If you thought solving a Rubik鈥檚 cube was difficult, you were right and maths can back you up. A recent study shows that the question of whether a scrambled Rubik鈥檚 cube of any size can be solved in a given number of moves is what鈥檚 called NP-complete 鈥� that鈥檚 maths lingo for a problem even mathematicians find hard to solve.

To prove that the problem is NP-complete, Massachusetts Institute of Technology researchers Erik Demaine, Sarah Eisenstat, and showed that figuring out how to solve a Rubik鈥檚 cube with any number of squares on a side in the smallest number of moves will also give you a solution to another problem known to be NP-complete: the Hamiltonian path problem.

That question asks whether there is route that visits each vertex exactly once in a given graph consisting of a collection of vertices connected by edges, like a triangle, pentagram, or the vast connections in a social network such as Facebook.

It鈥檚 reminiscent of the travelling salesperson problem, which aims to find the shortest route that visits several cities only once, probably the most famous NP-complete question of all. 聽鈥淚t鈥檚 very clever. The description of how it works is very clean,鈥� says Jeff Erickson of the University of Illinois Urbana-Champaign.

NP-complete problems are easy to check, if you鈥檙e given a proposed solution, but the amount of time it takes to solve them explodes as the number of inputs goes up, at least with the algorithms we know about today. On the other hand, problems that have algorithms that run their course in a more reasonable amount of time based on the number of inputs are called P.

Researchers are still unsure whether algorithms exist that can solve NP-complete problems faster. This question, often called the P vs NP problem, is one of the most important unsolved maths problems and will net the solver from the Clay Mathematics Institute of Cambridge, Massachusetts.

So if you鈥檙e frustrated about how long it鈥檚 taking you to solve a Rubik鈥檚 cube, it鈥檚 not just you. 鈥淵ou have the excuse now: Rubik鈥檚 cubes are legitimately hard,鈥� Rudoy says. You鈥檙e not going to figure it out quickly, so you might as well sit back and enjoy the puzzle.

Playing God

Every standard 3x3x3 Rubik鈥檚 cube can be solved in a maximum of 20 moves from any starting position, no matter how scrambled. In 2010, programmers found that 20 is the colourful puzzle鈥檚 so-called 鈥淕od鈥檚 number鈥�, a name they selected to suggest that even a deity couldn鈥檛 solve a Rubik鈥檚 cube any faster.

A year later, Demaine, Eisenstat and their colleagues devised a formula to solve a Rubik鈥檚 cube with sides of any length and found that the number of moves required for a cube of side n is proportional to n²/log n.

Finding God鈥檚 number for a cube with n=3 took several years of computing time and Demaine estimates that the n=4 case would take billions of times longer. 鈥淚 conjecture it will never be fully solved,鈥� he says.

God鈥檚 number is the upper limit for the most scrambled cubes, but many cubes will not take that long. Figuring out whether any given configuration of a cube will take fewer moves is tricky. 鈥淲e know an algorithm to solve all cubes in a reasonable amount of time,鈥� Demaine says. 鈥淏ut if I give you a particular configuration of the cube, and then you want to solve it with the fewest moves for that configuration, that鈥檚 really tough.鈥�

Demaine started working on the question after seeing it in a post from Erickson on a computer science forum, although Erickson says the question had been floating around for a while. 鈥淚t鈥檚 nice to see it finally closed off,鈥� he says.

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