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I know this is a very general question, but these kinds of puzzles are the only ones I can't figure out on my own. Rubik's cube, Twiddle, 16... (the last two are in Simon Tatham's Puzzle collection.)
Is there a general thinking principle that applies to puzzles like these? If not, can you give me an example of how to arrive at the solution for any one of these puzzles? I just can't even begin to think about something with so many interconnected parts.
Permutation puzzles are puzzles in which a random permutation needs to be restored to the identity permutation by using a limited (or 'clumsy') set of moves.
For example, with the Rubik's cube, you would like to be able to swap two corners by themselves, but you can't do this directly - you can only rotate faces.
A key feature is then to find - and hopefully optimize - the solving process.
For example, consider that you are only allowed to rotate three elements of a permutation forwards, e.g. $abc\to cab\to bca\to abc$. The three elements do not need to be consecutive.
Then $1234\to4321$ can be done in four steps:
1234 3124 3412 3241 4321
Or:
1234 1423 2143 4213 4321
But can it be done in three moves or less? And if not, why not?
It can:
1234 4132 4213 4321
And, as the OP found, a solution in 2:
1234 4213 4321
The challenge is now to find the shortest solution to reverse n numbers, but that's a different question!
My example shows how the straightforward approach is not necessarily the shortest, but the easiest to construct. So to solve a permutation puzzle, construct the 'obvious' solution first, and then try to find shortcuts.
