Skip to Content
MIT News magazine

Rubik's Cube Math

Solving a cube won’t take more than N2/log N moves
October 25, 2011

In 2010, an international team of researchers proved that no matter how scrambled a Rubik’s cube got, it would require no more than 20 moves to solve it. Their proof, however, relied on the equivalent of 35 years’ worth of number crunching on a good modern computer.

For cubes bigger than the standard Rubik’s cube, adequately canvassing starting positions may well be beyond the computational capacity of all the computers in the world. But in September, Erik Demaine (right), an associate professor of computer science and engineering, led a team including his father, CSAIL visiting professor Martin Demaine (left), that demonstrated the mathematical relationship between the number of squares in a cube and the number of moves in the shortest solution to its most scrambled state.

The standard way to solve a Rubik’s cube is to find a square that’s out of position and move it into place while leaving the rest of the cube as little changed as possible. That yields a worst-case solution whose number of moves is proportional to N2, where N is the number of squares per row. But the team saw that under some circumstances, a single sequence of twists could move multiple squares into place.

Describing those circumstances mathematically was no easy task. “In the first hour, we saw that it had to be at least N2/log N,” Erik Demaine says. “But then it was many months before we could prove that N2/log N was enough moves.”

Keep Reading

Most Popular

A Roomba recorded a woman on the toilet. How did screenshots end up on Facebook?

Robot vacuum companies say your images are safe, but a sprawling global supply chain for data from our devices creates risk.

A startup says it’s begun releasing particles into the atmosphere, in an effort to tweak the climate

Make Sunsets is already attempting to earn revenue for geoengineering, a move likely to provoke widespread criticism.

10 Breakthrough Technologies 2023

Every year, we pick the 10 technologies that matter the most right now. We look for advances that will have a big impact on our lives and break down why they matter.

These exclusive satellite images show that Saudi Arabia’s sci-fi megacity is well underway

Weirdly, any recent work on The Line doesn’t show up on Google Maps. But we got the images anyway.

Stay connected

Illustration by Rose Wong

Get the latest updates from
MIT Technology Review

Discover special offers, top stories, upcoming events, and more.

Thank you for submitting your email!

Explore more newsletters

It looks like something went wrong.

We’re having trouble saving your preferences. Try refreshing this page and updating them one more time. If you continue to get this message, reach out to us at customer-service@technologyreview.com with a list of newsletters you’d like to receive.