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 Post subject: Even Less Gears! by OSKAR
PostPosted: Sun Dec 12, 2010 10:41 am 
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Hi Twisty Puzzles fans,

Did you like the Gear Cube with 12 geared edges? And did you find Gear Cube Extreme harder to solve, even though it just has eight geared edges? Well, here is an even harder variation called "Even Less Gears!" with only four geared edges.

As Even Less Gears! has only four geared edges, four faces can make quarter turns like a Rubik's Cube, whereas the fifth and sixth face are coupled through the gears. However, one scrambled, more faces get coupled. The fact that sometimes faces are coupled and sometimes not, makes this puzzle arguably harder than Gear Cube Extreme.

One way of obtaining Even Less Gears! is by buying a Gear Cube from Mefferts and add a set of eight Shapeways-printed non-geared edges. Another alternative is to buy two Gear Cube Extreme from Mefferts and do some mix-and-match.

Here is a combinatorial question for you to solve.
How many essentially different variations does this puzzle have?
Note the term "essentially different". Replacing a single geared edge from Gear Cube by a non-geared one does not change the puzzle from a solving perspective, given the parity limitations of this puzzle. Nick Baxter's paper on the Burnside Lemma is probably a good place to start, if you want to solve the combinatorial question.

Variations to this question are
    How many Gear Cube Extreme do you have to buy in order to build allpossible variations?
    How much would that cost you?

Watch the YouTube video.
Buy the non-geared edges from my Shapeways Shop.
Read more at the Shapeways Forum.
Check out the photos below.

Enjoy!

Oskar
Attachment:
Even Less Gears - prototype - view 1.jpg
Even Less Gears - prototype - view 1.jpg [ 53.21 KiB | Viewed 3100 times ]

Attachment:
Even Less Gears - prototype - view 2.jpg
Even Less Gears - prototype - view 2.jpg [ 52.1 KiB | Viewed 3100 times ]

Attachment:
Even Less Gears - prototype - view 3.jpg
Even Less Gears - prototype - view 3.jpg [ 48.58 KiB | Viewed 3100 times ]

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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Sun Dec 12, 2010 10:55 am 
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I've been waiting for this for a while. I was going to make one for myself, if I ever got another Gear Cube Extreme. Could you ask Meffert about producing this?

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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Sun Dec 12, 2010 10:57 am 
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What's next, a puzzle without gears? :lol:

But seriously, another great puzzle Oskar!

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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Sun Dec 12, 2010 12:20 pm 
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Wow, this looks like a really neat puzzle (:
Is this variation easier or harder than the Gear Cube extreme? I'd imagine it to be easier but it does have some irregular 'bandaging' (?) on the gear parts.
great puzzle! :D

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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Sun Dec 12, 2010 12:25 pm 
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Monopoly wrote:
Wow, this looks like a really neat puzzle (:
Is this variation easier or harder than the Gear Cube extreme? I'd imagine it to be easier but it does have some irregular 'bandaging' (?) on the gear parts.
great puzzle! :D


Oskar wrote:
did you find Gear Cube Extreme harder to solve, even though it just has eight geared edges? Well, here is an even harder variation called "Even Less Gears!" with only four geared edges.

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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Sun Dec 12, 2010 10:52 pm 
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Does this mean that the fun of different combinations have different :)


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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Mon Dec 13, 2010 1:53 pm 
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Easily understood but nobody made it before => genius.
Oskar wrote:
How many variations does this puzzle have?
144
Please note that I made the question a bit easier. 8-)


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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Wed Dec 15, 2010 1:33 am 
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Sorry for the double post. But I thought this deserves it.
Oskar wrote:
Here is a combinatorial question for you to solve.
How many essentially different variations does this puzzle have?
Note the term "essentially different". Replacing a single geared edge from Gear Cube by a non-geared one does not change the puzzle from a solving perspective, given the parity limitations of this puzzle.
Less then I expected.
There are no more than 7 different variants. These are:
  • [] The cube without gears.
  • [FR] The cube with a single gear.
  • [FL BR]
  • [FL FR BR]
  • [FL FR BR BL] "Even less Gears"
  • [UL UF UR UB DL DF DR DB] Gear Cube Extreme aka Anisotropic Cube
  • [UL UF UR UB FL FR BR BL DL DF DR DB] Gear Cube (I don't dare to say "normal")
All other variants are identical in behaviour to one of these.


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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Wed Dec 15, 2010 2:54 am 
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Andreas Nortmann wrote:
There are no more than 7 different variants. These are: <...> All other variants are identical in behaviour to one of these.
Sorry Andreas, but I have to disagree with you.

Even though I do not know the correct answer myself, I do know that there are at least two functionally different puzzles with three gear edges.
    One with the three gear edges each in a different plane
    Another where some or all of the gear edges are in the same plane
It is not too hard to prove that if all geare edges are in a different plane, they will always stay in a different plane.

Oskar

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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Wed Dec 15, 2010 3:16 am 
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Oskar wrote:
Even though I do not know the correct answer myself, I do know that there are at least two functionally different puzzles with three gear edges.
    One with the three gear edges each in a different plane
    Another where some or all of the gear edges are in the same plane
Oskar


I think the one with three gear edges each in a different plane is similar to the Gear Cube (12 gear edges).

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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Wed Dec 15, 2010 6:37 am 
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Chilen wrote:
I think the one with three gear edges each in a different plane is similar to the Gear Cube (12 gear edges).


Looks like I got the difference. The first move is similar to the 12 geared cube, but when moved, some edges get aligned in the same plane, so it enables free moves on other layer. Am I correct, Oskar?

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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Wed Dec 15, 2010 7:14 am 
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fbgrac wrote:
Am I correct, Oskar?
No. When three gears are in three different planes, they stay in three different planes. My question is whether that 3-gear puzzle has the same state space as the 12-gear version. Maybe Andreas is right after all. Analysis needed.

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 Post subject: Re: Even Less Gears! by OSKAR
PostPosted: Wed Dec 15, 2010 3:12 pm 
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I know that there are two different versions with 3 gears.
Oskar recognized correctly that one of them has one gear in every slice all the time. This means this specific variants allows only slice-moves, like the GearCube. There are similar arguments for every other variant which might be missing.
When somebody considers only which moves are allowed at which time, there are only those seven variants.


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