# Curvy Copter algs



## mrCage (Sep 30, 2011)

Hi!

I challenge anyone to come up with a shortest possible way (please provide sequence or description) to flip all the 12 middle edges of the curvy copter. So far i am down to 72 turns but still improving. It is quite fun but aggrevating also ... 

Per


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## mrCage (Oct 3, 2011)

One can use the following 26 turns to make up a complete 12-flip solution in 58 turns. I show part 1 only. Part 2 is same on the other half.
It is done like so: part1 + setup + part2 + unsetup.

Part1:

/*000000*/BD,
/*000001*/RB,
/*000002*/DR,
/*000003*/BD,
/*000004*/DR,
/*000005*/BD,
/*000006*/DR,
/*000007*/BD,
/*000008*/RB,
/*000009*/BD,
/*000010*/DR,
/*000011*/RB,
/*000012*/DR,
/*000013*/FU,
/*000014*/RF,
/*000015*/UR,
/*000016*/FU,
/*000017*/UR,
/*000018*/FU,
/*000019*/UR,
/*000020*/FU,
/*000021*/RF,
/*000022*/FU,
/*000023*/UR,
/*000024*/RF,
/*000025*/UR,

Copy and paste into gelatinbrain applet to verify. It flips 6 edges and twists 2 corners. Yay!!

Per

PS! I really miss a "play" feature on gelatinbrain. Should be quite easy to implement i guess ...


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## mrCage (Oct 4, 2011)

*Curvy copter 2-flip with no jumbling.*

No, this is not another "fyou have been framed" post. This is serious! It has previously been proven that 2 adjacent edges cannot be flipped without jumbling. However 2 opposite edges (FR and BL) can be flipped, and i will constructively show how. For sure not the shortest way possible, but it does the trick.

First i will flip 6 edges with 2-part sequence:
DB DR UR UF FR [UR UF]3 FR UF UR FR DR DB (17 turns)
RF RD DL BL BD [DL BL]3 BD BL DL BD RD RF (17 turns)

Now we have a 6-flip!

Next we flip 4 of those edges:
[UR UF]3 UB UL BL FR DR DF [DB DL]3 DF DR FR BL UL UB (24 turns)

Altogether a dreadful 58 turns, same as i currently have for my 12-flip (also no jumbling).

Per

PS!

Copy and paste this ugly sequence into gelatin curvy copter applet to verify my sequence(s).

/*000000*/BD,
/*000001*/RD,
/*000002*/UR,
/*000003*/FU,
/*000004*/RF,
/*000005*/UR,
/*000006*/FU,
/*000007*/UR,
/*000008*/FU,
/*000009*/UR,
/*000010*/FU,
/*000011*/RF,
/*000012*/FU,
/*000013*/UR,
/*000014*/RF,
/*000015*/RD,
/*000016*/BD,
/*000017*/RF,
/*000018*/RD,
/*000019*/DL,
/*000020*/LB,
/*000021*/BD,
/*000022*/DL,
/*000023*/LB,
/*000024*/DL,
/*000025*/LB,
/*000026*/DL,
/*000027*/LB,
/*000028*/BD,
/*000029*/LB,
/*000030*/DL,
/*000031*/BD,
/*000032*/RD,
/*000033*/RF,
/*000034*/UR,
/*000035*/FU,
/*000036*/UR,
/*000037*/FU,
/*000038*/UR,
/*000039*/FU,
/*000040*/UB,
/*000041*/LU,
/*000042*/RF,
/*000043*/LB,
/*000044*/RD,
/*000045*/DF,
/*000046*/DL,
/*000047*/BD,
/*000048*/DL,
/*000049*/BD,
/*000050*/DL,
/*000051*/BD,
/*000052*/DF,
/*000053*/RD,
/*000054*/RF,
/*000055*/LB,
/*000056*/LU,
/*000057*/UB,
Per

It can be reduced by at least 12 turns when inserting the 4-flip into the 6-flip. Please feel free to work it out ...


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## mrCage (Oct 4, 2011)

My lemma. This is the only 2-flip scenario possible on the curvy copter with no jumbling sequences!!

