## Saturday, July 26, 2014

### Bandage Cube Kit—Bandaged Fortress

In October 2012 Burgo posted on the twistypuzzles.com Bandage Cube Kit thread:
The Bandaged Loop above won't prove too much of a challenge to the experienced solvers. So I made a version in which I sought to block known sequences. It looks like a castle so I called it the Bandaged Fortress.
Block known sequences is right! It wasn't too hard to come up with some sequences which make for double swaps of edges, and a couple useful 3-cycles. And I noticed that the corners couldn't scramble at all. And it seemed that edge-flipping was not an issue either.

But I had trouble scrambling it. The only way I had to scramble it was to use the sequences for solving. That made for nice neat scrambles. I needed more. A trip to the twistypuzzles.com Bandage Kit thread revealed a very nice scrambling sequence that Andrea came up with back in November 2012.
I found a nice way to scramble the bandaged fortress :
Top face is white, front face is red
( the fortress looks like a table)
Turn the following sequence one or more time.
Replace the 'x' with 1 or 3.
e.G
Rx (replacee with => 1 = R)
Rx (replace with 3 => R3 = R' )
Now, the sequence :
L2 B U2 F2 Dx R2 D2 Rx F2 R2 Fx D2 F2 Dx R2 U2 Lx B2
each x can be replaced with different numbers.
Easiest case: replace all 'x' with 1
L2 B U2 F2 D R2 D2 R F2 R2 F D2 F2 D R2 U2 L B2
It scrambles the cube very much.
It's not recommended, trying to solve the fortress with this sequence !
Much fun with a good scrambled fortress.
Cheers,
Andrea
This opened a can of worms for me. My first scramble using this sequence flipped lots of edges! I decided not to look at Andrea's scrambler any more lest it give me any hints about how to flip edges. In setting out to develop an edge flipper I kept coming up with a sequence that totally scrambled the cube. I just could not think of any other way to flip edges. Finally I decided to solve the scrambled cube, and, lo and behold, when I got everything permuted, 4 edges were flipped. I tried several slight modifications, but always came out with 4 flipped edges.

What could I do? I used Andrea's scrambler to scramble the cube, then permuted all the edges. 4 needed flipped, so I used my edge flipper algorithm, taking by faith that it flipped the edges while remixing everything up. Then I solved it, and indeed, it worked! I tried her algorithm again only going through it twice before starting my solve. Same idea—permute everything, see what needs flipped, flip 4 edges, permute everything again. This time I had to use other algorithms to move the 4 edges into place before flipping them. It worked.

Then I took a good close look at Andrea's scrambler. It is identical to my edge flipper algorithm!

My Algorithms
AR01: (R2 F2) x 3 double swaps edges UF < > DF and UR < > DR
AR02: R2 F (DE)2 F2 D2 F2 (DE)2 F R2 double swaps UR < > DL and UF < > DF
AR03: AR02 followed by AR01 3-cycles DL > DR > UR

AL01: (L2 F2) x 3 double swaps edges UF < > DF and UL < > DL
AL02: L2 Fi (DE)2 F2 D2 F2 (DE)2 Fi L2 double swaps UL < > DR and UF < > DF
AL03: AL02 followed by AL01 3-cycles DR > DL > UL

AR04: R2 F (DE)2 F2 D (R2 D2) x 2 R2 Di F2 (DE)2 F R2 3-cycles FU > FD > FR
AR05: R2 Fi (DE)2 F2 D (R2 D2) x 2 R2 Di F2 (DE)2 Fi R2 3-cycles FU > FR > FD

AL04: L2 Fi (DE)2 F2 Di (L2 D2) x 2 L2 D F2 (DE)2 Fi L2 3-cycles FU > FD > FL
AL05: L2 F (DE)2 F2 Di (L2 D2) x 2 L2 D F2 (DE)2 F L2 3-cycles FU > FL > FD

AR06: R2 F (DE)2 F2 D R2 D2 Ri F2 R2 F D2 F2 D R2 (DE)2 Ri F2 flips UR DR DL and FR
AR07: R2 F (DE)2 F2 D (R2 D2) x 4 Ri F2 R2 F D2 F2 D R2 (DE)2 Ri F2 flips UR DR DL and DF

AL06: L2 Fi (DE)2 F2 Di L2 D2 L F2 L2 Fi D2 F2 Di L2 (DE)2 L F2 flips UL DL DR and FL
AL07: L2 Fi (DE)2 F2 Di (L2 D2) x 4 L F2 L2 Fi D2 F2 Di L2 (DE)2 L F2 flips UL DL DR and DF

Andrea's Scrambler (translated to my puzzle orientation)
R2 F (DE)2 F2 Dx R2 D2 Rx F2 R2 Fx D2 F2 Dx R2 (DE)2 Rx F2

I am reasonably sure I could solve this puzzle using my algorithms no matter who scrambled it,  but I'm not saying it would be easy. For example. On a solved cube I did Burgo's edge flipper. It was a pure double swap of edges that somehow flipped two of them. Nice. So with one double swap done I solved the cube and ended up with two flipped edges at DL and DB. This solve turned out to be slightly easier than the solves from totally scrambled, but still took some work. Now to flip just two edges what I need to do is flip 4 edges including only one of the two that need to flip. Then I'll flip the other one along with the same three. That should do it. It works, but oh my! What a lot of work to accomplish one double swap.

July 27, 2014
When I think of flipped edges, I think of, well, edges that are flipped! Like the blue yellow edge is flipped if when permuted the blue is on the yellow side. Simple. And in the Petrus Method I know how to tell good edges and bad edges and equate bad edges as being flipped, and know how to flip them. But when Burgo says that swapping two adjacent edges on U flips them even though the U color is still on U—huh?!? I think I've seen him explain this in a video once, but it was long ago, and I didn't have any reason to remember it. But today, upon reflecting on it, here is what I came up with, and it seems to work on the Fortress. After scrambling the puzzle using Andrea's scrambler, look at each edge. Imagine it being on an unbandaged cube and using twists of R, F, L, and B, mentally take it home. If it is flipped when it gets home, then it is flipped. Using this idea to flip the edges before solving the puzzle has made for some nice easy solves. Perhaps because my edge flipper algorithm is so similar to Andrea's scrambler algorithm. Maybe.

One more thing. Burgo's 2 edge flipper algorithm, when translated to my solving perspective, is very similar to my other algorithms.