Parity on higher order puzzles.

[Christopher Mowla]

For me, the legitimate big cube base puzzle is the smallest cube size which is able to illustrate the bigger picture for the nxnxn regarding algorithm translation. Since many algorithms that work on the 4x4x4 can't work on larger cube sizes, it is NOT the base puzzle. The 5x5x5 is much better, as it includes the central edge pieces and its centers are affected by an odd permutation in the wing orbits. However, see this spoiler of what I believe to be the optimal opposite wing edge swap algorithm in quarter turns.

Spoiler

WCA Notation
4x4x4 (WCA Notation)
r u Rw' Uw' r' Uw Lw Uw' r2 Fw' r' Fw r' Uw f' r' x

5x5x5 (WCA Notation)
(M'r) u Rw' Uw' r' Uw (LwM) Uw' r2 Fw' r' Fw r' Uw f' (Mr') x

6x6x6 (SiGN Notation)
Outer orbit
2-3r 2-3u 3r' 3u' 2R' 3u 3l 3u' 2R2
3R
3f' 2R' 3f
3R'
2R' 3u 2-3f' 2-3r' x

Inner orbit
2-3r 2-3u 3r' 3u' 3R' 3u 3l 3u' 3R2
2R
3f' 3R' 3f
2R'
3R' 3u 2-3f' 2-3r' x

7x7x7 (SiGN Notation)
Outer
2-4r 2-4u 3r' 3u' 2R' 3u 4l 3u' 2R2
3R
3f' 2R' 3f
3R'
2R' 3u 2-4f' 2-4r' x

Inner
2-4r 2-4u 3r' 3u' 3R' 3u 4l 3u' 3R2
2R
3f' 3R' 3f
2R'
3R' 3u 2-4f' 2-4r' x

As you can see, if we apply the algorithm to individual orbits of the 6x6x6 and 7x7x7, we need interior conjugates (3R for the outer orbit and 2R for the inner orbit) to not discolor the centers. Don't believe me? Just apply the same algorithm without those conjugates. Therefore, the 5x5x5 is not good enough either.

I also agree with what Per said about that the base puzzle should have all piece types. Therefore, the 6x6x6 would not qualify as the base puzzle for all cube sizes based off of that statement by itself (it doesn't have central edges or any middle center pieces). Here is a counterexample algorithm to anyone who thinks differently.

Spoiler

4x4x4 (WCA Notation)
x' Uw l Uw' Rw2 x Uw' l' Uw Lw' L' x Uw Rw' Uw' l' Uw (L x) Uw' x

5x5x5 (WCA Notation)
x' Uw l Uw' M2Rw2 x Uw' l' Uw Rw'M x L' x Uw M Rw' Uw' l' Uw Lx Uw' x

6x6x6 (SiGN Notation)
Outer
x' 3u
3L
2L 3u' 3r2 x 3u' 2L' 3u 3l' L' x
3L'
3u 3r' 3u' 2L' 3u L x 3u' x

Inner
x' 3u
2L
3L 3u' 3r2 x 3u' 3L' 3u 3l' L' x
2L'
3u 3r' 3u' 3L' 3u L x 3u' x

7x7x7 (SiGN Notation)
Outer
x' 3u
3L
2L 3u' 4r2 x 3u' 2L' 3u 4r' x L' x
3L'
3u 4r' 3u' 2L' 3u L x 3u' x

Inner
x' 3u
2L
3L 3u' 4r2 x 3u' 3L' 3u 4r' x L' x
2L'
3u 4r' 3u' 3L' 3u L x 3u' x

As you can see with the translations to the 5x5x5 and 7x7x7, middle center pieces are discolored. This time if you delete the interior conjugates (3L for the outer orbit and 2L for the inner orbit), it's even uglier. (Very complicated alg.)

So, from an algorithm point of view, the 6x6x6 is the base cube sizes for even cubes and the 7x7x7 is the base cube for odd cubes. However, if an algorithm works correctly on a 7x7x7, it definitely will work on a 6x6x6 (with minor adjustments should the algorithm move the just the central slice on the odd cube). So I'm in favor of the 7x7x7 being the base cube size and have stated my reasoning why I think so.

 

[riffz]

My idea of higher order is basically like so: more parallell layers in all directions than an otherwise equivalent puzzle.
Higher order should not introduce any new kind of external pieces, only more of them. What happens internally is not of interest to me.
The base puzzle should have all kinds of external pieces, that a lower order could possibly have. The 2x2x2 loses the 3x3x3's centers and does not qualify in my definition as the base puzzle for cubes.

7x7 is a base cube then, if you count obliques as a distinct piece type.

 

[mrCage]

Corner pieces has faces on more than 2 of the puzzle's faces. Edge pieces has faces on exactly 2 faces of the puzzle. Center pieces lie entirely on a single face. Orbitals is something else .... My DEFINITION ....

Per

 

[riffz]

Corner pieces has faces on more than 2 of the puzzle's faces. Edge pieces has faces on exactly 2 faces of the puzzle. Center pieces lie entirely on a single face. Orbitals is something else .... My DEFINITION ....

Per

That works then