Helicopter cube solving (and patterns)

[coinman]

Just got a 5:51.xx solve on my third attempt (the two first was hoers long). I bought two of them so i still have one that is solvable

 

[qqwref]

I wonder what the God's Algorithm count would look like for LL corners, just using UL/UB/UF/UR turns.

Well, actually, nevermind that, I did the calculations. Here are the number of positions at each depth:
0 moves: 1 (that's the solved case)
1 move: 4
2 moves: 10
3 moves: 20
4 moves: 34
5 moves: 52
6 moves: 74
7 moves: 100
8 moves: 130
9 moves: 128
10 moves: 74
11 moves: 20
12 moves: 1 (that's the U2 case)
The average case takes 7.39 moves to solve.

 

[Portponky]

Here is a 9 move Y-perm for the last layer: UR, UB, UR, UL, UF, UL, UR, UB, UR.

 

[mrCage]

I wonder what the God's Algorithm count would look like for LL corners, just using UL/UB/UF/UR turns.

Well, actually, nevermind that, I did the calculations. Here are the number of positions at each depth:
0 moves: 1 (that's the solved case)
1 move: 4
2 moves: 10
3 moves: 20
4 moves: 34
5 moves: 52
6 moves: 74
7 moves: 100
8 moves: 130
9 moves: 128
10 moves: 74
11 moves: 20
12 moves: 1 (that's the U2 case)
The average case takes 7.39 moves to solve.

Details??

Per

 

[qqwref]

What do you mean? What details do you want? I'm not gonna just go posting an optimal solution to each position

 

[mrCage]

What do you mean? What details do you want? I'm not gonna just go posting an optimal solution to each position

I mean how did you find it? GAP??

Per

I came to think of an interesting subgroup: [UF,UR,FR] - a 3 generator subgroup. What would gods algorithm be for this group? Rather "Gods distribution"

Per

 

[qqwref]

No, I don't have GAP, I used ksolve (+ some javascript stuff to generate cases).

The <UR,UF,FR> subgroup sounds interesting, only about 2600 cases so it's still doable with my primitive method I'll take a look.

EDIT: Tried it, here's what comes up:
Depth 0: 1
Depth 1: 3
Depth 2: 6
Depth 3: 12
Depth 4: 24
Depth 5: 48 [it stops doubling here, so if you do up to 5 moves there is exactly 1 optimal solution and it's the inverse]
Depth 6: 93
Depth 7: 180
Depth 8: 315
Depth 9: 489
Depth 10: 604
Depth 11: 522
Depth 12: 250
Depth 13: 42
Depth 14: 3 [RF UF RF UR UF UR UF RF UF RF UR UF UR UF, or a corner 2-2 cycle, and its rotations]
So that's 2592 total cases, with an average of 9.45 moves to solve.

 

[coinman]

My method (when it's been scrambled without jumbling) is as follows:
- First layer centers.
- Finish first layer by pairing the corners up with two centers each before inserting.
- Last layer corners (I solve one by one, then do last two at once).
- Last layer centers, do the 3-cycles first and then any 2-cycles in pairs.
I use three main algs: J-perm (UL UF UR UF UL), 2-2 cycle ((UF UR)3), 3-cycle ((UF FR UR FR)2) and inverse. And see, it works pretty well:
36.48, (1:01.38), (34.00), 51.96, 42.26, 41.95, 46.47, 57.55, 54.00, 45.83, 49.07, 47.68 => 47.32

How do you do this step? I can't find a way to pair the corners with centers easily and if manage i can't keep them paired up wile moving it to insert it in the right slot.

 

[Portponky]

- Finish first layer by pairing the corners up with two centers each before inserting.

How do you do this step? I can't find a way to pair the corners with centers easily and if manage i can't keep them paired up wile moving it to insert it in the right slot.

What I do is first I get the corner underneath its final location with the top layer colour facing downwards, so it's only one twist away from being solved. You want to get one center on either side of this corner. Find one of the centers (there are only four valid locations) and get it one twist away from aligning with the corner. Twist the corner away, twist the center in, and then twist the corner back. Repeat this for the other center, and then you can twist the corner up to the top layer along with the centers. Repeat again for the remaining corners and you've done that step.

 

[mrCage]

No, I don't have GAP, I used ksolve (+ some javascript stuff to generate cases).

