How do you get your computer to solve your 15 puzzle?  How about your
Rubik's Revenge?  Programs for such problems have long been
fundamental to computational investigations in group theory, one of
the first areas of pure mathematics to capitalize on computers.  More
recently, the underlying algorithms have themselves become an object
of investigation in computer science.  Are there faster methods?  Are
they parallelizable?  Surprisingly, the mathematical foundation for
the answers to these, and other, computational problems, has only
recently become available via the Classification of Finite Simple
Groups.  For the most part, the CFSG is invoked via succinct,
uncomplicated corollaries. although some complexity-theoretic results
demand a case-by-case analysis of the classes of simple groups.