Alex's Adventures in Numberland - Alex Bellos [103]
The most mathematically interesting cube-solving category is how to solve it in the fewest moves possible. The contestant is given an officially scrambled cube and has 60 minutes to study the position before describing the shortest solving sequence he can come up with. In 2009 Jimmy Coll of Belgium claimed the world record: 22 moves. Yet this was just how many moves a very smart human needed to solve a jumbled-up cube after 60 minutes of thinking about it. Might he have been able to find a solution from the same configuration in a smaller number of moves if he had had 60 hours? The question that has most intrigued mathematicians about the Rubik’s Cube is this: what is the smallest number of moves, n, such that every configuration can be solved in n moves or less? As a mark of reverence, n in this instance is nicknamed ‘God’s number’v hei>
Finding God’s number is extremely complex because the numbers are so large. There are about 43×1018 (or 43 followed by 18 zeros) cube positions. If every unique cube position were stacked on top of each other, the tower of cubes would go to the sun and back more than eight million times. It would take far too long to analyse each position one by one. Instead, mathematicians have looked at subgroups of positions. Tomas Rokicki, who has been studying the problem for around two decades, has analysed a collection of 19.5 billion related positions and found ways of solving them in 20 moves or less. He has now looked at a million or so similar collections, each containing 19.5 billion positions, and found again that 20 moves was sufficient for a solution. In 2008 he proved that every other remaining Rubik’s Cube position is only two moves away from a position in one of his collections, giving an upper bound for God’s number of 22.
Rokicki is convinced that God’s number is 20. ‘I’ve solved, at this point, approximately 9 percent of all cube positions, and none of them has required 21 moves. If there are any positions that require 21 or more moves, they are exceptionally rare.’ Rokicki’s challenge is not so much theoretical but logistical. Running through sets of cube positions uses an incredible amount of computer memory time. ‘With my current technique, I would need about 1000 modern computers for about one year to prove [that God’s number] is 20,’ he said.
Cube maths has been a long-term hobby for Rokicki. When I asked him whether he has thought of investigating the maths of other puzzles, like Sudoku, he joked: ‘Don’t try to distract me with other shiny problems. Cube math is challenging enough!’
Ernö Rubik still lives in Hungary and rarely gives interviews. I did, however, get a chance to meet one of his former students, Dániel Erdély, in Atlanta. We met in a room in the hotel devoted to ‘mathematical objects’. Origami models, geometrical shapes and elaborate puzzles were laid out on tables. Erdély was there looking after his own creations: light blue objects about the size of cricket balls, ridged with intricate, swirling patterns. Erdély treated them with the affection that a dog breeder has for a litter of his puppies. He picked one up, pointed to the palm-sized planet’s crystalline landscape and said: ‘Spidrons.’
Erdély, like Rubik, is not a mathematician. Rubik is an architect, and Erdély is a graphic designer who studied graphic design at the Budapest College of Applied Arts, where Rubik was a professor. In 1979 Erdély attended classes given by Rubik. As homework for these classes he devised a new shape made out of a sequence of alternate, and shrinking, equilateral and isosceles triangles. He called the shape a ‘spidron’ because it curved like a spiral. By the time he left university, spidrons were his obsession. He played around with them endlessly, noticing that they could fit together like tiles in many aesthetically satisfying ways, in both two and three dimensions. About five years ago, a Hungarian friend helped write a program to generate spidrons on the computer. Their tessellating properties have subsequently captivated mathematicians,