The <UR,UF,FR> subgroup sounds interesting, only about 2600 cases so it's still doable with my primitive method I'll take a look.

EDIT: Tried it, here's what comes up:
Depth 0: 1
Depth 1: 3
Depth 2: 6
Depth 3: 12
Depth 4: 24
Depth 5: 48 [it stops doubling here, so if you do up to 5 moves there is exactly 1 optimal solution and it's the inverse]
Depth 6: 93
Depth 7: 180
Depth 8: 315
Depth 9: 489
Depth 10: 604
Depth 11: 522
Depth 12: 250
Depth 13: 42
Depth 14: 3 [RF UF RF UR UF UR UF RF UF RF UR UF UR UF, or a corner 2-2 cycle, and its rotations]
So that's 2592 total cases, with an average of 9.45 moves to solve.

How do you make ksolve run so fast?? Hmm...

Per

 

[qqwref]

Hm, what do you mean? It took about 20 seconds for ksolve to generate those cases (but I do have optimal solutions for them all now ) so at about 100 cases/sec it would take a LOT of time for anything serious like pyraminx or 2x2. I imagine an algorithm which goes through the state tree once would take a lot less time than one which separately computes each optimal solution, though.

 

[mrCage]

Hm, what do you mean? It took about 20 seconds for ksolve to generate those cases (but I do have optimal solutions for them all now ) so at about 100 cases/sec it would take a LOT of time for anything serious like pyraminx or 2x2. I imagine an algorithm which goes through the state tree once would take a lot less time than one which separately computes each optimal solution, though.

I remember the ridiculousslow times of ksolve for some domino cube cases when the optimal solution was more than 10 turns. Maybe i had bad definition file or used old ksolve version??

PS! How would you do a pure U-layer supertwist?? (UFR+,UFL-,UBL+,UBR-)

Per

 

[coinman]

For the case in the end of this video i think the jumbling move is really useful http://www.youtube.com/watch?v=9RyBURilLzY&feature=related

 

[coinman]

Has anybody fund a good alg for flipping two corners? I found one but it's 24 moves, i like something shorter.
Right now i'm trying to find algs for al corner cases so i can solve F2L then al centers and finally the four remaining corners, this method might be worth trying.

I also found a short alg that swapped two corners without turning them. I think it was les then ten moves + a double jumble to swap four centers back as seen in the video above, but i totally forgot what i did

 

[qqwref]

There's [DF, FR UR UB UL UB UR UF FR], 18 moves, but it's still pretty long. I seriously think you are going to have better luck (from an efficiency standpoint) doing the corners before the centers.

 

[mrCage]

There's [DF, FR UR UB UL UB UR UF FR], 18 moves, but it's still pretty long. I seriously think you are going to have better luck (from an efficiency standpoint) doing the corners before the centers.

Helicopter solving is still in its infancy. Lets not kill off any ideas prematurely. I have alredy posted an alg for twisting 2 corners:

[FR FD FR , FU] [FD FR FD , FL] (16 turns)

Skip the first turn (cyclical shift) for twisting 2 diagonally opposite cornera inatead, also 16 turns. Still working on the super twist. I'm down to 22 turns, due to partial cancellation (still twicw the 12 turn 3- twist)

Per

 

[Portponky]

You can solve all the last layer corners in 16 turns with the method I posted so unless you are making big savings on the centers this doesn't seem worth it. What is your method for the last layer centers?

 

[mrCage]

You can solve all the last layer corners in 16 turns with the method I posted so unless you are making big savings on the centers this doesn't seem worth it. What is your method for the last layer centers?

I am talking about twisting 4 corners purely, with centers unaffected. Personally i do corners first then centers. I am not fast at all, nor is that a priority I guess i need 5-10 mins for each solve. I willwork more on some pretty patterns as time goes by ...

Per

 

[coinman]

[FR FD FR , FU] [FD FR FD , FL] (16 turns)

Skip the first turn (cyclical shift) for twisting 2 diagonally opposite cornera inatead, also 16 turns. Still working on the super twist. I'm down to 22 turns, due to partial cancellation (still twicw the 12 turn 3- twist)

Per

After posting the above i also found a 16 move corner flip (maybe optimal?). It's basically a combination of a 3-cycle Per came up with and one i cam up with my self. But after studding the two 3-cycles i see they are the same from different angels, as is the whole corner flip alg the same Per came up with but in U instead of F
Pers 3-cycle (RU FU RU BU)*2 my 3-cycle (FU RU FU LU)*2 .

