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 Post subject: Solving David's Gear CubePosted: Sun Feb 02, 2014 2:29 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Update: Completely rewritten, now with less fail!

Here's the David Gear Cube:

http://www.youtube.com/watch?v=KU3e4_LtOZQ

Notation: rotating all the corners clockwise at once is G. Counterclockwise is G-. Everything else is normal.

First, solve the corners ignoring the centers, using any solution to the 2x2x2. If you find that the parity is off, then if what's needed is rotating a single corner clockwise by 1, do a G-. If you need to rotate one counterclockwise, do a G. In the next step you'll completely mess up the corners again. The whole point of this step is to get the corner parity right. If you added the extra stickers then you don't have to do this step, then shame on you for cheating.

Next up is getting the center pieces on the right parts of the right corners. To do this you use this sequence:

G-2 U R F2 U R F2 R U F- R G2

This sequence does two 3-cycles, one where it rotates the centers on just the DBL piece, and one where it cycles the left center of FUL, the down of DRF, and the back of BRU. You should mostly use that second cycle to solve things, and put either a completely unsolved corner, or one which has the right pieces but rotated the wrong way, or if absolutely necessary an already solved corner in the BDL. If you find yourself with only a single corner which is off, and that one just has to have its center pieces rotated, then put it in the BDL and put centers of all the same color in the other 3-cycle positions.

(I had to 'cheat' and use http://www.jaapsch.net/puzzles/cube2.htm to find a short sequence for the 2x2x2 portion of that. If anyone can find a more understandable presentation or version of that sequence I would much appreciate it. I can come up with one myself, but it would be a lot longer.)

Once you've gotten the center pieces on the right positions on the right corners, just solve the whole thing as a 2x2x2.

I have now done this solution myself, and it's downright pleasant and not at all confusing, unlike the extremely long sequences which tend to come up when finding solutions to this puzzle.

Last edited by Bram on Mon Feb 17, 2014 10:29 pm, edited 1 time in total.

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 Post subject: Re: Solving David's Gear CubePosted: Sun Feb 02, 2014 3:16 pm

Joined: Fri Dec 20, 2013 11:47 pm
Several people do in fact have the cube already.

viewtopic.php?f=15&t=26495&hilit=david+gear+cube

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 Post subject: Re: Solving David's Gear CubePosted: Mon Feb 03, 2014 2:56 pm

Joined: Mon Mar 07, 2011 11:08 pm
Uh, this doesn't help me at all...

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 Post subject: Re: Solving David's Gear CubePosted: Mon Feb 03, 2014 3:01 pm

Joined: Mon Mar 30, 2009 5:13 pm
Crazy56U wrote:
Uh, this doesn't help me at all...
Have you tried turning it off and back on again?

_________________
If you want something you’ve never had, you’ve got to do something you’ve never done - Thomas Jefferson

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 Post subject: Re: Solving David's Gear CubePosted: Mon Feb 03, 2014 3:39 pm

Joined: Fri Feb 18, 2011 5:49 pm
Location: New Jersey
I had made this post in the original thread for the Polo Gear Cube:

http://twistypuzzles.com/forum/viewtopi ... 15&t=23999

DKwan wrote:
Some days ago, there was a brief discussion about the difficulty of solving this puzzle in the TP chatroom. You can solve the little rectangular edge-stickers as a normal 2x2x2, and then since all corner gears must have the same amount of turning-offset from their corners, you can just make a single gear twist to fix all of the corner gears.

The tricky piece is the split center-gears, and indeed they are very difficult pieces. I came up with what I believe should be a super-long [10,14] pure 3-cycle:
X = [G, R',D',R, U', R',D,R, U, G'] (conjugates a [3,1] corner perm with a gear turn to affect only 1 piece out of the DBR and DBL corner groupings)
Y = [R',U2,R,B,U2,B', D', B,U2,B',R',U2,R, D] (a [6,1] corner orient for the previously mentioned corners)

(G = rotate corner gears 120 degrees clockwise)

I can't really comment on how short an alg would be possible for these pieces, but I think finding a nice short one wouldn't be easy! Perhaps there is another more convenient method for solving this puzzle.

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 Post subject: Re: Solving David's Gear CubePosted: Mon Feb 03, 2014 8:38 pm

Joined: Sat Sep 03, 2011 3:11 pm
This does not apply when the smaller rectangular pieces are not stickered. This adds to the difficulty of figuring out the correct orientation of each piece.

