Known Distance-25 in the Quarter-Turn Metric

[swipymam]

Using the "Rubik's Cube Signature" (R'CS) program, 2 more positions at a distance of 25q have been discovered:

 

[rokicki]

These are the two already-known distance-25 positions.

 

[swipymam]

These are the two already-known distance-25 positions.

Perhaps. I was checking against your findings (https://www.cube20.org/distance20s/): “There are essentially only two known distance-25 positions, both immediate neighbors of the above distance-25 positions; scrambles for them are:
U1U1F1U1U1R3L1F1F1U1F3B3R1L1U1U1R1U1D3R1L3D1R3L3D3 Scramble: ccciiiihhhhqqqqttt (vibcatch24.github.io/R-CS/)
U1F1U1U1R3L1F1F1U1F3B3R1L1U1U1L1U1D3R3L1D1R3L3U1U1” Scramble: bdbuvuvtetemnmnhhh

The two positions I mentioned were not there.
Scramble: cccffffhhhhrrrrttt R L' U B D R' F B U R' B' R' U D2 R L B2 U2 D2 R L
Scramble: uvudbdbnmnmtetehhh R2 L2 F2 D' F B' U D' F' U' D' L' B2 U D' R2 F' R' L' D'

It is difficult for me to verify with my imperfect solver at the moment whether these additional scrambles require fewer than 25 qtm, or if they are simply absent from the source mentioned above. Am I possibly overlooking something? If so, what is it?
I see some symmetries in these positions. Perhaps you excluded them intentionally?

 

[rokicki]

The two positions you give are just rotated versions of the ones in the files. For brevity the files only include one symmetry representative. But the positions you list above are indeed distance 25, just not new.

To see this without any code, just put your scrambles into any visualization program, and observe that the positions are exactly one quarter-turn away from the known distance-26 position.

 

[swipymam]

To see this without any code, just put your scrambles into any visualization program, and observe that the positions are exactly one quarter-turn away from the known distance-26 position.

Following your advice, I have visualized all of these positions, without a doubt that they have long been known to you.

Could you please explain something? I see that the top two positions are one move (D, D') away from the single 26q* position located in the center. But for the bottom two positions, I don't see this move. Could you elaborate on this, as it seems to be related to symmetries?

Perhaps the explanation for this is the existence of another 26q* position:

Scramble: bdbdbdbtetetetehhh R2 L2 F2 D' F B' U D' F' U' D' L' B2 U D' R2 F' R' L' U' D' (18t, 26q*, 21f) ↩

Unfortunately, these cubes are not equivalent, so one of them cannot be discarded as redundant. Their solutions do not completely coincide (under the substitution U <-> F & D <-> B) out of the 10 verified solutions for the first scramble ( ccccccchhhhhhhhttt ) and the 9 solutions for the second scramble ( bdbdbdbtetetetehhh ). At least four solutions for the first have no corresponding solutions in the second, and three solutions for the second have no corresponding solutions in the first.

 

[rokicki]

Just take your first 26q solution and conjugate by x to get a solution to your last position:

 Twizzle link 

x F U2 R' L F2 U F' B' R L U2 R U D' R L' D R' L' U2 D2 x'

So, clearly, they are equivalent. (They are also equivalent just by inspection; clearly both are superflip plus fourspot, just a different fourspot.)

-tom

 

[swipymam]

So, clearly, they are equivalent.

Yes, the cubes (ccccccchhhhhhhhttt) & (bdbdbdbtetetetehhh) are equivalent. The imperfect solver was overlooking an additional R<->L swap, which is why some of the solutions converged while others did not. Thank you very much for your time and attention.

Let's summarize the search for 25q positions. Out of all ≈ 4.3*10^19 possible positions of the 3x3x3 Rubik's Cube, there are only 3 equivalent positions that require 26q* moves to solve (in the QTM metric):
I. Scramble: ccccccchhhhhhhhttt F U2 R' L F2 U F' B' R L U2 R U D' R L' D R' L' U2 D2 (26q*)
II. Scramble: bdbdbdbtetetetehhh R F2 U' D R2 F R' L' U D F2 U F B' U D' B U' D' F2 B2 (26q*)
III. Scramble: dbdbdbdeteteteteee U L2 F B' U2 L U' D' F B L2 B R' L F' B R F' B' R2 L2 (26q*)

Despite being equivalent to each other, they are still distinct positions (which is clearly visible from their Catch-24 position codes).
In turn, each of these 3 positions leads to 12 more positions at a distance of 25q (there are 36 different 25q signatures in total).
The move for changing the above-mentioned positions is shown after the < symbol

