A shuffle of two strings is formed by interspersing the characters into a new string, keeping the characters of each string in order. For example, MISSISSIPPI is a shuffle of MISIPP and SSISI. Lets call a string a square if it is a shuffle of two identical strings.
In November 2012, Sam Buss and I have succeeded in proving that the problem of determining whether a string can be written as a square shuffle is NP complete. This applies even over a finite alphabet with only 7 distinct symbols, although our proof is written for an alphabet with 9 symbols. This question is still open for smaller alphabets, say with only 2 symbols.
This problem was first posted on the CS Theory Community Blog in August of 2010. The problem received a lot of attention at the Theoretical Computer Science StackExchange. Our solution has been published in December 2013, the Journal of Computer and System Sciences:
http://www.sciencedirect.com/science/article/pii/S002200001300189X
(here is the corresponding StackExchange post)