Per


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## mrCage (Oct 6, 2011)

I'm gonna sketch a proof for my own lemma. Make complete map of the 4 center orbitals. The only place that 2 center orbitals "cross" over another edge (after first flip) is on the opposite side of the puzzle. This can be done like so (disregarding corners, that can trivially be restored):

/*000000*/UB,
*/*000001*/UR,*
*/*000002*/LU,*
*/*000003*/RF,*
*/*000004*/FL,*
*/*000005*/BR,*
*/*000006*/LB,*
*/*000007*/RD,*
*/*000008*/DL,*
/*000009*/DF,
/*000010*/DL,
/*000011*/RD,
/*000012*/LB,
/*000013*/BR,
/*000014*/FL,
/*000015*/RF,
/*000016*/LU,
/*000017*/UR,





Bolded part is setup turns! So: first swap around UB, setup, then swap around DF. Etc ....

Per


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## mrCage (Oct 7, 2011)

I completed the above sequence with solving the 4 corners also. Slight optimisations gave me this:

/*000000*/UB,
/*000001*/RF,
/*000002*/FU,
/*000003*/RF,
/*000004*/UR,
/*000005*/RF,
/*000006*/FU,
/*000007*/RF,
/*000008*/LU,
/*000009*/RF,
/*000010*/FL,
/*000011*/BR,
/*000012*/LB,
/*000013*/RD,
/*000014*/DL,
/*000015*/DF,
/*000016*/DL,
/*000017*/RD,
/*000018*/LB,
/*000019*/BR,
/*000020*/FL,
/*000021*/RF,
/*000022*/LU,
/*000023*/FU,
/*000024*/UR,
/*000025*/UB,
/*000026*/UR,
/*000027*/FU,
/*000028*/UR,
/*000029*/UB,
/*000030*/UR,
/*000031*/FU,
/*000032*/UR,
/*000033*/FU,

34 non-jumbling turns for an edge 2-flip!! 

Per


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## mrCage (Oct 10, 2011)

*Curvy copter middle edges flips (and fliptwists).*

Here i will post any (best cases only) edge flipping algorithms that i have found for the curvy copter. These are invisible on the normal helicopter cube!!

First one out of the sack is the 12-flip. This one has a long story behind it, but i will spare you the details and just post the sequence:

/*000000*/UB,
/*000001*/UR,
/*000002*/RF,
/*000003*/BR,
/*000004*/RD,
/*000005*/LU,
/*000006*/FL,
/*000007*/LB,
/*000008*/DL,
/*000009*/BD,
/*000010*/DF,
/*000011*/LU,
/*000012*/FL,
/*000013*/LU,
/*000014*/FL,
/*000015*/LU,
/*000016*/FL,
/*000017*/DL,
/*000018*/LB,
/*000019*/FL,
/*000020*/LU,
/*000021*/DL,
/*000022*/LB,
/*000023*/DL,
/*000024*/LB,
/*000025*/DL,
/*000026*/LB,
/*000027*/UR,
/*000028*/RF,
/*000029*/UR,
/*000030*/RF,
/*000031*/UR,
/*000032*/RF,
/*000033*/RD,
/*000034*/BR,
/*000035*/RF,
/*000036*/UR,
/*000037*/RD,
/*000038*/BR,
/*000039*/RD,
/*000040*/BR,
/*000041*/RD,
/*000042*/BR,
/*000043*/FU,






Per

PS! More flips will follow when i get the time, and please feel free to contribute.


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## Godmil (Oct 10, 2011)

It seems like your pretty much on your own with the love for the Curvy Copter 
I'm curious now though, think I'll buy one, which do you recommend?


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## mrCage (Oct 10, 2011)

Right. I guess i'm a slow solver  But if you should choose between a helicopter cube and a curvy copter one. Get the curvy, unless the extra edge solving annoys you. Cheaper and better turning quality 

Per


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## mrCage (Oct 14, 2011)

Ok, i will add more flip cases. Now a 2-flip from another thread.