DKwan wrote:
I had made this post in the original thread for the Polo Gear Cube:

http://twistypuzzles.com/forum/viewtopi ... 15&t=23999

DKwan wrote:
Some days ago, there was a brief discussion about the difficulty of solving this puzzle in the TP chatroom. You can solve the little rectangular edge-stickers as a normal 2x2x2, and then since all corner gears must have the same amount of turning-offset from their corners, you can just make a single gear twist to fix all of the corner gears.

The tricky piece is the split center-gears, and indeed they are very difficult pieces. I came up with what I believe should be a super-long [10,14] pure 3-cycle:
X = [G, R',D',R, U', R',D,R, U, G'] (conjugates a [3,1] corner perm with a gear turn to affect only 1 piece out of the DBR and DBL corner groupings)
Y = [R',U2,R,B,U2,B', D', B,U2,B',R',U2,R, D] (a [6,1] corner orient for the previously mentioned corners)

(G = rotate corner gears 120 degrees clockwise)

I can't really comment on how short an alg would be possible for these pieces, but I think finding a nice short one wouldn't be easy! Perhaps there is another more convenient method for solving this puzzle.

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 Post subject: Re: Solving David's Gear CubePosted: Mon Feb 03, 2014 9:11 pm

Joined: Sat Sep 15, 2012 7:42 am
zackz115 wrote:
This does not apply when the smaller rectangular pieces are not stickered. This adds to the difficulty of figuring out the correct orientation of each piece.

Can't you just turn the gears until it's restored to cube shape, then solve as a 2x2x2? Since all the gears move in the same direction, this should always be possible. So the tiny stickers really don't add or subtract anything from the solve. You're just doing (restore to cube ---> solve 2x2) instead of (solve 2x2 --> restore to cube).

_________________
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 Post subject: Re: Solving David's Gear CubePosted: Mon Feb 03, 2014 9:24 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
zackz115 wrote:
This does not apply when the smaller rectangular pieces are not stickered. This adds to the difficulty of figuring out the correct orientation of each piece.

Barely. A clockwise gear-turn does 8 clockwise corner twists which is equivalent to a single counter-clockwise twist. Just solve like a 2x2x2 and if the last corner needs a clockwise twist, do a counter-clockwise gear turn and re-solve. If the last corner needs a counter-clockwise twist then do a clockwise gear twist and re-solve.

For additional analysis:

A 2x2x2 move changes the parity of the corners and the parity of gear-wedges. A gear-twist doesn't change the parity of any piece. Therefor the parity of the corners and the gear-wedges stay linked so even a Super-David's Gear Cube wouldn't run into any unexpected trouble beyond just a single corner twisted (which is easy to fix as described above).

_________________
Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Solving David's Gear CubePosted: Mon Feb 03, 2014 11:56 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
DKwan, I don't follow. How do you put together X and Y to form a 3-cycle? Is your X without the G just a 2-cycle of two corners? If so it's the same as my A.

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 Post subject: Re: Solving David's Gear CubePosted: Tue Feb 04, 2014 12:16 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Bram wrote:
DKwan, I don't follow. How do you put together X and Y to form a 3-cycle? Is your X without the G just a 2-cycle of two corners? If so it's the same as my A.

His X without the use of G and G' is a 3-cycle of corners. His Y twists two adjacent corners.

The full sequence would be the commutator X, Y, X', Y'

After looking at it, the G portion moves a gear-wedge in the bottom layer into the D side of the FRD corner. Then the middle portion of the X sequence replaces the FRD corner with the LFU corner and the G' moves the now-replaced gear-wedge back into either the D side of the BDR corner or the D side of the BDL corner. This effectively isolates a gear-wedge so that the Y sequence that twists the BDR and BDL corners will make a pure commutator.

I can appreciate the commutator-approach and that's certainly how I'd approach it but the sequence can trivially be reduced in length by using the optimal equivalent of the 2x2x2 moves.

_________________
Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Solving David's Gear CubePosted: Tue Feb 04, 2014 7:20 am

Joined: Fri Feb 18, 2011 5:49 pm
Location: New Jersey
Brandon's explanation of my alg is accurate. Saying it was [10,14] was my way of explaining that it was a commutator [X,Y].

When I first made that alg 1.5 years ago, I was happy enough to just have a solution... But with regards to making it shorter while using the same basic structure... I think the X part can be shortened to [G, R', F, R, F', G'] since I believe the URB corner is also allowed to be affected. This would make it a [6,14].

I'd appreciate it if someone could confirm that my alg works on the actual puzzle.