I.
Scramble: iiiccccqqqqhhhhttt F U2 R' L F2 U F' B' R L U2 R U D' R L' D R' L' U' D2 ( 25q) < U
Scramble: fffccccrrrrhhhhttt F U2 R' L F2 U F' B' R L U2 R U D' R L' D R' L' U D2 ( 25q) < U’
Scramble: ccciiiihhhhqqqqttt F U2 R' L F2 U F' B' R L U2 R U D' R L' D R' L' U2 D’ ( 25q) < D
Scramble: cccffffhhhhrrrrttt F U2 R' L F2 U F' B' R L U2 R U D' R L' D R' L' U2 D ( 25q) < D’

Scramble: ccxtccxohhhohhhptt U2 D2 F2 L' U D' R' L D' F' B' R' F2 U' D R2 F' U' D' L' ( 25q) < R
Scramble: ccwsccwphhhphhhott R U D F R2 U D' F2 R F B D R L' U' D L F2 D2 U2 ( 25q) < R’
Scramble: xtccxtchhohhhohtpp U2 D2 F2 L' U D' R' L D' F' B' R' F2 U' D R2 F' U' D' R' ( 25q) < L
Scramble: wsccwschhphhhphtoo L U D F R2 U D' F2 R F B D R L' U' D L F2 D2 U2 ( 25q) < L’

Scramble: tccccxthohhhohhnnt R2 L2 U2 F' U D' F' B U' F' B' L' D2 F' B R2 U' R' L' B' ( 25q) < F
Scramble: sccccwshphhhphhmmt F R L U R2 F B' D2 L F B U F B' U' D F U2 L2 R2 ( 25q) < F’
Scramble: cxtxtcchhhohhhottn R2 L2 U2 F' U D' F' B U' F' B' L' D2 F' B R2 U' R' L' F' ( 25q) < B
Scramble: cwswscchhhphhhpttm B R L U R2 F B' D2 L F B U F B' U' D F U2 R2 L2 ( 25q) < B’


II.
Scramble: uvudbdbnmnmtetehhh R2 L2 F2 D' F B' U D' F' U' D' L' B2 U D' R2 F' R' L' D' ( 25q) < U
Scramble: vuvdbdbmnmntetehhh U R L F R2 U' D B2 L U D F U' D F' B D F2 L2 R2 ( 25q) < U’
Scramble: bdbvuvutetenmnmhhh R2 L2 F2 D' F B' U D' F' U' D' L' B2 U D' R2 F' R' L' U' ( 25q) < D
Scramble: bdbuvuvtetemnmnhhh D R L F R2 U' D B2 L U D F U' D F' B D F2 R2 L2 ( 25q < D’

Scramble: bdwsbdwpetepeteohh U2 D2 F2 R' F B' R' L F' R' L' D' B2 R' L U2 F' U' D' L' ( 25q) < R
Scramble: bdxtbdxoeteoetephh R U D F U2 R L' B2 D R L F R L' F' B R F2 D2 U2 ( 25q) < R’
Scramble: wsbdwsbtepetepehoo U2 D2 F2 R' F B' R' L F' R' L' D' B2 R' L U2 F' U' D' R' ( 25q) < L
Scramble: xtbdxtbteoeteoehpp L U D F U2 R L' B2 D R L F R L' F' B R F2 D2 U2 ( 25q) < L’

Scramble: hdbdbjhtktetkteqqh U F2 R L' U2 F U' D' R L F2 L F B' R' L B R' L' F' B2 ( 25q) < F
Scramble: edbdbgetltetlterrh U F2 R L' U2 F U' D' R L F2 L F B' R' L B R' L' F B2 ( 25q) < F’
Scramble: bjhjhdbtetktetkhhq U F2 R L' U2 F U' D' R L F2 L F B' R' L B R' L' F2 B’ ( 25q) < B
Scramble: bgegedbtetltetlhhr U F2 R L' U2 F U' D' R L F2 L F B' R' L B R' L' F2 B ( 25q) < B’


III.
Scramble: vuvbdbdmnmneteteee R2 L2 B2 U' F' B U' D B' U' D' L' B2 U D' R2 F' R' L' D' ( 25q) < U
Scramble: uvubdbdnmnmeteteee U R L F R2 U' D B2 L U D B U D' F B' U B2 L2 R2 ( 25q) < U’
Scramble: dbduvuvetetmnmneee R2 L2 B2 U' F' B U' D B' U' D' L' B2 U D' R2 F' R' L' U' ( 25q) < D
Scramble: dbdvuvuetetnmnmeee D R L F R2 U' D B2 L U D B U D' F B' U B2 R2 L2. ( 25q) < D’

