, she said, is that it is so “accessible and fun.”
“It’s a good way to get people to want to learn math,” he added.
By now, all the Platonic solids have been transformed into puzzle variants. Starting with the original, there is the 4-by-4-by-4 Rubik’s Revenge, the 5-by-5-by-5 Professor’s Cube, and then the 7-by-7-by-7 cube, the largest cube used at World Cube Association meetings. The 21-by-21-by-21 is the largest cube currently available on the market ($1,499.99). The 256-by-256-by-256 cube exists only in the virtual world, where it was solved by a team of six people in 633,494 moves in a cumulative time of about 96 hours.
During the Q&A, 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 core cubes and inner mechanism that holds Rubik’s iconic invention together. This got Rubik waxing philosophical again. “Perfection is an idealistic encounter,” he said. He understood curiosity-driven explorations — adding something, taking something away. But he preferred the classic combination of cubes and colors. “I also love the sound of the cube, the movement,” he said.
Rubik then added that he was not so fond of puzzles designed simply to be puzzles. “I love the content of the puzzle of life and the universe as it is.”
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