Erik Demaine, an associate professor of computer science and engineering at MIT; his father, Martin Demaine, a visiting scientist at MIT’s Computer Science and Artificial Intelligence Laboratory; graduate student Sarah Eisenstat; Anna Lubiw, who was Demaine’s PhD thesis adviser at the University of Waterloo; and Tufts graduate student Andrew Winslow showed that the maximum number of moves required to solve a Rubik’s cube with N squares per row is proportional to N2/log N. “That that’s the answer, and not N2, is a surprising thing,” Demaine says.
The standard way to solve a Rubik’s cube, Demaine explains, is to find a square that’s out of position and move it into the right place while leaving the rest of the cube as little changed as possible. That approach will indeed yield a worst-case solution that’s proportional to N2. Demaine and his colleagues recognized that under some circumstances, a single sequence of twists could move multiple squares into their proper places, cutting down the total number of moves.
But finding a way to mathematically describe those circumstances, and determining how often they’d arise when a cube was in its worst-case state, was no easy task. “In the first hour, we saw that it had to be at least N2/log N,” Demaine says. “But then it was many months before we could prove that N2/log N was enough moves.”