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 Post subject: Variomatic Cube by OSKARPosted: Fri Aug 10, 2012 1:40 pm

Joined: Mon Nov 30, 2009 1:03 pm
Hi Twisty Puzzles fans,

Variomatic Cube is a twisty puzzle with gears. All three pairs of opposing faces are connected by gears. And they are not just some gears, they are gears with a varying gear ratio. Some sometimes, opposing gears will turn with a 1:1 gearing ratio, some with a 1:2 ratio, and sometimes with a 2:1 ratio. The operation of the variable-ratio gears is explained and illustrated here and here. Watch a demo of the inner gearing mechanism here.

The name of this puzzle is a tribute to the variomatic transmission system by the Dutch car manufacturer DAF. Note that the variable-gear-ratio mechanism is very different from the DAF variometic system.

Bram wrote:
Post subject: Re: question: what are the hardest puzzles (to solve)
Posted: Mon May 14, 2012 4:41 pm

The Paso Doble is probably extremely difficult.
Variomatic Cube may be significantly harder than Paso Doble.

Buy the puzzle at my Shapeways Shop.
Read more at the Shapeways Forum.
Check out the photos below.

Enjoy!

Oskar
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 Post subject: Re: Variomatic Cube by OSKARPosted: Fri Aug 10, 2012 1:50 pm

Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland
Brilliant. I really like your gear-based line of puzzles you've been outing lately.

I wonder if these will be an addon to Carl's Multi Gear Cube Kit?

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 Post subject: Re: Variomatic Cube by OSKARPosted: Fri Aug 10, 2012 2:29 pm

Joined: Sun Mar 15, 2009 12:00 am
Location: Jarrow, England
Mind boggling as usual Oskar. How does this rank on the difficulty scale for 3x3x3s?

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 Post subject: Re: Variomatic Cube by OSKARPosted: Fri Aug 10, 2012 2:35 pm

Joined: Sat Jan 15, 2011 10:22 am
Gus wrote:
Mind boggling as usual Oskar. How does this rank on the difficulty scale for 3x3x3s?

Over the top

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 Post subject: Re: Variomatic Cube by OSKARPosted: Sat Aug 11, 2012 3:33 am

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Is it possible to have the cube externally solved but with the gears in different positions than they initially are?

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 Post subject: Re: Variomatic Cube by OSKARPosted: Sat Aug 11, 2012 6:46 am

Joined: Mon Nov 30, 2009 1:03 pm
Jared wrote:
Is it possible to have the cube externally solved but with the gears in different positions than they initially are?
My bet is that "Jared's conjecture" is true. However, it remains yet to be proven or falsified.

Oskar

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 Post subject: Re: Variomatic Cube by OSKARPosted: Wed Aug 15, 2012 4:55 pm

Joined: Tue Aug 11, 2009 2:44 pm
There was some debate at IPP as to whether this puzzle is at all human-solvable. There was a bet proposed between Oskar (nobody will solve it within 30 days) and Bram (somebody will), but I'm not sure whether the bet was actually made.

My feeling is that it should be substantially easier to solve than the Latch Cube. I solved that with a couple days' hard effort at IPP 30, and I'm sure others have solved it with less effort since then. However, the Variomatic Cube does need some center markings so the internal gear orientation can be identified. Oskar and I agreed on what is probably the best way to do that.

The intuition here is that the challenge with the Latch Cube is to first get it into a canonical configuration, with all the arrows pointing the same way as in the solved state. Then, transforms can be built up that preserve this property. But it is quite challenging to reach such a configuration from a scrambled state, and even then, the puzzle exhibits only a low degree of symmetry, resulting in very long, composed, error-prone sequences of transforms.

The Variomatic Cube, on the other hand, is trivial to put info a canonical configuration (where each pair of opposing faces is in a reference orientation, at a fixed cw / ccw gearing for the next turn). Also it has much more symmetry than the Latch Cube. So, it should just be a matter of some thought + experimentation to find a set of sufficient transforms.

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 Post subject: Re: Variomatic Cube by OSKARPosted: Wed Aug 15, 2012 4:58 pm

Joined: Tue Aug 11, 2009 2:44 pm
Oskar wrote:
My bet is that "Jared's conjecture" is true. However, it remains yet to be proven or falsified.

I also bet this is true. In fact, I think we implicitly assumed it's true when deciding on how to mark the centers.

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 Post subject: Re: Variomatic Cube by OSKARPosted: Wed Aug 15, 2012 6:12 pm

Joined: Mon Mar 30, 2009 5:13 pm
Oskar wrote:
Jared wrote:
Is it possible to have the cube externally solved but with the gears in different positions than they initially are?
My bet is that "Jared's conjecture" is true. However, it remains yet to be proven or falsified.
If that is true, which I think it probably is, then it means the cube can't be scrambled and solved the same way twice, unless all the gears happen to go back to the same starting positions. It would be interesting to calculate how many different gear permutations there are in total, and how many of these are possible for the solved state...

So, I calculate that there are 3 pairs of opposite faces ^ 3 gear positions per pair = 27 different gear configurations in total. Is that correct?

Now, how do we figure out how many of these 27 different gear configurations are compatible with the solved state? Presumably there are some parity issues in the orientations of the centers, so some but not all of those gear configurations will be possible...

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Last edited by KelvinS on Wed Aug 15, 2012 6:41 pm, edited 3 times in total.

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 Post subject: Re: Variomatic Cube by OSKARPosted: Wed Aug 15, 2012 6:30 pm

Joined: Mon Nov 30, 2009 1:03 pm
KelvinS wrote:
... unless all the gears happen to go back to the same positions.
There is only a 1:27 chance for that, as for each of the opposing center pairs there are three states, only one of which is the canonical one.

Oskar

P.S. Bob Hearn just taught me the word "canonical". When applied to twisty puzzles, I suspect that it means something like "unique distinguished state".

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 Post subject: Re: Variomatic Cube by OSKARPosted: Wed Aug 15, 2012 6:46 pm

Joined: Mon Mar 30, 2009 5:13 pm
Oskar wrote:
KelvinS wrote:
... unless all the gears happen to go back to the same positions.
There is only a 1:27 chance for that, as for each of the opposing center pairs there are three states, only one of which is the canonical one.

Oskar

P.S. Bob Hearn just taught me the word "canonical". When applied to twisty puzzles, I suspect that it means something like "unique distinguished state".

Yep, the 27 gear configurations makes sense, however only some of these may be compatible with the solved state due to parity issues.

I know the term canonical from chemistry, as it is used to describe distinct but equivalent resonance structures of conjugated or aromatic compounds like benzene.

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Last edited by KelvinS on Wed Aug 15, 2012 7:03 pm, edited 1 time in total.

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 Post subject: Re: Variomatic Cube by OSKARPosted: Wed Aug 15, 2012 7:03 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Canonical is a pretty good term here. I like it. For jumbling puzzles you can say "back in shape" but for bandaged puzzles it's hard to describe the subset where the bandaging is the same as the solved state.

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