A New Puzzle Turns Earth Into a Rubik’s Cube, But More Complex

Another orbit around the sun and here we are again: back where we started but spun about — changed, perhaps deranged.

Henry Segerman, a British American mathematician and mathematical artist at Oklahoma State University, has invented just the puzzle for this disorienting annual event: Continental Drift, a 3-D sliding puzzle that made its debut earlier this year. The underlying geometric concept is holonomy: When you travel a loop on a curved surface and return to the starting point, you arrive somewhat turned around, rotated, perhaps by 180 degrees.

“Take a mathematical idea, can you make it real?” — this question, Dr. Segerman said, is what motivates his inventions.

He is keen on visualizing mathematics, whether with 3-D printing (he has written a book on the subject) or through non-Euclidean virtual reality experiences. But Dr. Segerman has aphantasia, an inability to construct mental pictures, or “visually hallucinate images at will,” as he puts it. This might explain his passion for making concrete pictures, especially the impressive collection he produced in 2022.

Continental Drift is Earth in miniature, mapped onto a truncated icosahedron — a soccer ball — with its regular patchwork of 12 pentagonal faces and 20 hexagonal faces.

The conceptual inspiration was a Victorian craze: the classic 15 Puzzle, wherein square tiles numbered 1 to 15 are scrambled on a 4-by-4 grid, with one square left empty; you solve the puzzle by sliding tiles around into numerical order.

In Continental Drift, a spherical version of the 15 Puzzle, it’s the hexagonal tiles that are scrambled. (The pentagons are recessed and remain stationary.) “One of the hexagons, this one in the South Pacific, comes out,” Dr. Segerman explains on his YouTube channel. “We can then activate the San Andreas fault and slide California south into the ocean. And we can keep going, mixing up all of the continents.”

Holonomy happens when a tile travels a full loop along the curved surface of the puzzle: Slide the tile featuring, say, Greenland all the way around the perimeter of a single pentagonal tile — perhaps the tile featuring the North Atlantic. After a complete loop, the Greenlanders return to their starting position rotated by 60 degrees. If the loop encompasses two adjacent pentagons, then the tile returns to the starting point rotated 120 degrees. And so on.

And as it turned out, the die can “roll along its path,” Dr. Segerman noted. Given the right slope, gravity and a nudge, the die wiggles along a perfect chronological countdown. “That was a surprise,” Dr. Segerman said. “Reality does tend to bite back.”

Nonetheless, for a rough measure of Continental Drift’s complexity, he calculated that it has 7 × 10³¹ states, or possible configurations. (The Rubik’s Cube, with roughly as many moving parts, has only around 4 × 10¹⁹ states.) A YouTube viewer calculated that exactly half of Continental Drift’s states are attainable.

To Dr. Segerman’s knowledge, only one person has solved Continental Drift so far. “I solve it by unscrewing the removable part of the frame that lets you take the tiles out,” he said. Then he reorients himself and the tiles, and screws the puzzle back together.

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