You didn't think I had any particular reason for rambling about the Rubik's Cube? I started talking about it because in the recent issue of Scientific American, there is a very interesting article about the Rubik's Cube and similar puzzles. Sorry, it's subscription only, but I'll explain the basic ideas.
Permutation puzzles, as I previously explained, are puzzles where you have to arrange a number of items (like cubies or numbers) into the correct order. The underlying mathematics of permutation puzzles is called Group Theory. That's right, you're doing advanced mathematics when you turn a Rubik's Cube! Basically, all the possible positions of a permutation puzzle constitute a "group". For a Rubik's Cube, the size of this group is 43,252,003,274,489,856,000. But don't be intimidated by the size. As I said, the vast majority of permutation puzzles are easy. I mean, sure, you've got a 4-dimensional Rubik's Cube that boggles the mind, but they're easy in principle.
The mathematical reason that most permutation puzzles are easy is because they are based on symmetric or alternating groups. In a symmetric group, every single permutation is possible. In an alternating group, half of the permutations are possible. Maybe the previous three sentences made no sense to you, so allow me to translate to a concrete example: the Rubik's Cube. If the Rubik's cube is based on a symmetric group, that means you can find a way to switch two cubies without changing anything else. If it is based on an alternating group, that means you can find a way to switch three cubies without changing anything else.
But what happens if you've got a puzzle based on a much more complicated group? What if the simplest algorithm possible must switch 4 cubies at a time, or more? If we look to the mathematical research, there are lots of different finite simple groups. There are exactly 16 families of groups, plus 27 oddball groups, called the sporadic groups. (What's really amazing to me is that mathematicians can prove that there are no other finite simple groups.) So what happens if we build a puzzle based on one of those sporadic groups?
Well, that's what the Scientific American article did. They built puzzles based on the M12 sporadic group, the M24 sporadic group, and the Co0 group (aka the dotto group). Try them yourself!
The M24 puzzle and the dotto puzzle look ridiculously complicated, and I don't, at the moment, have enough patience to solve them. But let's look at the M12 puzzle.
Basically, you've got two moves called Invert and Merge. You can also create a custom macro, if you ever find an algorithm that you like. To solve this puzzle, I followed the same basic steps as I explained for the Rubik's cube. However, they are no longer easy. It is impossible to find an algorithm that switches exactly 2 or 3 numbers. In fact, you must switch at least 8 numbers at a time! Nevertheless, I came up with some useful algorithms.
So do you want some tips on solving the M12 puzzle? Try these simple algorithms. I came up with cryptic names for them too!
Once you figure out what they do, you'll have to come up with a super-algorithm that combines them to solve the puzzle. It isn't easy, even with my hints.