Hey forum,

Here is my Geared Redi cube puzzle. Although it may appear average, It has something very unique about it!

I think it comes across best in the video:

Video


For those of you who don't have youtube available, four of the eight cuts are geared together so that they turn in synchronization. The other four cuts are free to turn independently.

I would like to thank Oskar for taking the time to show me how to design a proper involute. that made his puzzle possible!



Here are some pictures.

Solved




a normal turn




Mid gear turn



Scrambled!

It's really beautiful, great job. It turns great. It sure doesn't look like your first real gear project.

As for solving, I agree I think it's probably very hard. Not because a 3-cycle is hard (there are 4 of them) but because the setup moves to use a non-geared twist as a 3-cycle will be very challenging.

Perhaps there is a class of easy 3-cycles we haven't spotted?

For what its worth, I'm pretty sure the position count for this puzzle is:
? (12! / 2) * 3^4 * 3
% = 58198694400

58B positions is within reach of a "god's number" calculation with some creative encoding and use of disk space. Possibly a patricia trie as the state-storage data structure would help?

Edit: I thought about it some more. A patricia trie is the wrong approach. It would be better to fully encode the state as an index into a bit array. That way you don't have to store the state at all, the state is the index into the array and the position can be reconstructed from any given index (and vice-versa).

Also, using a trick from Jaap of storing the number of positions from solved, modulo 3 (totally genius idea btw), you only need 2 bits per state. So 58B states would take up a minimum of 14.5 GB of memory. Totally reasonable for a modern machine.

I still struggle to come up with an algorithm to go from state -> index without any holes. I've done it once but that was a pretty easy puzzle.

Congratulations on getting such an original and gear-heavy concept to work, especially on the first try.

Not quite. The obvious 3-cycle also reorients a corner. A 3-cycle which leaves the corner orientations alone is much more challenging to find.

Very cool Eric. Would you mind posting a picture of the mech?

My assumption was that if you could figure out how to 3-cycle arbitrary pieces without much issue then twisting a corner is just a matter of setting up the right configuration to twist the corner in the way you want.

This is a pretty big assumption and may not turn out to be as "easy" in practice as I assume it would be.

As I understand, the corners never move. They only rotate. So the first step of my algorithm is to turn them back to the correct orientation. This step is trivial. They second step is to cycle the edges.

To find out 3-cycles, I use Gelatinbrain 3.2.4 to simulate the edges. Gear move G = [URF, ULB, FDL, RBD] is allowed. The other four moves LUF, BRU, DFR, and DBL are allowed. Then the [3,1] commutator [G LUF G', BRU] is a 3-cycle. GB notation:

[URF, ULB, FDL, RBD],LUF,[RBD', FDL', ULB', URF'],
BRU,
[URF, ULB, FDL, RBD],LUF',[RBD', FDL', ULB', URF'],
BRU',

This algo doesn't turn corners. With setup moves this should be sufficient.

Very good puzzle.
And the video prooves the influence of Oskar in another way...GAP comes to the same result.

Cool! So we have one more geared puzzles designer here at last

Looks realy nice Congrats!

Very nice! But I wish the gear corners were marked somehow, for example by truncating a tiny bit off the tip.

I suppose you probably considered a version with both pairs of corners geared, but couldn't make it work...

(About the theoretical double-geared version - how many positions would it have compared to this one or the regular Redi Cube?)

Great puzzle Eric, will you be selling this on Shapeways? BTW, I think that Oskar has copyrighted Hello, my name is ...

Great puzzle!

I wish to continue on Jareds topic, marking the geared corners.
Would it be possible to make a tetrahedral version of this, where all the faces are geared and all the corners are normal, or vice versa?

If you saw my post previously answering this question and claiming it had a lot of states, I had a bug. I've found the bug. The correct answer is that the double-geared version has 36 states and "god's number" is 4:

0 turns: 1
1 turns: 4
2 turns: 8
3 turns: 14
4 turns: 9

This would be pretty similar to a Gear Shift is my guess.

Edit:
The double-geared version would be super simple to solve. If one edge is solved then they all are. There are only 12 possible configurations for the edges and each configuration happens 3 times with different corner twists. You could probably solve the puzzle optimally by hand every time.

For each of the 9 corner configurations, there are 4 different edge configurations.

Impressive!

Oskar

Gearing has always been some of my favorite puzzle techniques.
I am beyond thrilled to have another gear designer!
Eric, fantastic job! The creativity is mind blowing!
I really love how you put such a interesting tweak on such a doctrine puzzle!

-Doug