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 Post subject: Does the Rex Cube have two solutions?
PostPosted: Sun Jan 02, 2011 4:36 pm 
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I know that the Dino Cube has two solutions (mirror images of each other), and the Rex Cube edges are equivalent to those pieces. So... if you solve the edges "backwards" on a Rex Cube, can you still complete the solve, or do you end up with a parity problem?


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 Post subject: Re: Does the Rex Cube have two solutions?
PostPosted: Sun Jan 02, 2011 4:39 pm 
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You end up with two centers switched because you cannot have two centers switched if you solve it normally. (I have no idea what the technical term is for this, or why it should be. I just know that that's the way it is...)
Therefore, remember your color scheme or you will have a hard time digging yourself out later!

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 Post subject: Re: Does the Rex Cube have two solutions?
PostPosted: Sun Jan 02, 2011 4:57 pm 
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Monopoly wrote:
You end up with two centers switched because you cannot have two centers switched if you solve it normally. (I have no idea what the technical term is for this, or why it should be. I just know that that's the way it is...)
Therefore, remember your color scheme or you will have a hard time digging yourself out later!
On the Rex Cube, Dino Cube, and many other puzzles, every move you do is an even permutation of the pieces. On the Dino cube, when you twist a vertex you move 3 edges in a cycle. All 3-cycles are even permutations because a 3-cycle swaps two pairs of edges (in a 3-cycle each pair shares a piece). On the Rex cube the centers also 3-cycle. Since the only move you can do is vertex twists (a slice is just two vertex twists) the edges and centers must stay in an even permutation.

Solving into the opposite color scheme is equivalent to swapping two opposite faces. Since there are no 3 color/sided pieces to uniquely define the color scheme you can accidentally do this with the edges like on the Dino Cube. If you count the number of swaps, it is just 4 pairs of opposite edges and 4 swaps is an even permutation.

But in the centers, swapping two faces requires swapping just two centers which is an odd permutation (takes an odd number of swaps). Since you can only do moves that swap centers in pairs you won't be able to solve the centers into the opposite color scheme.

For what it's worth, this is also the cause of the edge parity on the void cube. If you move a middle layer then you have 4-cycled the edges which is an odd permutation (needs 3 swaps). This is how you can wind up with two edges swapped at the end but the corners are solved.

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 Post subject: Does the Rex Cube have two solutions?
PostPosted: Mon Jan 03, 2011 2:52 am 
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About a Rex Cube read here: http://twistypuzzles.com/forum/viewtopic.php?p=240925#p240925

Kind of a Rex Cube at wrong orientation of sides rather each other:

Image

Read more attentively messages in this section!


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 Post subject: Re: Does the Rex Cube have two solutions?
PostPosted: Mon Jan 03, 2011 4:33 pm 
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Thank you! I didn't read that thread closely because I didn't want to spoil anything for when I got my own...


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