Tuesday, July 29, 2014

Bandage Cube Kit—Pendulum

Andrea of Germany came up with this one and posted it on the twistypuzzles.com forum back in November 2012. Burgo rated it as difficult. I confer. I've given it three good scrambles. On the first one, after getting down to needing a corner 3-cycle to finish up, I retiled for further analysis. After coming up with some more corner 3-cycles and a way to flip edges and twist corners, I solved it completely a couple times. The last time I had to flip all 4 edges and twist 5 corners at the end.

After solving the bandaged pieces, I placed the edges. It is an easy task and seeing which ones that need to flip is so much easier for me when they are in place. Using 90° turns of the white bar and the red bar, pairs of edges can be flipped. More specifically, with orange up and yellow on front, M2 (F U F2 Ui F) (Ri F2 R F2) M2 flips UR and FR. It also 5 cycles corners. The first (section) does the flipping, and the second (section) puts them back in place. During the solve I don't really put the edges at UR and FR and do the algorithm. It doesn't really matter which side the one on top is on, and it doesn't matter which side the one on bottom is on. Turn the one on bottom that needs flipped down with either F or Fi. Turn the one on top that needs flipped to UF. Swap them with F2 and they are flipped when the bandaged pieces are put back home. After the edges are flipped and solved the hard part comes—the corners.

During the experimentation and analysis stage I came up with 4 key sequences that can be used to solve the corners. I'm thinking that perhaps with further work the method could be refined and simplified, but I'm ready to move on. Annie wants to help me tile a new puzzle. :D

S1: M R U2 Ri U2 Mi = FR > UR > UL; 18327; x3 pure cycles 12873; doesn't move 4
S2: M Li U2 L U2 Mi = FL > UL > UR; 18274; x3 pure cycles 17842; doesn't move 3
S3: M2 Ri F2 R F2 M2 = UR > FR > FL; 14723; x3 pure cycles 12437; doesn't move 8
S4: M2 L F2 Li F2 M2 = UL > FL > FR; 14238; x3 pure cycles 13482; doesn't move 7

In practice you don't have to do all the M moves between sequences of these cycles. Just move the white bar either to the top or front, wherever it is needed.

Stringing these together in different combinations yields pure corner 3-cycles. After doing the first two, choose the third one by seeing what needs to happen to restore the edges.

S1, S2, S4 cycles corners 1 > 3 > 7
S1, S3, S2 cycles corners 1 > 2 > 7
S1, S4, S3 cycles corners 1 > 4 > 3
S2, S1, S3 cycles corners 2 > 4 > 8
S2, S3, S4 cycles corners 2 > 3 > 4
S2, S4, S1 cycles corners 1 > 8 > 2
S3, S1, S4 cycles corners 1 > 2 > 3
S3, S2, S1 cycles corners 1 > 8 > 7
S3, S4, S2 cycles corners 1 > 7 > 3
S4, S1, S2 cycles corners 2 > 7 > 8
S4, S2, S3 cycles corners 1 > 4 > 2
S4, S3, S1 cycles corners 2 > 8 > 4

When looking for the edge flipper I made a happy little discovery.

M2 F U F2 Ui F S3 S2 S3 S4 S1 S3 S2 Mi. Of course there are M moves between the sequences when necessary. What does this do? It flips UR and UF, and it twists corner 3 clockwise and corner 7 anti-clockwise. That means if you do the whole thing twice the flipped edges get unflipped and the corners get twisted again, so the net result is to twist 3 anti and 7 clock. Granted, it is a lot of work to twist a couple corners, and if the corners aren't at 3 and 7 it requires some crazy setup moves, but it worked. Perhaps someday I will look at Pendulum again and find a better way.

July 31, 2014
Burgo gave me some feedback on the forum, so I went ahead and looked at his solution method. He used a commutator to cycle corners home Domino style. That meant he had to orient the corners before permuting them. This inspired me to do some more experimenting in two areas. Orienting corners, and cycling corners. It led to a Pure 3-cycle for Edges that I will probably never use. I don't know. It might be useful during the orientation phase if I translate it into Orange Up Yellow Front twists. Here it is:

M (U2 R U2 Li U2 Ri U2 L) x 2 Mi cycles FL > UL > FR without scrambling corners.