/*000000*/UB,
/*000001*/RF,
/*000002*/FU,
/*000003*/RF,
/*000004*/UR,
/*000005*/RF,
/*000006*/FU,
/*000007*/RF,
/*000008*/LU,
/*000009*/RF,
/*000010*/FL,
/*000011*/BR,
/*000012*/LB,
/*000013*/RD,
/*000014*/DL,
/*000015*/DF,
/*000016*/DL,
/*000017*/RD,
/*000018*/LB,
/*000019*/BR,
/*000020*/FL,
/*000021*/RF,
/*000022*/LU,
/*000023*/FU,
/*000024*/UR,
/*000025*/UB,
/*000026*/UR,
/*000027*/FU,
/*000028*/UR,
/*000029*/UB,
/*000030*/UR,
/*000031*/FU,
/*000032*/UR,
/*000033*/FU,






Per


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## mrCage (Oct 14, 2011)

And a "slice" 4-flip. Actually much shorter than the previous 2-flip.

/*000000*/UB,
/*000001*/FU,
/*000002*/LU,
/*000003*/UR,
/*000004*/LB,
/*000005*/BR,
/*000006*/FL,
/*000007*/RF,
/*000008*/DL,
/*000009*/RD,
/*000010*/DF,
/*000011*/BD,
/*000012*/DL,
/*000013*/RD,
/*000014*/FL,
/*000015*/RF,
/*000016*/LB,
/*000017*/BR,
/*000018*/LU,
/*000019*/UR,






Per


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## qqwref (Oct 14, 2011)

Here's a shorter 2-flip alg.
UB (RU RB RD RF) (LB LU LF LD)
DF (LD LF LU LB) (RF RD RB RU) [18]

EDIT: 8 flip
(UF UR UB UL)3 FL FR BL BR
(DF DR DB DL)3 FL FR BL BR [32]

EDIT: my attempt to figure out your superflip:
UB UF
(RU RF RB RD)
(LU LF LB LD)
DB DF
(LU LF)3 LD LB LF LU (LD LB)3
(RU RF)3 RD RB RF RU (RD RB)3


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## mrCage (Oct 16, 2011)

qqwref said:


> Here's a shorter 2-flip alg.
> UB (RU RB RD RF) (LB LU LF LD)
> DF (LD LF LU LB) (RF RD RB RU) [18]
> 
> ...



Nice 2-flip!! Mine was dirty. Trying to do too much early on gave ugly corner fixup job 

I was about to post a 8 flip also, but now i don't bother ....

I will instead post an adjacent 2-flip, involving jumbling.

RF+ UR RF UR RF UR RF++
UF RF UF RF UF
UR RF UF RF UR RF UF

Per


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## mrCage (Oct 16, 2011)

Actually, i'm going to improve the 8-flip.

(UR UB UL UF)3 - *LF LB RB* - (DR DB DL DF)3 -* RB LB LF*

Saving a graceful 2 turns  The 3 bolded turns can be done in any order !!!

Per


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## mrCage (Oct 16, 2011)

Now a U layer flip-twist.

/*000000*/UR,
/*000001*/UB,
/*000002*/LU,
/*000003*/FU,
/*000004*/UR,
/*000005*/UB,
/*000006*/LU,
/*000007*/FU,
/*000008*/UR,
/*000009*/FU,
/*000010*/UR,
/*000011*/UB,
/*000012*/FU,
/*000013*/LU,
/*000014*/FU,
/*000015*/UB,
/*000016*/UR,
/*000017*/UB,






Per


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## mrCage (Oct 18, 2011)

A new contribution, the superfliptwist. 48 turns (was hoping for better cancellations. Slice 4-flip (20) + twice fliptwist(36).)
(Corners easily were in the wrong configuration ... )

/*000000*/RF,
/*000001*/BR,
/*000002*/RD,
/*000003*/DF,
/*000004*/BD,
/*000005*/DL,
/*000006*/UR,
/*000007*/FU,
/*000008*/UB,
/*000009*/LU,
/*000010*/LB,
/*000011*/FL,
/*000012*/LU,
/*000013*/FU,
/*000014*/LU,
/*000015*/FU,
/*000016*/UR,
/*000017*/UB,
/*000018*/LU,
/*000019*/FU,
/*000020*/UR,
/*000021*/FU,
/*000022*/UR,
/*000023*/UB,
/*000024*/FU,
/*000025*/LU,
/*000026*/FU,
/*000027*/UB,
/*000028*/UR,
/*000029*/UB,
/*000030*/DL,
/*000031*/DF,
/*000032*/DL,
/*000033*/DF,
/*000034*/RD,
/*000035*/BD,
/*000036*/DL,
/*000037*/DF,
/*000038*/RD,
/*000039*/DF,
/*000040*/RD,
/*000041*/BD,
/*000042*/DF,
/*000043*/DL,
/*000044*/DF,
/*000045*/BD,
/*000046*/RD,
/*000047*/BD,






Per


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## mrCage (Oct 19, 2011)

Could not resist. I made a 10-flip. Using qqwref 2-flip and 2 times mono-layer 4-flip. 16 turns cancelled out for good measure.

