Tony Fisher wrote:

You pull the puzzle open and make a move, Literally any fraction of rotation is possible and it will still snap back in place for the next move. The more random moves you make the more it becomes mixed. Gods algorithm is surely infinity?

I think there is a flaw in your logic. Hang with me for a minute. First, assume that you can make an infinitely precise twist.

Now imagine a puzzle that is 2 "gears" that are actually rubber wheels that can be engaged and disengaged with each other.

Here is a diagram:

ooAnd here they are disengaged:

o oNow assume they have been scrambled. That is, each wheel has some twist independent of the other.

You can solve them in this order:

Step 1: disengage wheels

Step 2: rotate wheel 1 until it perfectly matches the orientation of wheel 2

Step 3: re-engage wheels

Step 4: rotate wheels until they are solved

Now, this solution can be extended even if the wheel diameters are in some other ratio. Here is a diagram:

ooStep 1: disengage wheels

Step 2: do some basic math (ratio of circumferences) to determine what position you need to put wheel 1 in so that when twisted it solves both

Step 3: put wheel 1 in that position

Step 4: re-engage wheels

Step 5: rotate wheels until they are solved

In general I think this can be extended to any chain of wheels.

I think the flaw in your logic is that even though each piece can be in an infinite number of orientations, you must assume that you can be infinitely precise in your movements and that negates the infinite degrees of freedom.