# How to think about permutation puzzles?

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.

• Maybe this previous question will help: Rubik's cube without any algorithms. Here's another: How can one solve a Rubik's cube without relying on guides/algorithms? Mar 26, 2020 at 22:49
• Mar 26, 2020 at 22:51
• So does that mean there is no trick to thinking about these puzzles in general? Each one is completely different? Mar 26, 2020 at 22:56
• The answers to the two questions I linked to are applicable to almost all permutation puzzles. The second link is better cause I gave an answer :-) Mar 26, 2020 at 23:01

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.

• Well, I assume that question was directed to me; getting the "1" to its position in itself takes 3 moves. So it can't be less than 3 moves. It can't be 3 moves either because "4" has to move in place as well and the first click to move "1" will not be able to move "4". I think this reasoning proves that the solution can be - at minimum - 4 moves. Mar 27, 2020 at 22:16
• You can rotate 124 or 134 first (to get 4132 or 4213). @alexoland
– JMP
Mar 28, 2020 at 3:56
• Well then 2 moves is clearly possible: 1234 4213 4321 It can't be 1 because there is no way to move the "1" to its position in 1 move. But I don't feel like I am solving this in the proper way. I am just trying things out and stumble into a solution. Is there something else you want me to realize? Mar 28, 2020 at 18:28