Scramble: dbjhdbjktetktetkee U D B2 R2 L2 U D R2 L' U' D R B L D' F B R D' B' D' ( 25q) < R
Scramble: dbgedbgltetltetlee U' F R F U' R' L' F D' R' U' F' B D' B' F' R2 B' F' D2 R L2 ( 25q) < R’
Scramble: jhdbjhdetktetktekk F B U2 R2 L2 F B L2 R' F B' L U R F' U D L F' U' F' ( 25q) < L
Scramble: gedbgedetltetltell U D' R F L U' F B R U' F' U' R L2 D U F2 L2 R2 D U ( 25q) < L’

Scramble: sbdbdwsepetepetmme R2 L2 D L2 F' B D2 L U' D' F B L2 F R' L F B' R B’ ( 25q) < F
Scramble: tbdbdxteoeteoetnne F R' F' B R L' F' L2 F' B' U D L' D2 F B' L2 D' R2 L2 ( 25q) < F’
Scramble: dwswsbdetepetepeem R2 L2 D L2 F' B D2 L U' D' F B L2 F R' L F B' R F' ( 25q) < B
Scramble: dxtxtbdeteoeteoeen B R L U R2 F B' D2 L F B D F' B U D' B D2 L2 R2 ( 25q) < B’

 

[Herbert Kociemba]

Eventually it is noteworthy that there are essentially only two different known 25q* positions. For example for your first scramble
S1 = F U2 R' L F2 U F' B' R L U2 R U D' R L' D R' L' U' D2 (26q*)
S1*U, S1* U', S1*D and S1*D' all result in equivalent positions.
S1*R, S1*R', S1*L and S1*L', S1*F, S1*F', S1*B and S1*B' also are all equivalent to each other and give the other position.

 

[swipymam]

Eventually it is noteworthy that there are essentially only two different known 25q* positions.

Reducing 12 actual positions, each one move away from the initial state, to just two is quite a puzzle, but I suppose I can agree with you, if we call the 'two distinct positions' two distinct categories of positions that are solved using mirrored or inverse algorithms.

 

[Herbert Kociemba]

It is not difficult if you look at the symmetries of the 26q* position S1 with Cube Explorer.

The blue dot on the top means that the position has a 90° rotational symmetry around the UD axis. This implies that R, F L and B moves lead to equivalent positions. The red dot on the R face means that there is a 180° rotational symmetry around the RL axis. This implies that U and D moves give equivalent positions. There is a horizontal mirror symmetry which maps the U to the D face (and the D face to the U face). This shows that U' and D' moves give equivalent positions to U and D moves. Another mirror plane maps the R face to the L face and another the F face to the B face. This shows that R', L', F' and B' moves lead to equivalent positions as R, L, F and B moves. So there are only two equivalence classes, one with 8 positions and one with four positions.

 

[swipymam]

It is not difficult if you look at the symmetries of the 26q* position S1 with Cube Explorer.

A beautiful conclusion to my research, which I believe was not redundant and helps to better understand the (categories) classes of symmetry.

Can one more question about non-spatial symmetries be clarified?

It says: "Only 79 780 positions of distance-24 in the quarter-turn metric are known. Of these, 78,820 have some form of symmetry; there are 3,324 different positions with respect to symmetry and antisymmetry. " - Of these 3,324 (3341) different positions, at least 1/6 (596) are reversible, for which the scratch sequence is their same solution (as for the S1-superflip consisting of four spots).

That is, the antisymmetry property, or in other words the presence of a position having symmetry with respect to the inversion operation, is quite large (1/6). I wonder if this is inherent for states "far" from the solution (like distance-24) or if there is some general pattern?

 

[Herbert Kociemba]

Your observation is right. Symmetry and antisymmetry happen much more often for positions which have a large distance to solved in both QTM and HTM. I have no good explanantion for this.
By the way antisymmetry is more complicated than symmetry. While there are 33 types of symmetry, there are 131 types of antisymmetry. While there are exactly 164,604,041,664 symmetric cubes - with exactly 1,091,994 of them with 20 moves in HTM, which is about 600,000 times more than the ratio of 20 move positions for all cubes - we only know the number of cubes for each of the 131 antisymmetry types. But we do not know the number of 20f*/26q* positions among these. See https://kociemba.org/antisymmetric.html if you want to know more about antisymmetry.