The Serious Whimsy of Recreational Mathematics at JMM 2024

They swarm the Rubik’s Cube booth, opening night at JMM 2024.

All hands are on deck, swirling and swooping and spinning cubes in an organized frenzy.

Once solved, each $3\times 3\times 3$ cube is stacked purposefully in a large, square wooden frame, creating an elaborate mosaic of Ernő Rubik’s namesake in honor of its fiftieth birthday.

More about that mosaic design later from event organizer Tomas Rokicki, who is too busy to talk now, running around, corralling “cubers.”

Scanning the buzzing scene is co-organizer Robert Hearn of H3 Labs and Gathering 4 Gardner, an educational nonprofit devoted to recreational math named for writer and polymath Martin Gardner.

Hearn observes the speedy solve rates of the 384 cubes with mock consternation. “I thought it would take longer, at least an hour or two,” he said. “Constructing that frame took me two days.”

In a mere 25 minutes, the mosaic is completed.

Figure 1.

At JMM 2024, cubers solve twisty puzzles.

Seemingly whimsical topics such as toys, games, and puzzles can lead to significant discoveries. Recreational mathematics has inspired findings in probability and graph theory, and the aperiodic monotile of 2023 known as “the Hat.”

This thinking underscored JMM 2024’s Serious Recreational Mathematics special sessions, organized by Erik Demaine of the Massachusetts Institute of Technology (MIT), along with Rokicki and Hearn. Demaine, who met Rokicki at a “Gathering 4 Gardner,” was co-advisor to Hearn’s PhD from MIT.

“I got interested in recreational mathematics through my father [MIT mathematician Martin Demaine], who got interested in it from Martin Gardner’s ‘Mathematical Games’ column in Scientific American,” Demaine said. “In the late 1990s when I was a new PhD student, we started working in the field of computational origami, which was just getting started, and its motivation was essentially recreational: How to design cooler origami art. Later it turned into a practical field with many applications to science and engineering.”

“Ever since, I’ve been motivated by mathematical problems that I find fun—be it origami or video games or puzzles.”

To celebrate the Rubik Cube’s fiftieth anniversary, Demaine’s magician friend Mark Mitton suggested organizing a panoply of events in 2024. “A JMM special session seemed like a great venue to get more mathematicians excited about recreational mathematics,” said Demaine. At JMM, he presented a talk called “Puzzles and Games Meet Algorithms and Complexity,” while Hearn discussed “The Puzzling Origins of Compound Symmetry Groups” and Rokicki presented “Twenty Moves Suffice for Rubik’s Cube.”

Let’s get back to that completed mosaic: a colorful cube with what appears to be an aesthetically appealing, random pattern.

Figure 2.

The Rubik’s Cube mosaic at JMM 2024 was composed of 384 $3\times 3\times 3$ cubes.

Not so fast, according to Hearn.

The pattern created by the 384-cube mosaic is the so-called “superflip,” where every one of the 12 edge pieces on the cube is flipped. “Tom (Rokicki) chose this pattern, I suspect, because it is one of a very small number of patterns that requires a full 20 moves to solve, as he was the first to prove,” Hearn says.

In 1999, Rokicki made a bold claim that he would determine the maximum number of moves required to solve Rubik’s Cube, referred to as “God’s number.” This required analyzing solutions for all 43,252,003,274,489,856,000 possible positions of the cube. Nearly 10 years later, “with help from some very bright people and a donation of computer time from Google,” Rokicki ultimately determined God’s number to be 20.

Rokicki corrects Hearn, giving credit where due. “Michael Reid was the first to show that superflip required 20 moves and it was the first position that was shown to require 20 moves. My result was that this is a maximum, that every position can be solved in 20 moves or less.”

Serious Recreational Mathematics special-session topics ran the gamut. Donald Knuth of Stanford University spoke of “Recreational Computer Programming” while Susan Goldstine of St. Mary’s College of Maryland discussed “Counting Stitches: Enumerative Problems in Knitting.” Martin Demaine presented “Fun with Fonts: Algorithmic Typography.” Ryuhei Uehara of the Japan Advanced Institute of Science and Technology presented his research on common shape puzzles, while Klara Mundilova, a recent PhD graduate from MIT advised by Erik Demaine, discussed “Art-Inspired Curved-Crease Origami Analysis and Design.”

“We were fortunate to be able to recruit a very high-quality selection of speakers, and I’d call the session an unqualified success,” said Hearn, citing “much larger attendance than we could have imagined. It seems possible this could become a regular JMM session going forward, as the Mathematics and the Arts sessions have been for a while.”

Figure 3.

Steve Butler of Iowa State University explains the mathematics of juggling.

Mundilova added, “Hearing how accomplished researchers playfully apply various disciplines of mathematics to recreational areas, such as puzzles, games, art, origami, juggling, tilings, and knitting, was informative, entertaining, and very inspiring. I would definitely attend a Serious Recreational Mathematics session again.”

In a session called “The Mathematics of Discrete Periodic Patterns …or How I Learned to Stop Worrying and Love the Throw,” Steve Butler of Iowa State University used colorful round props.

“Juggling is about patterns and we can apply math to juggling by exploring these patterns,” Butler said. “This can mean discovering new patterns, connecting patterns together, and finding the limits of what is possible.”

In their talks, the presenters used humor to good effect. As Butler told the crowd, “Every math talk should have one proof and one joke. They should not be the same thing.”

For the three session organizers, a pinnacle of Serious Recreational Mathematics was the Saturday morning interview with Ernő Rubik himself: via Zoom, with the Hungarian inventor’s image splayed huge across screens like the Wizard of Oz.

“It was great talking with Ernő!” Rokicki said. “I thought he had some great answers and stories!”

Figure 4.

Ernő Rubik, via Zoom, at JMM 2024.

“Really, Rubik shaped a lot of the direction of my life. To be able to ask him questions for an hour was really something special,” noted Hearn, who did his PhD on games and puzzles. Erik Demaine added that they had hoped to reveal something of Rubik the person behind the cube.

The three interviewers covered a lot of ground with their questions. “Do aliens have Rubik’s Cubes? What is the meaning of life? Why did the $3\times 3\times 3$ appear before the $2\times 2\times 2$? What was it like living behind the Iron Curtain as the cube spread throughout the world? and much more,” Hearn said.

Mundilova attended the 8 a.m. interview session. “It was very interesting to hear about the thought processes and the historical contexts of the development and design of the Rubik’s Cube,” she said.

“I find it fascinating how this puzzle has captivated people across generations. My parents enjoyed solving the Rubik’s Cube decades ago, despite it initially being a rare and sought-after item in Czechoslovakia. I find it remarkable that the same concept, though now in various forms, continues to engage and challenge people today,” she said.

“The Rubik’s Cube has been a wonderful way to get the general population to experience math without telling them that is what they are doing,” Butler said.

“Playing with the Rubik’s Cube makes people think about algorithms, symmetries, and group actions,” he said. “As a result, the Rubik’s Cube has been a wonderful tool for the math community.”

But Butler chose a different highlight for his JMM experience this year.

“The best part was getting communities back together again. It felt like we were back to normal.”