M ((U2 R U2) Li (U2 Ri U2) L) x 3 Mi does a double swap of corners. 14 and 38.
M ((U2 Li U2) R (U2 L U2) Ri) x 3 Mi does a double swap of corners. 23 and 47.

Holding Yellow Up and Orange Front gives even more possibilities. 27 and 38, and 18 and 47. These numbers are from the point of view of Orange Up and Yellow Front.

As far as the orientation goes, the same technique that flips edges also twists corners. With M2 F U F2 Ui F, Corner 3 twists clockwise and moves to 7. Corner 7 twists anti and moves to 1. The edges adjacent to the corners are the ones that flip. The corner edge pairs swap during F2. Different combinations of F and U can result in different corner edge pairs being reoriented.

Of course, all of this orientation of corners is based on the fact that the moves for placing them do not change the orientation. How do I know if an edge or a corner is oriented correctly before it is placed? With Orange Up and Yellow Front, Blue and Green are on the left and right. The edges and corners without blue or green on the left or right need to be flipped or twisted.

The pure edge flipper and corner twister that I gave a couple days ago can be shortened to simply orientation sequences if you don't care that the pieces move around. I suppose the best case scenario is when two edges need flipped and two corners twisted and you can easily arrange them so that doing one F U F2 Ui F orients them all at once. If no edges need to flip and four corners need to twist, the following sequence will twist 1 and 7 anti and 3 and 4 clock.

((F U F2 Ui F)(Ri F2 R F2)) x 2

And doing it x 4 instead of x 2 twists only two corners: 2 anti and 4 clock.

I'm thinking that a good strategy of solution would be to solve the bandaged pieces first, then with Orange Up and Yellow Front proceed. Orient all the edges and corners. Permute the edges. Permute the corners.

If I were planning to solve this particular puzzle on a regular basis, I might want to master Burgo's method of permuting corners at least, if not also his way of orienting them. 

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.
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.
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.

Wednesday, July 23, 2014

Bandage Cube Kit—Bandaged YZ Family

I don't know why exactly, but I like this configuration a lot. It is just fun to play with. Maybe it is the three 2x2x1 blocks. Maybe it is because edges can be solved with EPS and corners permuted with Triple-EPS, and oriented with Sune/Mirror. Perhaps because although setups are needed to get around the bandaging, they are not overly complex. There is just something about the feel that is enjoyable.


After a couple solves I decided to share my enjoyment of this puzzle on the twistypuzzles.com thread. I noticed that Burgo listed the Bandaged YZ as a difficult puzzle. I couldn't believe it. So I checked the pictures and noticed that the three little tiles across the middle of the green side are supposed to be one long tile. Maybe I should graduate from my very simplified one to his Unbandaged one before attempting the terribly bandaged one!

Burgo's Unbandaged YZ (below) didn't seem difficult either, although he lists it as difficult. Perhaps I should have scrambled it and solved it several times to see if there were any hidden surprises. But the blue and white faces could be turned from the get-go so 3 of the 5 edges and 4 of the 5 corners that need to be solved are easy to handle. And turning the blue face 180° makes the orange yellow corner edge pair easily accessible. Furthermore doing blue' yellow' puts all little ones on top in such a way that all manner of EPS and Sunes can be done. I even had to swap two edges and it was no problem at all with my standard edge swapper algorithm—U' R U' R' ....


At first Burgo's Bandaged YZ (see below) seemed like it was going to be another relatively easy solve.

First Solve:
  1. Bandaged pieces—easy
  2. Edges—easy
  3. Permuting Corners—easy
  4. Orienting Corners—required coming up with an all new technique that I've never seen before which uses a combination of Edge Piece Series and a Corner 3-cycle.
Second solve:
  1. Bandaged pieces—easy
  2. Edges—easy
  3. Permuting Corners—not easy, but not difficult
  4. Orienting Corners—interesting
Orienting Corners—
  1. Need to twist 1, 2, 5, 7
  2. Did 1 with 3-cycle and 3-cycle' (2)
  3. Did 5 with 3-cycle and 3-cycle' (4)
  4. 3-cycle 2 to 3
  5. Double swap 3 to 1 and 7 to 2 (Triple-EPS)
  6. Twist 2 and 7 with 3-cycle and 3-cycle' (2)
  7. Noticed I now needed an n=1 3-cycle, so didn't have to undo steps 4 and 5. :D
Throughout the solve I hold green on the left and white up.