/*000000*/UB,
/*000001*/RF,
/*000002*/RD,
/*000003*/UR,
/*000004*/BR,
/*000005*/UR,
/*000006*/BR,
/*000007*/UR,
/*000008*/BR,
/*000009*/RD,
/*000010*/RF,
/*000011*/BR,
/*000012*/UR,
/*000013*/RD,
/*000014*/RF,
/*000015*/RD,
/*000016*/RF,
/*000017*/DL,
/*000018*/FL,
/*000019*/LB,
/*000020*/LU,
/*000021*/LB,
/*000022*/LU,
/*000023*/LB,
/*000024*/LU,
/*000025*/FL,
/*000026*/DL,
/*000027*/LU,
/*000028*/LB,
/*000029*/FL,
/*000030*/DL,
/*000031*/FL,
/*000032*/DL,
/*000033*/DF,
/*000034*/FL,
/*000035*/DL,
/*000036*/LB,
/*000037*/LU,
/*000038*/RD,
/*000039*/RF,
/*000040*/UR,
/*000041*/BR,






Per


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## mrCage (Nov 2, 2011)

Yet another short U-D 8-flip:

[UR UL UF UB]3 FR FL BR BL [DR DL DF DB]3 FR FL BR BL (32 turns)

Basically same idea as previous 32/30 mover ...

Per


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## mrCage (Nov 9, 2011)

*Curvy copter pure corner twist algs.*

Hi!

Qqwref already posted an extensive list of corner twisting algs for the helicopter cube. All of them still work for the curvy copter except the 3-twist. [UF UR UB UL]3 obviously flips the 4 upper layer middle edges. I was looking for a pure version that actually twist only 3 corners, and nothing more. I came up with this neat algorithm:

[UR UF]2 UB UL UF UR [UL UB]2 UF UR UB UL (16 turns)

Per


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## cuBerBruce (Aug 26, 2012)

I found a different 32-move 8-flip maneuver. It flips U and E layer edges.

(FR UF UR FR UR UF FR UR y)4

Also, using Copter Explorer, I found a local optimization for Per's superflip maneuver. This reduces the upper bound for a superflip maneuver to 42.

UB UL BL FL UR UB UL UB BR FR DR DL DF FL
DB DL BL FL DL UL BL DL BL DL BL UR FR UR
FR UR FR DR BR FR UR DR BR DR BR DR BR UF


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## cuBerBruce (Sep 6, 2012)

I just found a "two-phase" solution for superflip - only 32 moves!

Phase 1 (Double X-Cross):
DF DB DL FL DF BL DB DL DR FL DF FL BL DB BL DL

Phase 2 (Rest of Cube):
UF UB UR FR UF BR UB UL UR FR UF FR BR UB BR UR

I note that phase 2 is essentially the same maneuver as phase 1, only applied at a different angle.


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## mrCage (Feb 14, 2017)

An improved 2-flip using jumbling:

UF+ FR UF FR UF FR UF++
UR FR UF UR UF FR UR (14 turns)

I am very sure this is optimal. But I have no idea how to prove it ...

Per : D


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## rokicki (Apr 9, 2021)

Using twsearch, I've found a 32-move superflip, and also proved there is no 30-move superflip (ignoring
jumbling moves).

UL UB UF FL UR UB BR DL FR DF UL UF FL DL UR UL DR BR UB DB UR FL UF BL DR DF BR DL DF DB DR DB





__





Twizzle Explorer ᴬᴸᴾᴴᴬ







experiments.cubing.net





This is a good candidate for a non-jumbling antipode on this puzzle; I know of no position at a higher distance.


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## rokicki (Apr 10, 2021)

Oops; a related sequence was already found nine years ago by cuBerBruce two posts up. My apologies. But I did at least show that there is no 30-move non-jumbling superflip on the curvy copter.


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