Beginner question about puzzles with parities

[kikenyoy]

I'm new to puzzles and so far I don't find solving parity issues to be very fun (4x4x4). I'm planning on buying some new cubes and I'd like to know if the following cubes have parity issues and if so, roughly how often do they occur and how easy are they to solve. Thanks.

5x5x5
3x3x5
Megaminx (none on the 12 color, right?)
Fisher cube
Curvy Copter
Traiphum 6 color ball

 

[guysensei1]

5x5x5: No OLL/PLL parities, but there is edge pairing parity which may or may not be considered parity depending on who you talk to. A simple algorithm can fix this.
3x3x5:idk
Megaminx:None on 12 color.
Fisher: Yeah. Very easy if you think about it.
Curvy copter: No.
6 color ball: Same as fisher cube.

 

[LNZ]

There might be parity on a 3x3x5 cuboid.

 

[Wilhelm]

On the 3x3x5 there is no parity because the centers are fixed. On 3x3x4 for example there is parity because 4 of the 6 centers are not fixed but on most of the cubiods they're pretty easy to solve^^

 

[TankRed]

The curvy copter has actually the same kind of parity that the Fisher cube has: it happens that there are only two pieces swapped, but it happens because two pieces of the same colour are in opposite orbits. Anyway parities are not too bad to deal with: they teach you how the puzzles work and you can also learn some group theory from that

 

[Kirjava]

On the 3x3x5 there is no parity because the centers are fixed. On 3x3x4 for example there is parity because 4 of the 6 centers are not fixed but on most of the cubiods they're pretty easy to solve^^

You can certainly get inner edge permutation parity on 3x3x5.

I mean, it really depends on what methods you're using. You can get parity on 3x3x3 if you want to.

 

[TMOY]

5x5x5: No OLL/PLL parities, but there is edge pairing parity which may or may not be considered parity depending on who you talk to.

Wooooooooooow. The cluelessness of cubers about parities will definitely always surprise me.

If you're referring to the edge pariring parity in the reduction method, this is certainly a parity since you're swapping 2 wings while preserivng everything else (at least in appearence). And 2 is not even or odd depending on tue person you talk to.

 

[guysensei1]

Wooooooooooow. The cluelessness of cubers about parities will definitely always surprise me.

If you're referring to the edge pariring parity in the reduction method, this is certainly a parity since you're swapping 2 wings while preserivng everything else (at least in appearence). And 2 is not even or odd depending on tue person you talk to.

Yeah. This parity thing confuses me. Sometimes situations where 2 pieced are swapped while preserving everything else is a parity, sometimes it isn't. :/

 

[cmhardw]

Yeah. This parity thing confuses me. Sometimes situations where 2 pieced are swapped while preserving everything else is a parity, sometimes it isn't. :/

Which puzzle?

 

[elrog]

I know the mars circle cube is one of them, and so is any big cube. You can make it appear as if you swapped 2 center pieces on a big cube by doing a 3 cycle of centers when 2 are the same color. 3-cycles take an even number of moves in QTM and are not parity in a mathematical sense. I'm sure there are others, but I don't own any others.

Here is a proof for the mars circle cube. I started holding the cube so that the circles that don't turn with the side are on the left, right, and bottom. Movecounts are in QTM.

1 - (R2 U2)3 y // 12
2 - U R L U2 R' L U2 L2 U2 R L' U2 R' L' U // 20
4 - (R' U' L' U' L R) D2 (R' L' U L U R) // 14
5 - (L' U' R' U' L R) D2 (R' L' U R U L) // 14
6 - U R U2 R' L' U2 R U2 R' U2 R L U2 R' U y' // 20
Total movecount: 80

Doing this solution swaps the part of the UL edge that is also part of the circle on the left side with the part of the UR edge that is also part of the circle on the right side. Mathematically, it is not parity because an even number of moves (In QTM) were used.

4 and 5 are conjugates. I used cube explorer for an optimal {R,U,L} solution for 2 and 6. I am aware that I could combine 2 and 6 into one part, but this way it is easier to see what I have done with the conjugates.

On a supercube (holding the cube how I started), the F and U centers are both turned 180 degrees and the part of the FR edge that is also part of the right circle is swapped with the part of the BR edge that is also part of the right circle, as well as the swap you would see on a regular cube.

 

[cmhardw]

I know the mars circle cube is one of them, and so is any big cube. You can make it appear as if you swapped 2 center pieces on a big cube by doing a 3 cycle of centers when 2 are the same color. 3-cycles take an even number of moves in QTM and are not parity in a mathematical sense. I'm sure there are others, but I don't own any others.

Here is a proof for the mars circle cube. I started holding the cube so that the circles that don't turn with the side are on the left, right, and bottom. Movecounts are in QTM.

1 - (R2 U2)3 y // 12
2 - U R L U2 R' L U2 L2 U2 R L' U2 R' L' U // 20
4 - (R' U' L' U' L R) D2 (R' L' U L U R) // 14
5 - (L' U' R' U' L R) D2 (R' L' U R U L) // 14
6 - U R U2 R' L' U2 R U2 R' U2 R L U2 R' U y' // 20
Total movecount: 80

Doing this solution swaps the part of the UL edge that is also part of the circle on the left side with the part of the UR edge that is also part of the circle on the right side. Mathematically, it is not parity because an even number of moves (In QTM) were used.

4 and 5 are conjugates. I used cube explorer for an optimal {R,U,L} solution for 2 and 6. I am aware that I could combine 2 and 6 into one part, but this way it is easier to see what I have done with the conjugates.

Very cool! Is the swapping of those two pieces truly the only effect on the cube in the supercube sense?

If so that's very neat! I am not as familiar with the puzzles that have these circular parts like this. This makes me want to get one of these puzzles.

 

[elrog]

I edited my first post to include supercubes. On a supercube, you swap other pieces and turn some centers.

 

[cmhardw]

I edited my first post to include supercubes. On a supercube, you swap other pieces and turn some centers.

Ah, that makes more sense to me. So it seems that this "parity" is a parity in the same sense that having two t-centers swapped on a 5x5x5 is a "parity".

I would venture to say that permutation parity for the circular pieces on this puzzle has no meaning because of the indistinct nature of those pieces of the same color. This is the same way that we would say that the concept of permutation parity on a 5x5x5's t-centers has no meaning since the pieces on the same face, of the same color, are indistinct.

Once you make the puzzle a supercube (for the mars cube or the 5x5x5), then the piece orbits once again follow all the rules of permutation parity that we would expect of a Rubik's style puzzle.

 

[elrog]

Ah, that makes more sense to me. So it seems that this "parity" is a parity in the same sense that having two t-centers swapped on a 5x5x5 is a "parity".

This is correct. I would still encourage you to get one though because they are fun to figure out how to solve. I wish I would have gotten sickerless ones though because the small stickers don't like to stay on very well.

I would love to get a 4x4 circle cube (the one with the larger circle) because I can foresee this "parity" happening at the same time as actual 4x4 parity and making a fun solve.

 

[kikenyoy]

Thank you for all the replies.