Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host. And now, on to the math.

Do you remember the Rubik’s Cube, that 3x3x3 cube puzzle of multicolored squares, with each side able to rotate independently, that was a fad in the 1980s? The goal was to take a cube that has been scrambled by someone else with a few rotations, and get it back to a configuration where all squares on each side match in color. Originally developed by Hungarian architecture professor Erno Rubik in 1975, he had initially intended it to model how a structure can be designed with parts that move independently yet still hold together. The legend is that after he demonstrated the independent motion of a few sides and had trouble rearranging it back to the original configuration, he realized he had an interesting puzzle. In the early 1980s, it started winning various awards for best toy or puzzle, and quickly became the best selling toy of all time. (A title which it apparently still holds.) It was insanely difficult for the average person to solve, though typically with some trial and error you could get one or two sides done, the key to holding people’s interest. To get a sense of how popular it was back then, there were cube-solving guidebooks that sold millions of copies, and even a Saturday morning cartoon series about a cube-shaped superhero. But by 1983 or so sales were dropping off, and the fad was considered over.

Like everyone else who was alive in the early 80s, I spent some time messing around with cubes, but found it too frustrating after a while, and eventually solved it with the aid of a guide book. I remember being impressed by a classmate who swore he hadn’t read any guidebooks, but could take a scrambled cube from me, go work on it in a corner of the room, and come back with it fully solved. His secret was to remove the colored stickers from the squares and put them back in the right configuration, without rotating the sides at all. But the non-cheating solutions in the guidebooks typically revolve around identifying sequences of moves that can move around known sets of squares while keeping others in their current configuration, then getting the desired cubes in place layer by layer. These sequences are tricky in that they appear to be completely scrambling the cube before restoring various parts, which is a key reason why average cubers would fail to discover them— you need to mess up your cube on the way to completing the solution. It’s actually been mathematically proven that any reachable configuration can be solved in 20 moves or fewer.

One aspect of the cube that is always fun to mock is its marketing campaign. Typically the cube was sold and advertised with the phrase, “Over 3 Billion Combinations!” But if you think about it for a minute, there are a lot more. A cube has 8 corner pieces, so you could combine those in 8 factorial (8!) ways, which is 8 * 7 * 6 … down to 1. And since each of these combinations can have each corner piece in 3 different rotations, you need to multiply by 3 to the 8th power. Similarly, the 12 side pieces can be arranged in 12 factorial (12!) ways, then you need to multiply by 2 to the 12th power. So to find the total possibilities, we need to multiply 8! * 3^8 * 12! * 2^12. It turns out that only 1/12 of these positions are actually reachable from a starting solved state, so we need to divide the result by 12. But we still get a total a bit larger than 3 billion: 4.3 * 10^19. So the marketing campaign was underestimating the possibilities by a ridiculous amount— in fact, if you square the 3 billion that they gave, you still don’t quite reach the true number of cube configurations. Perhaps they were afraid that the terms for larger numbers, such as the quintillions needed for the true number, would confuse the average customer.

I was surprised to read recently that Rubik’s Cube actually has actually gained in popularity again in the modern era, fueled by daring YouTubers who speed-solve cubes, solve them with their feet, and perform similar feats. Lots of enthusiasts take their cubing very seriously, and there are Rubik’s Cube speed-solving championship events held regularly. If you’re a professional-class cuber, you can buy specially made Rubik’s lubricants to enable you to rotate your sides faster. The latest championship, this year in Toronto, included standard cube solving plus events on varying size cubes (up to 7x7x7), blindfolded cube solving, and one-handed cube solving. (Apparently they eliminated the foot-only solving, though, concerned that it was unsanitary.) Champion Matty Inaba fully solved a 3x3x3 cube in 4.27 seconds, which sounds like about the time it usually takes me to rotate a side or two. Author A.J. Jacobs, in “The Puzzler”, also points out that if you’re too intimidated to compete yourself, there are Fantasy Cubing leagues, similar to Fantasy Football, where you can bet on your favorite combination of winners.

So, what does the future hold for the sport of Rubik’s Cubing? Well, even though the big leagues are only competing up to the 7x7x7 level this year, Jacobs tracked down a French inventor who has put together a 33x33x33 one. As you would guess, the larger cubes are pretty challenging to build— this one involved over 6000 moving parts and ended up the size of a medicine ball. Experienced cubers do say, however, that the basic algorithms for solving are fundamentally the same for all cube sizes. Due to the home manufacturing capabilities enabled by modern 3-D printing, one of Jacobs’ interviewees points out that we are in a “golden age of weird twisty puzzles”. Hobbyists have invented many Rubik’s like variants that are not perfect cubes, and have numerous asymmetric parts, to create some extra challenge. Personally I’m not sure I would ever have the patience to deal with anything beyond a basic 3x3x3 cube, though maybe I’ll look into joining a fantasy league sometime.

And this has been your math mutation for today.

References:

- https://en.wikipedia.org/wiki/Rubik%27s_Cube
- https://www.worldcubeassociation.org/competitions/NAC2022
- https://ajjacobs.com/books/the-puzzler/
- https://web.mit.edu/sp.268/www/rubik.pdf