3-cycle: (RUi RiU) x n Ri RUiRi Di RURi D R (UiR URi) x n
  • n = 0 moves UFL > URF > RDF
  • n = 2 moves UFL > RFU > RDF
  • n = 4 moves UFL > FUR > RDF
  • n = 1 moves UFL > FRD > RUB
3-cycle': (RUi RiU) x n Ri Di RUiRi D RURi R (UiR URi) x n

The idea is that you can twist corner 1 by cycling it to 2 with n = 0, then cycle it back with n = 2 or 4 depending on how it needs to twist. Notice the UFLs and RDFs above? But the piece at 2 twists. That is the key. Same thing works for corner 3. Cycle 3 to 2, then back to 3 with the proper twist. 2 takes care of itself since you can't have just one twisted corner.


Monday, July 21, 2014

Bandage Cube Kit—Recap of Summer Solves So Far

Traffic Jam
So named on rline’s youtube video.
Bandaged 3
Had to get a hint from the forum. Get F2L then use LDL’ to put the vertical post on the bottom. 
Had to get a couple hints from rline’s blog. Edges first. 3x3x2 CPS to solve corners. Bars on L and R.
This one is the easiest one yet for me. Reminds me of the F2L part of the Petrus method.
Right-handed Triple-EPS (RUR’U’) followed by left-handed (F’U’FU) 3-cycles corners on top (on U). 
Edge Road
Simple using the techniques described above.
Corner Road

Belt Road


Stalagmites and Stalactites

Unbandaged Big Block v1
Similar to all the rest of the Big Blockish types so far. i.e. get block then orient edges…
Unbandaged Big Block v2
Orienting edges was tricky but possible.
Unbandaged Big Block v3

Bandaged Clock Mars
Edges First
Bandaged Clock Saturn

3 Bar
Similar to 3x3x2
3 Bar Clock
Edges First; 3x3x2; 1x2 corners first; last 3-cycle tricky—needed setup.
Mr T
Swap of corner-edge pairs needed at end. Never did codify how to do it, but did it. 
Unbandaged Wall-i
Very easy
Got it down to needing to twist two corners but they are not in corners that can be twisted using the sune/mirror dune method. Finally got it! 3-cycle the corners into the correct positions, then twist them, then 3-cycle them back.
Detiled Corners Only Fuse-3
Could get close but not finish it, so finally sat down with a notebook and figured it out.
Fuse-3 minus 5 edges
Used the Corner 4-cycle to hide the edge. It worked.
Fuse-3 minus 4 edges
Worked out a fairly thorough solution and documented it on Puzzled 2.
Worked out a fairly thorough solution and documented it on Puzzled 2.
Bandage Loop
Documented on old blog post.
Unbandaged Flying Carpets
I struggled to get the middle layer, but finally achieved it by trying to organize the 2x1 yellows somewhat. Then when I looked at the yellows all I needed to do was twist 3 corners to have a pattern that pleases me. I have "iii" on one side, and "iö" on the other side. (from PM to Burgo)
budlcuber 01
I made a puzzle and solved it. Then I extended it to budlcuber 02 and am totally baffled.
budlcuber 02
Documented on blog
budlcuber 03
Documented on blog
budlcuber 04
Fully tiled version of 01-03. Documented on the blog.
Big Block Clock
A couple solves. The first one fell together a bit easier, but the second wasn’t all that bad.
Unbandaged Big Block Clock v1
Scrambled, solved down to a tricky 4-cycle, gave up. Approached it from the “analyze what CAN be done” perspective and figured it out. Scrambled and solved.
Unbandaged Big Block Clock v2
This one solves like the others but looks sort of like budlcuber 04. Sort of.
Very similar to Unbandaged Big Block Clock v2 but with a 2x1x1 column under the 2x2x1 roof.
Bandage 3+
When it was down to 2 corners needing twisted with the 2x2x1 on D and 1x1x1 at 3, RU’R’ F’U2F makes it so the Sune/Mirror twister works.

