I have a puzzle in my office called the Topsy Turvy, designed by M. Oskar van Deventer, based on ideas of Igor Kriz, a topologist at the University of Michigan. (You can read the Scientific American article by Kriz and undergraduate Paul Siegel here.) The puzzle has 12 tokens numbered 1 through 12, which start off in order. There are two possible "moves", called Left and Right. Left (L), applied to the starting configuration, permutes them to 11 9 7 5 3 1 2 4 6 8 10 12. Right (R), applied to the starting configuration, permutes them to 2 4 6 8 10 12 11 9 7 5 3 1. The group generated by these permutations is one of the sporadic simple groups, called the "Mathieu 12" group, and is index 5040 in S_12.
In the fall of 2010, I asked my first quarter honors algebra (Math 120) class how to solve the puzzle. I was pleased and surprised to get a number of solutions. I'd asked for a computer implementation, basically because I thought a non-computer solution (one that could be communicated in text) would be unreasonable, but three people gave me explicit algorithms that are human-readable.
The computer solvers (along with their language of choice, where I remember) were the following. Many find the optimal solution.
[solvable, solution] = M12_solve(BST_M12, input)
Explicit (text) solutions were given by
If you have a topsy turvy puzzle that needs solving (and I know there are a double digit number of you out there), some of their solutions are available; just click on their names. Some are web-based; others you can run on your machine.