# looking for nxnxn supercube "atom" algorithms

#### tongjunhui

##### Member
Hi guys,

I'm looking for efficient nxnxn "atom" algorithms, here "atom" means a set of orthogonal algorithm with minimal possible "effect" on the cube. With these algorithms I can make any pattern on the cube (without breakout everything and build from bottom up).

For example, for 3x3x3 cube, the set of "atom" algorithms are:
- corner 3-cycle. (1 algorithm + conjugates => be able to loop any 3 corners)
- edge 3-cycle.
- corner swap + edge swap.
- flip 2 edges.
- rotate 2 corners by 120 degree.
- rotate 2 center pieces by 90 degree.

I will be very pleased if I can extend the above to nxnxn, my current progress are
- [done] 3-cycle of any pieces (corners, edge, centers, with the same kind) by: [aR' bD aR, U], where a and b are between 1 and n
- [done] flip 2 center edges (also by commutator, e.g. [R' E' R2 E2 R', U])
- [done] rotate 2 corners by 120 degree. (just like 3x3x3 does)
- [done] rotate 2 center pieces by 90 degree. (by [M' E M, U])
- [NEED HELP] swap 2 non-center edges with only one corresponding "+center" swap.
1. The traditional parity algorithm (2R2 B2 U2 ... one) on nxnxn has "(n-2)" additional center pieces swap: for even n, they are all unnecessary, for odd n, only one out of them (the "+center one") is necessary.
2. I extended the "super cube safe" 4x4x4 parity algorithm on wiki to 6x6x6 but a few arc center pieces affected suprisingly: https://alg.cubing.net/?alg=2R-_U-_2&#45;3u_2R_U-_2R_U_2R_2&#45;3u-_2R-_2&#45;3u_2R_U_2R_U-_2R_U-_2&#45;3u-_2R-_U2&puzzle=6x6x6
3. apparently I can do a (2R2 B2 U2 ...) one followed by several center pieces fix-up, e.g. (on 5x5x5) :
(2R2 B2 U2 2L U2 2R' U2 2R U2 F2 2R F2 2L' B2 2R2) (2B L' 2R' 2D 2R U' 2R' 2D' 2R U L 2B') y' (2B L' 2R' 2D 2R U' 2R' 2D' 2R U L 2B') y
but a big thinks if someone can provide shorter algorithms!

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#### Christopher Mowla

2. I extended the "super cube safe" 4x4x4 parity algorithm on wiki to 6x6x6 but a few arc center pieces affected suprisingly: https://alg.cubing.net/?alg=2R-_U-_2&#45;3u_2R_U-_2R_U_2R_2&#45;3u-_2R-_2&#45;3u_2R_U_2R_U-_2R_U-_2&#45;3u-_2R-_U2&puzzle=6x6x6
In the supercube single dedge flip algorithm part of the wiki, which I see you already had a peak at, specifically, in the yellow (highlighted) side note, I provide a link to a post of mine which is a mathematical expression which tells you the minimum number of center pieces which must be unsolved in the nxnxn supercube when a certain number of sets (orbits) of non-middle edge pieces (we call them "wings") are in an odd permutation.

I think you will enjoy this paper that I wrote about this topic. (It's not my final revision. I improved on it years ago, but that improvement . . . and additional content . . . is in my to-be published book.)

3. apparently I can do a (2R2 B2 U2 ...) one followed by several center pieces fix-up, e.g. (on 5x5x5) :
(2R2 B2 U2 2L U2 2R' U2 2R U2 F2 2R F2 2L' B2 2R2) (2B L' 2R' 2D 2R U' 2R' 2D' 2R U L 2B') y' (2B L' 2R' 2D 2R U' 2R' 2D' 2R U L 2B') y
but a big thinks if someone can provide shorter algorithms!
I stated in the introduction in the wiki that the //Safe occurs in various portions of the wiki. There are other 2-cycles (non-middle edge swaps) of wings covered on that page. One of the algorithms (from the non-symmetrical part of this section) is only 20 moves, and it works on the 5x5x5 (and nxnxn):
2L U 2L U' 2L' U' 2L U 2L U 2L' D' 2B2 2L 2B2 2L' U2 D 2L' U//Safe

You can see similar //Safe algorithms in other 2-cycle sections on the wiki page.

But if you want it in the form of the single dege flip, you can shift this algorithm and conjugate with two moves. (24 moves instead of 39)
2B2 L2 D 2L' U 2L U 2L U' 2L' U' 2L U 2L U 2L' D' 2B2 2L 2B2 2L' U2 L2 2B2

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#### tongjunhui

##### Member
provide a link to a post of mine which is a mathematical expression
Yes I found this post previously and I planned to have a deeper look later. After carefully examined your formular, I've found a mistake I made. see below.

But if you want it in the form of the single dege flip, you can shift this algorithm and conjugate with two moves. (24 moves instead of 39)
2B2 L2 D 2L' U 2L U 2L U' 2L' U' 2L U 2L U 2L' D' 2B2 2L 2B2 2L' U2 L2 2B2
Thanks! Both for the alrorithm ifself and for the correction of my wrong understanding that "for even n, only the corresponding +center need to match the wing edge parity". -- I examined the effect of this algorithm and found that for your 7x7x7 case, there are 2 "unexpected" center swap besides the expected +center one. Finally I relized the 2 "unexpected" swap can not be reduced (because they are in different orbits) , so they are actually expected. I made some mistake and will revise my goal with the new understanding.

BTW are there any online super cube? alg.cubing.net don't have supercube so I need to write some code to analysis the algorithm impact