Saturday, July 19, 2014

Bandage Cube Kit—Big Block Clock Family including Stonehenge

I solved Big Block Clock once in December 2012 and had a terrible time with it. LINK to blog entry.

Today it was much easier. A little Petrus to get the F2L, and Sune to get the LL edges. Then Triple-EPS to place the LL corners and some creative use of Sune/Mirror Sune to twist 2 corners at a time. It is a bit tricky because of the clock hands but not that hard. Solved 2 or 3 times today.

Unbandaged Big Block Clock v1 was next. At first it seemed easy enough, but I got stuck at the end. After re-tiling the last 4 corners and experimenting a little I realized that it wasn't that hard after all. So I scrambled and solved it a time or two.

Now for Unbandaged Big Block Clock v2. I haven't solved it yet, but it reminds me a bit of budlcuber 04. I don't think they will solve anything the same, but they look similar.


July 21, 2014
v2 done yesterday. Today Stonehenge was made and solved. I'm including it in the Big Block Clock
Family because it is identical to v2 only with the orange blue edge/orange blue white corner replaced by a 2x1x1 block. It was surprisingly easy.

Since v2 looks sort of like budlcuber 04 and Stonehenge is a close kin to v2, I have to add a 2x1x1 to budlcuber 04 and make budlcuber 05.

(Later) Tried it and decided it makes it impossibly difficult. At least it makes it a whole new challenge that I am not up to at this point, so I went on to Bandaged 3+.

Friday, July 18, 2014

Bandage Cube Kit—budlcuber 03 & 04

In the last post I meandered my way from the Unbandaged Flying Carpets to budlcuber 02 with a quick mention of budlcuber 01, which was very similar to budlcuber 02. budlcuber 03 continues the series by adding the remainder of the edges. In my opinion it is not much more difficult than 02.

Here it is—budlcuber 03


And here is budlcuber 04—


I found this fully tiled version to be extremely challenging and so far have only solved it once. Had to come up with a brand new sequence to orient two corners at the end and then restore the edges that moved. I have this niggling feeling that there is probably an easier way.

Recap of versions
budlcuber 01—A green, orange, white frame, and a red, blue, yellow frame
budlcuber 02—Fill the blue, yellow, red frame with a 3x1x1 block
budlcuber 03—Add all the missing edges
budlcuber 04—Add all the missing corners

Green Front; Red Up—R FiLF2 RiDR B2 Orange Front; Yellow Up— From solved to working state

R2 Green Front; Red Up RiDiR F2LiF Ri— From working state to solved state

R2 U2 R U2 R2— FR > UR > BR > UL

(U2 Li U2 L) x 3— 12487

(U2 Li U2 L) x 5— FL > UL > UR

RiFR Ui RiFiR— Flip UL and UF (and move stuff all around)

(RiFR U RiFiR Ui) x 2— Flips FL UF UR UB

L-Sune U L-antisune ULiUiL (U2 Li U2 L) x 5 LiULU2—Twists 2 clock and 7 anti (no longer use)

L-Sune Ui L-antisune U2 LiUiL U2 (U2 Li U2 L) x 5 U2 LiULUi—1 anti and 2 clock (no longer use)

(L-Sune U2 L-antisune U2) x 2—8 clock and 2 anti (came up with this 8/11)

August 11, 2014
After working through most of Burgo's list, I decided to review my creation and wait until I have time to concentrate before moving on to the Very Hard ones. While doing mine this time I was trying to recall how the long corner twister works. Somewhere along the line I started wondering if there were any simpler Sune-based pure corner twisters that used only 2 sides. Came up with one, and added it above at the bottom of the list of algorithms.

