Candy Crush is a phenomenon, if only for the amount of time the world has wasted on it. It seems to be a simple game but judging by the the number of downloads it has a quality that makes it addictive. Now we know why - it's NP-hard.
Toby Walsh at the University of New South Wales, Australia has a proof that puts Candy Crush in its place and might even suggest ways to make the game better.
If you don't know Candy Crush then you can't be one of the estimated half a billion downloaders at Facebook or on iOS or Android. It is a variant of a class of match-three games. The game that was analysed has a board that isn't fixed in size, but otherwise it is the same game:
The board is filled with a selection of candies that come in six different colors. The challenge is to swap over neighboring candies to create runs of three of the same color. When you get a set of three of one color they disappear and the candies above fall down to replace them. You score one for each set of three you make disappear.
The challenge is to achieve a set score s in k swaps.
It is fairly easy to show that the problem is in NP. If you are given a set of k swaps that scores s then you can check that this is so in polynomial time and hence the problem is in NP.
In his paper, Walsh explains how to map the 3-SAT problem, which is of course known to be NP Complete, in polynomial time, to Candy Crush, which proves that Candy Crush is at least NP-hard.
NP-hard problems are roughly speaking at least as hard as the hardest problems in NP. The reduction from 3-SAT follows the usual route of creating "gadgets" that convert the game to a set of Boolean functions.
From here the paper goes on to consider some of the modified game play in later levels of Candy Crush, such as having layers covered up - these too are NP-hard.
The final question concerns whether there are multiple solutions to a given Candy Crush problem. Presumably the game is better to play if there is just one solution and so it would be nice to find out if a given problem has a unique solution. It turns out that this problem is co-NP-hard - that is proving that there are many solutions is NP-hard.
Practical thoughts on Candy Crush?
NP-hard is a worst case classification. How can we vary the difficult of Candy Crush problems? Could there be a "phase change" in parameters that turn easy problems into hard ones? The author of the paper has a nice idea:
"Many millions of hours have been spent solving Candy Crush. Perhaps we can put this to even better use by hiding some practical NP-hard problems within these puzzles?"