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 Post subject: Re: Solving David's Gear CubePosted: Tue Feb 04, 2014 2:17 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
I think I see. My sequence is still shorter (32 regular moves instead of 40), and also a lot more brainless to do.

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 Post subject: Re: Solving David's Gear CubePosted: Tue Feb 04, 2014 10:41 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Come to think of it, given the nature of the 3-cycle, it's probably a lot easier to get all the center parts associated with the correct corner parts first and then solve the 2x2x2. There's a funny thing with the parity of the corners though, where you need to rotate the corners properly to get their parity right. It's extremely annoying for a human to look over everything and try to calculate this in their head, so my preferred approach would be to first try to solve the 2x2x2, then if necessary use the big rotation to get the corner parity right, then use the 3-cycle to get all the center pieces on the correct corresponding edges, then solve it as a 2x2x2.

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 Post subject: Re: Solving David's Gear CubePosted: Thu Feb 06, 2014 10:31 pm

Joined: Sat Sep 03, 2011 3:11 pm
I'm still lost with this, lol. I have been trying to solve the centers first, but it always seems to mess up previous ones. I don't fully understand the notation for the algorithms a few of you have posted. Thanks

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 Post subject: Re: Solving David's Gear CubePosted: Mon Feb 10, 2014 3:44 pm

Joined: Mon Mar 07, 2011 11:08 pm
KelvinS wrote:
Crazy56U wrote:
Uh, this doesn't help me at all...
Have you tried turning it off and back on again?
>Has legitimate problem with proposed solution method.
>Gets sarcasm in response.

(begins laughing obnoxiously/sarcastically)

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 Post subject: Re: Solving David's Gear CubePosted: Sun Feb 16, 2014 1:24 am

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Okay, I have a David Gear Cube now, and it turns out I bungled this solution pretty badly, mostly because I misremembered what my X sequence does. An appropriate substitute which would work is RU2R-U-RU2L-UR-U-L. I don't particularly like that sequence, it's long and non-intuitive. I had to look it up to find it (it's a standard PLL).

I have a better approach which is dependent on having a reasonable sequence for a pretty pattern. The pattern in question is to have FUL, FDR, and BRU all be 3-cycled, but with their colors relative to each other still correct, while everything else remains exactly in place. If we call that sequence Z, then doing G2ZG-2Z- does exactly two 3-cycles, and if you're doing the cube in two phases where first you get the center parts associated with the correct corners and then solve the 2x2x2 then you can just use G2ZG-2, which will cause a useful 3-cycle of center pieces and another somewhat extraneous 3-cycle which just rotates the pieces on the BDL corner. To use this to solve the whole puzzle, first solve one corner and then use it as the garbage corner while you solve everything else, then if the garbage corner happens to have the wrong orientation of its center pieces position the other corners such that their 3-cycle is all of the same color and do the standard sequence to get the garbage corner back into position. Then solve the 2x2x2.

So, does anybody know a nice sequence for the pretty pattern? It seems like it should be possible to do it with some combination of R-FRF-, RU-R-U, and UF-U-F. I need to mess with a puzzle a bit more to find it though.

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 Post subject: Re: Solving David's Gear CubePosted: Mon Feb 17, 2014 10:30 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
I've now rewritten the original post to reflect a vastly better solution I came up with.

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 Post subject: Re: Solving David's Gear CubePosted: Tue Feb 18, 2014 4:20 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Why not G' instead of G- to go with standard cube notation?

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 Post subject: Re: Solving David's Gear CubePosted: Fri Feb 28, 2014 2:24 am

Joined: Mon Jun 07, 2010 11:24 am
Location: Long Beach, CA USA
I've come up with my own solution. I think it's quite easy.

Here's the added notation I came up with: C and C'
(C stands for "corners." C is corners clockwise one twist and C' is counterclockwise one twist.)

Here's the alg I came up with: (R C R' C') U2 (C R C' R') U2

It's a simple alg that swaps the centers on the top face in the URB and ULF, along with swapping the centers on the bottom face in the DLB and DRF.

Step 1: Solve corners to make sure corners are in the correct orientation. (Easier with tiny 2x2 stickers)
Step 2: Reduce the puzzle to a 2x2 using the alg above.
Step 3: Solve like a 2x2.

In Step 2, you just need to put two corners in the top two slots that would benefit from swapping centers, and put two corners in the bottom slots that would either benefit from a swap or at least two corners that won't be destroyed by a swap. Just repeat until all centers are matched with their corners.