Tuesday, July 15, 2014

Bandage Cube Kit—Unbandaged Flying Carpets & budlcuber 02

Today I solved the Unbandaged Flying Carpets, by Burgo. It was fun. He had posted—a long time ago—on the twistypuzzles.com forum. The interesting thing is that he posted pictures of the scrambled state, not solved. So part of the challenge was making your way to a place without knowing where you were going. I struggled to get the middle layer, but finally achieved it by trying to organize the 2x1 yellows somewhat. Then when I looked at the yellows all I needed to do was twist 3 corners to have a pattern that pleased me. I got "iii" on one side, and "iö" on the other side.

After solving Unbandaged Flying Carpets I felt like making my own bandaged cube, scrambling it, and posting it online. budlcuber 01 was very simple to solve. Well, when approached correctly it was. So I decided to make it a teeny bit more complex. Now I can't solve it! Here it is in scrambled state:


budlcuber 01 was the same puzzle minus the 3x1x1 block.

budlcuber 02 is almost solved, but even if I can figure out how to flip the last edge, I won't be able to tell you how I got to this point.

Here is what it looks like now. Perhaps it is time to just swap the tiles and start slowly and carefully analyzing it.

July 16, 2014
I tried last night but got nowhere and it was soon a mess again. I had just about decided to detile it and start over where I left off with the ones on Burgo's list on the forum. Another possibility was to break the 3x1x1 into a 2x1x1 and a 1x1x1, but I didn't really want to do that. The problem with this puzzle was that it seemed that the best I could do was have two layers that could twist. Sometimes only one. And even when I could twist two layers I couldn't get any familiar algorithms to work. I was so close to giving up, but decided to fiddle with it one more time today. This time instead of worrying about what I couldn't do, I just looked for things I could do, and wrote them down and recorded the affects. (I'm using Ri for R anti clockwise, because I don't like the way the F' looks.)

With Green on front and Red up, the first thing I tried was—
I wasn't overly impressed with the results so went back to solved. Next I tried—

R FiLF2 RiDR y
Surprise! Now L, M, and R can all move! Follow with
and now L, M, R, and U can all move! So altogether we have—

R FiLF2 RiDR yxR2 from the solved state puts the

3x1x1 at the bottom in R;
The white 2x1 on the bottom at the back of the M slice;
The orange 2x1 on the front at the bottom of the M slice;
The 2x2x1on the bottom back in L.

And what can I do now?
  • The Sune Family! The cube, or at least the bottom layers need to be spun around for some. Example—(DE)2 Ri or y2 sets it up for a right-handed Sune.
  • LiUi LU and y2Ri RU RiUi.
Shortly after this I started experimenting with a solved 3x3x3 cube so I could better see where pieces were moving.
  • (Right-hand Sune) (DE)2 L (Left-hand Sune) Li (DE)2 RU RiU moves UF > UB > UR and twists corners 2, 7, and 8 clockwise. 
  • With the 3x1x1 in R do R2 U2 R U2 R2 to cycle FR > UR > BR > UL.
So now I am somewhat hopeful that with this knowledge I can maybe make some progress toward solving budlcuber 02. (Later) Almost there. The one corner isn't in the correct place. Maybe tomorrow. Good night.

July 17, 2014
Solved! But getting the last corner was a matter of trying the algorithms above in different combinations until something worked. Not real confident about it. I need a corner twister that doesn't move edges.

Breakthrough on another front. Mi U2 M Ri F2 R does a 5 cycle of corners and a 3 cycle of edges. After doing it for awhile I realized it could be shortened to these:
  • (U2 Li U2 L) x 3 cycles 1>2>4>8>7 A Pure Corner Cycle!
  • (U2 Li U2 L) x 5 cycles UL > UR > FL A Pure Edge Cycle!
To get back from the working state to solved state—
  • R2x'y' RiDiR F2LiF Ri
As far as the corner twister goes, Once all the edges are solved, get the corner to 1, twist it using Li Ui L U times 2 or 4, if necessary. Then move it into place using the pure 5 cycle. Then Use Li Ui L U times whatever until the edges are solved.

I think I've got everything I need to solve this little bugger. Ain't so impossible after all.

The finished product—

Monday, July 14, 2014

Bandage Cube Kit—Fuse-3




After RUF' RU2R'

So solve to these and do

RU2R' FU'R' and its done!

Last night I discovered a way to twist 4 corners without cycling edges at the same time! Yay!

