This is an idea I want to throw out there. To be honest I haven't entirely thought it through and I am sure it must have been considered before and probably rejected. It was inspired from a recent YouTube comment.
Take for example a Gear Shift Cube. A small number of cogs were chosen for each gear but it could have been more. In fact it could have been an infinite number i.e. perfect circles turning on each other. From a practical point of view rubber could be used to ensure enough friction.
I am trying to get my head round what this would mean. 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 can see lots of other puzzles working in a similar way.
Now someone tell me I am talking rubbish and missed something really obvious. If I haven't, wouldn't it be the hardest puzzle ever?

No, the lowest ratio between the gears' sizes would dictate the possibilities of the puzzle. For example, imagine two pieces of rubber, one with a circumference of 4pi and one with a circumference of 7pi. After 28 turns of the large rubber gear, even if it's circular, you will end up at the solved position.

Interesting idea/question.

I think it depends on the relative circumferences of the connected wheels, and how many complete 360' rotations will be required to bring the wheels back into the same register, within the limits of tolerence.

For example, if the circumference of one wheel is 10% bigger than a connected wheel, you would need 10 complete 360' rotations to bring them back into register.

Does that answer your question?

EDIT: NType3 answered while I was typing, but we seem to be saying the same thing: in summary, you would need many more turns, but never infinite.

P.S. I don't think it would be the hardest puzzle ever from a logical perspective, but it may well be the most tedious - just think of having to turn each individual layer 10s or 100s of times before you can get it aligned properly.

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:

oo

And here they are disengaged:

o o

Now 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:

oo

Step 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.

Yes, agreed, in fact that's what I meant by: .
Still, a nice thought experiment, thanks for sharing, Tony.

You need to be careful when you say god's algorithm is infinity. god's algorithm can be unbounded (ie: the only solution is to reverse the scramble and the scramble can arbitrarily long) and a puzzle where the solution can require an infinite amount of moves (such as a permutation puzzle on the natural numbers). Maybe even then you need to distinguish between countable and uncountable infinite solutions. A very trivial puzzle with an uncountable god's algorithm would be the permutation of ]0, infinity[ given by e^x where you need to unscramble the points by doing transpositions of two arbitrary points.

One of the things I have been thinking about is an infinitely-unbandaged Helicopter Cube. It could be defined by the permutations of points in the three-dimensional sphere. They could be colored according to a map of XYZ-coordinates to RGB. Given that each move is defined by an invertible rotation of a half-space, it would be easy to simulate and backtrack each point on the screen to its original before the puzzle was scrambled. It would require keeping track of what moves were done exactly to scramble the puzzle and it would impossible to determine if it was solved, but it would be a very good simulation of arbitrary unbandaging.

Because the effects are additive, it essentially winds up keeping of the count of the number of times it's gone around. It would be neat to have a mechanism which kept such a count, and I think someone made a rubik's cube with LEDs which counted that one time, although that wasn't a very interesting puzzle.

The puzzle of mine which has the wheels connected by bars would benefit greatly from a mechanism which counted the number of complete rotations a circle has done, but I don't know of an elegant one.