It's a fun puzzle!

_________________
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 Post subject: Re: Solving David's Gear CubePosted: Sun Mar 02, 2014 9:49 pm

Joined: Sat Feb 22, 2014 12:34 am
Thank you, Bram and Kenneth! I'm a new forum member; I recently bought a David Gear puzzle, and searched for solutions, and came across this page.

I eventually got Bram's solution to work, and just now I tried Kenneth's and that worked nicely too. I wanted to note, though, that I was a little bit confused by a couple of things in Bram's at first, so I figured in case others were running into the same problem, it might be worth posting a followup. So this post is for anyone who ran into the same confusion I did, especially if you're starting out with your cube scrambled and thus you can't necessarily easily see what effect the sequences are having.

Note: To be completely clear, I didn't create either of the solutions here, and I'm not trying to claim any credit for them. This is just an attempt to clarify a few details.

Both of these solutions have the same basic structure:

Step 1: Make sure the corners have the correct parity, to avoid having only one corner misrotated at the end.

Step 2: On each cubie, make the center colors on all three faces match the corner colors, resulting in a (scrambled) 2x2x2 cube.

Step 3: Solve as if it were a normal 2x2x2, with no gear rotations.

In a little more detail:

Step 1: If you have the tiny stickers on your cube, then skip this step. If you don't have the stickers, then check the parity by ignoring the centers and temporarily solving the corners as a 2x2x2 cube. (You'll be messing this up again in a minute.) If it turns out that you have one corner twisted counterclockwise at the end of that, then do a G (clockwise corner twist); if, instead, you have one corner twisted clockwise, then do a G' (counterclockwise corner twist). The parity is now correct.

This initial 2x2x2 solving was *only* to get the parity right. You'll now proceed as if the cube is completely scrambled; don't try to preserve anything you've just done.

Step 2: The key insight here (which took me a while to understand, which is why I'm spelling it out here) is that if each corner color matches its corresponding center color, then what you've got is a scrambled 2x2x2 cube. So the goal of this step is to move centers around (relative to corners) in order to get to a state where each cubie's center colors match their corresponding corner colors, without caring about trying to get the cubies into the right places yet.

Bram and Kenneth gave two different sequences for completing step 2. For both sequences, the basic idea is that you start by using standard 2x2x2 moves (not described here) to put complementary corner/center pairs that need fixing into the positions that are modified by the sequence; then you use the sequence.

Kenneth's sequence swaps the URB/U and ULF/U centers (I'm using "/U" to mean "the U surface of that cubie"), and also swaps the DRF/D and DLB/D centers. It leaves everything else alone. It goes like this:

(R G R' G') U2 (G R G' R') U2

Bram's sequence cycles three centers: ULF/L, URB/B, and DRF/D. (Technically, I think it's leaving those centers in place and cycling the cubies, but for practical purposes it's simpler to think of it as cycling as the centers.) It also cycles three other centers: the three centers on the DLB cubie. (DLB/D, DLB/L, and DLB/B.) It goes like this:

G'2 (U R F2) (U R F2) (R U F' R) G2

Bram's sequence is more powerful; it can potentially fix six corner/center pairs at once (though in practice I usually used it to fix three, or sometimes two), where Kenneth's sequence can only do four at once (though in practice I mostly ended up using it to fix two, or sometimes only one). But Bram's is also a little harder to set up for, at least in my experience so far. Anyway, you can certainly mix and match, using whichever sequence you need at any given time. (As long as you're careful about the cycling DLB cubie in Bram's sequence.)

(Bram noted that his sequence is based on a 2x2x2 sequence from http://www.jaapsch.net/puzzles/cube2.htm that cycles three cubies. Adding the gear moves is what makes the centers cycle.)

Step 3: This is the easy part: after the centers match their corners, then you have a scrambled 2x2x2 cube, which you can solve in the usual way without doing any gear moves.

...Okay, I think that's all. Thanks again, Bram and Kenneth, for providing these great solutions! Very useful.

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 Post subject: Re: Solving David's Gear CubePosted: Tue Mar 25, 2014 3:06 pm

Joined: Sun Jul 22, 2012 2:46 pm
I use an very awkward and long pure 3-Cycle.

It cycles FRD, RBD and LFD and uses an algorythm for a corner swap for example:
A: (R2 U R2 U' R2) (F2 U' F2 U F2 U')

with this corner swap algorythm it goes:

(G A G' A) D (A G A G') D'

This is what I used solving it the first Time, maybe it can be useful in some situations.

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