I also now rediscovered how to swap two corners without turning them. It takes a four centers jumbling twist. Just do UR UL jumble UF UR UL. You can save a quarter turn on the first UL by stopping in the jumbling position.

You can solve all the last layer corners in 16 turns with the method I posted so unless you are making big savings on the centers this doesn't seem worth it. What is your method for the last layer centers?

I didn't study the method you got to solve the LL corners in 16 moves but maybe i should, is it hard to recognize the cases? I solve the corners intuitionally, this can take quit a lot of moves.
On the LL centers i mainly used the (RU FU)*3 and the (UF FR UR FR)*2 sometimes i also use (UR UF UL)*4 that moves six centers in two three cycles and the double jumble to swap four centers

The problem i se with solving the centers last is that it varies a lot from solve to solve how many centers you have to move. A lot of times they are also spread out in ways that force you to do set-up moves to solve them with the algs above. Or is there ways to control this?

Solving the centers after F2L is easy if you don't have to preserve corners so i was thinking that a method for LL corners that don't effects centers cud be more efficient.

My best time with centers last is so far 1:56.13 and after only a few solves 2:14.05 with corners last.

Edit: Now after only five-six solves with corners last i beat my PB, 1:43.47

 

[qqwref]

Hi, I computed some optimal 5-gen (allowing UF, UR, UB, UL, FR) edge twist cases. I tried all 4 orientations but allowing FR didn't improve the optimal solution at all so I guess using LL twists only is fastest. These are all LL-only optimal solutions I got, up to y rotations.

"T" orientation (adj. 2 twist):
- [UR (UF UB) UL (UF UB)]2 (12f*)
"U" orientation (adj. 2 twist, inverse of T):
- [(UF UB) UL (UF UB) UR]2 (12f*)

"L" orientation (opp. 2 twist):
- [(UF UR UF) (UB UL UB)]2 (12f*)
- [(UR UF UR) (UL UB UL)]2 (12f*)

"sune" orientation (3 twist):
- [UF UR UB UL]3 (12f*)
"antisune" orientation (3 twist):
- [UF UL UB UR]3 (12f*)

"H" orientation (symmetrical 4 twist):
- [(UF UB) UR UB (UR UL) UF UR]2 (16f*)
- [(UF UB) UR UF (UR UL) UB UR]2 (16f*)
- [UF (UL UB UL) (UR UB UR UB)]2 (16f*)
- [(UF UL UF) UB (UR UF UR) UL]2 (16f*)
- [UF (UR UB UR) (UL UB UL UB)]2 (16f*)
- [(UF UR UF) UB (UL UF UL) UR]2 (16f*)
- [(UL UB UL UB) (UF UL UF) UR]2 (16f*)
- [(UL UF UL UF) (UB UL UB) UR]2 (16f*)

"Pi" orientation (asymmetrical 4 twist):
- [(UB UR UB) (UF UR UF) (UR UL)]2 (16f*)
- [UB UR (UL UF UL UF) UB UL]2 (16f*)
- [(UF UR UF) (UB UR UB) (UR UL)]2 (16f*)
- [UF UR (UL UB UL UB) UF UL]2 (16f*)
- [(UR UB UL UF) (UB UR UB) UL]2 (16f*)
- [(UR UB UR) (UF UB) (UL UB UL)]2 (16f*)
- [UR UF (UB UR UB UR) UL UF]2 (16f*)
- [UR UB (UF UR UF UR) UL UB]2 (16f*)
- [(UR UF UL UB) (UF UR UF) UL]2 (16f*)
- [(UR UF UR) (UF UB) (UL UF UL)]2 (16f*)
- [(UR UL) (UB UL UB) (UF UL UF)]2 (16f*)
- [(UR UL) (UF UL UF) (UB UL UB)]2 (16f*)

I also now rediscovered how to swap two corners without turning them. It takes a four centers jumbling twist. Just do UR UL jumble UF UR UL. You can save a quarter turn on the first UL by stopping in the jumbling position.

Ah, of course. This is a quite nice way to do this. I don't mind that it involves jumbling because you'd have to jumble to get to this position anyway