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 Post subject: TetraVex Cubes, a sticker mod series
PostPosted: Mon Mar 16, 2009 9:33 pm 
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I’ve always been interested in sticker patters in which the correct solved position of specific cubies is not obvious when the cube is scrambled. The popular maze 3x3x3 sticker mod is a great example of this. A puzzle like this is considerably harder to solve than a standard cube since you often have to experiment with the placement of cubies while solving and may have to backtrack over and over until you get it right. Even the best cubing experts can find this kind of puzzle frustrating. My first experiment in creating a cube like this was my 16 color 4x4x4 Suduko Cube. I now present my second experiment in creating these kinds of puzzles.

TetraVex is a puzzle which uses square tiles divided into four quadrants split corner to corner. Each quadrant is a different color and the object of the puzzle is to arrange the tiles so that matching edges touch each other on all four sides of all the tiles. Eternity II is a board game version of this kind of puzzle which uses 256 pieces and offers a $2 million prize for the first to solve it (still unclaimed). These puzzles are known to be easy to start because at first it is really easy to fit together a bunch of tiles. But they get difficult to complete because you will find that eventually you can’t fit any more tiles and you have to backtrack. I have now applied this to a series of Rubik’s Cubes from 2x2x2 through 5x5x5.

For the sake of simplicity with my description when I refer to “tiles” I will be referring to one stickered surface of a cubie consisting of four stickers (three for corners), not to be confused with cubies although for centers they are one in the same.

I designed these patterns using the following rules:

- There is a neutral color for the borders (white in this case). Some versions of TetraVex such as Eternity II define edge and corner tiles by using neutral colors for the outer edges. Since the tiles as applied to edge and corner cubies are fixed with the other tiles on the same cubie, it really didn’t matter if I assigned those edges unique colors or not. I decided to go with the neutral color because I think it gives a unique look to the cubes.

- The number of colors is kept low enough so that there are multiple possible positions for each tile, but not so low that there are an over abundance of solutions and duplicated tiles. The 2x2x2 uses three colors, the 3x3x3 uses 4, and the 4x4x4 as well as the 5x5x5 use five colors (not counting the neutral edge color).

- Colors won’t be duplicated on the same tile. Each corner tile contains two unique colors, each edge tile has three, and all center tiles have four unique colors. This is not always the case with TetraVex where you may sometimes see the same color showing up two, three, or even four times on a tile.

- If a tile exists on a cube, it also exists on all the cubes bigger then it. All twenty-four tiles from the 2x2x2 are present on the 3x3x3. All fifty-four tiles from the 3x3x3 are present on the 4x4x4 and so on. I felt that this made for a good connection between the cubes in this series. I wanted it to seem like each puzzle builds upon the lower order puzzles, which I guess makes since because that is how I developed them (taking all of the tiles from a cube, rearranging them, and then adding additional tiles to fill the spaces).

-No two cubies can be exchanged without it breaking the solution. This means that all corner, edge, and center cubies are unique even if some of the tiles aren’t. On the 5x5x5 this also applies to centers, center edges, and center corners each as separate groups (you may see the same tile on a center edge and a center corner but since they can’t be exchanged it doesn’t break this rule).

- Orientation matters for all cubies. Corner cubies won’t contain three of the same tiles and edge cubies won’t contain two of the same tiles. Since all center tiles have four unique colors this isn’t an issue for them.

- Each cubie can be placed next to at least two other cubies. I didn’t want any obvious paring of cubies so when you solve this you will have to do some experimental placing of them and may have to backtrack (a common occurrence when playing TetraVex).

- Multiple solutions may be possible (also common with TetraVex). I didn’t design any into the puzzles but I didn’t take steps to prevent them from occurring.


Attachments:
tetravex cubes.jpg
tetravex cubes.jpg [ 272.64 KiB | Viewed 2185 times ]
2x2x2and3x3x3.jpg
2x2x2and3x3x3.jpg [ 107.67 KiB | Viewed 2175 times ]
4x4x4.jpg
4x4x4.jpg [ 125.37 KiB | Viewed 2173 times ]
5x5x5.jpg
5x5x5.jpg [ 178.06 KiB | Viewed 2170 times ]
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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Mon Mar 16, 2009 9:35 pm 
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Joined: Sun Dec 07, 2008 6:42 pm
That has to be a pain to solve, let alone actually get one layer =P

Very nice job

Edit: Cubedron in twisty cube form =P

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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Mon Mar 16, 2009 9:38 pm 
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:shock:

I'd love to see someone solve that 5x5.


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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Mon Mar 16, 2009 10:43 pm 
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Location: Orangevale, CA
Scramble it and send it to Michael (Gottlieb).

I'd get a headache from trying :shock:


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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Mon Mar 16, 2009 11:23 pm 
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These are kinda like the tartan cubes! Good job! Looks like a real pain!

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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Mon Mar 16, 2009 11:59 pm 
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Location: NY, USA
That looks like it would be quite crazy to solve. Even trying to figure out whether it's solved looks like a puzzle!

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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Tue Mar 17, 2009 2:20 am 
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That sticker pattern is just downright terrifying! I would like to buy a set for my 5x5 :D

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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Tue Mar 17, 2009 2:57 am 
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Location: Malibu, California
Another amazing sticker mod!

Since you mentioned TetraVex, I feel like calculating how many positions it has: 256 pieces with 4 orientations each gives 256!*4^256≈1.150*10^661.

However, since the neutral colors define the edges and corners, the number is less: 4 corners, 56 edges, and 196 centers all with 4 orientations each gives 4!*56!*196!*4^256≈1.162*10^596.

However, the corners and edges don't really have that many valid orientations. In any given position, the corners have 1 orientation, as do the edges. This then gives 4!*56!*196!*4^196≈8.744*10^559.

I also just realized that the orientation of the board doesn't matter so you then have to divide by 4 and you get ≈2.186*10^559.

Of course, these numbers are so big that the difference hardly matters.

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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Tue Mar 17, 2009 3:42 am 
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Joined: Mon Jul 30, 2001 2:50 am
Location: Nottingham, UK
Wow!
That is fantastic sticker variation.
I would love to have a set from 2x2 to 5x5... or even to 7x7 :D

Vadim


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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Tue Mar 17, 2009 12:56 pm 
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Fantastic puzzles. Tetravex fan here.


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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Tue Mar 17, 2009 1:11 pm 
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My brain hurts just looking at it. Nice work!


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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Tue Mar 17, 2009 2:01 pm 
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Scott Bedard wrote:
I would like to buy a set for my 5x5 :D
Unfortunately I don’t have any sticker sets to sell or even give away. These are just Cubesmith’s standard stickers which I hand cut using an X-Acto knife and metal straight edge. I may one day make some more of these cubes for use in trades, but that won’t be right away as I have some other projects to work on next. In the mean time I have provided the full pattern for you guys to use to make your own and I encourage you to do so. I would love to read everyone’s impressions of these as they try to solve them. But what I don’t want to see is someone making a bunch of these to sell on eBay (I would like the reserve the right to do so if I choose).


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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Tue Mar 17, 2009 9:45 pm 
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What kind of twisted (pun) mind thinks up nightmares like this? Speaking of which, I should make one at some point. :mrgreen:

Wonderful (in a rather demented way) puzzle!

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 Post subject: Re: TetraVex Cubes, a sticker mod series
PostPosted: Wed Mar 18, 2009 1:36 am 
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They do look tricky. I would prefer the 3x3x3 TetraVex over the maze cube (which I have never solved).

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