( ( R' F' ( R U R' U' ) F R ) (R' F2 y' ( R U2 R' U' R U' R' ) y ) ) x 2

I think of it as doing the RU corner swapper followed by an antisune, and doing it twice.
It twists 2 and 8 anti, and 4 and 6 clock. 2 is what I call UFR, 4 is DFL, 6 is DBR, and 8 is UBL.

Then later there was cause for more rejoicing. Whereas the first discovery was the result of deliberate documentation, the second one was stumbled upon while scrambling. After a few twists I realized that all three 2x1x1 posts were planted in the F2L. After solving it, further experimentation uncovered the way to get there from the place where the 2x1x1 pieces are solved—RUF' RU2R'. This state allows use of F RUR'U' F' and its buddy, for flipping edges. It also allows for the Sune family of cycling edges and twisting corners, which may make the previously mentioned discovery obsolete. 

While on the topic of Sune, I'd like to mention a very nice little Word document that Burgo put online that has a way to use Sune to do a pure 3-cycle of edges. Right-handed Sune, y, Left-handed Sune, y' cycles UF > UL > UB. After undoing the setup with RU2R' FU'R' it translates to FD > UB > UL.

In the previous post I included some algorithms. I don't know exactly how many I will actually use now, but here is a more complete list:

Solving the 2x1x1 Pieces
Something that can help in some situations to get the 3 2x1x1 pieces is R ( U R U' R' ) or ( R U R' U' ) R'.

3-cycle of Corners
3>4>2 followed by 3>6>2 followed by 4>3>2 cycles 2>6>4 and returns the 3 2x1x1 pieces to their solved state.

Corner Swappers and Twisters
R' F' ( R U R' U' ) F R swaps 2 and 4, and 6 and 8, and cycles edges.
R' F' ( R U R' U' ) x 2 F R twists 2 and 4 clock, and 6 and 8 anti, and cycles edges.
R' F' ( R U R' U' ) x 4 F R twists 2 and 4 anti, and 6 and 8 clock, and cycles edges.

F R ( F' U' F U ) R U R2 F' swaps only 2 and 4, and cycles edges.

( ( R' F' ( R U R' U' ) F R ) (R' F2 y' ( R U2 R' U' R U' R' ) y ) ) x 2 twists 2 and 8, and 4 and 6 without cycling edges.

3-cycle of Edges

R' F' ( R U R' U' ) F R swaps some corners and does FD > RB > UB.
R' F2 y' ( R U R' U R U2 R' ) y F2 R swaps some corners and does FD > RB > UB.

( RUF' RU2R' ), Right-handed Sune, y, Left-handed Sune, y' ( RU2R' FU'R' ) 

cycles FD > UB > UL with no corners affected.

Edge Flippers

( R' F' ( R U R' U' ) F R ) (R' F2 y' ( R U2 R' U' R U' R' ) y F2 R ) 
flips FD and UB and swaps corners.
( R' F' ( R U R' U' ) F R ) (R' F2 y' ( R U R' U R U2 R' ) y F2 R ) 
flips FD and RB and swaps corners.

Alternately, one could just solve to the above picture using familiar algorithms, and be 6 twists from solved! :D

  1. Solve the 2x1x1 pieces.
  2. Check corners and permute if necessary.
  3. With white up, solve the two middle layer edges.
  4. RUF' RU2R'
  5. Solve the edges.
  6. Solve the corners.
  7. RU2R' FU'R'
Although only a small handful of algorithms are actually needed, it was fun coming up with all of these and sorting through them to come up with a strategy.

Saturday, July 12, 2014

Bandage Cube Kit—Fuse-3 Partially Detiled

Note: This is more of a story than a solution guide. Check later posts for more about the Fused-3 Bandage Cube.

The Detiled Corners Only Fuse-3 was a challenge at first, but then became relatively simple compared to the monster I created by adding a couple edges to it. Here is what the Detiled Corners Only Fuse-3 looks like with centers and complete corners.


Below you can see it with one edge added. In this case the green-white edge. I solved this one at least once. Hiding the edge using the 4-cycle of corners, then putting it back after doing whatever needs to be done with the corners seems to be the ticket.


Next I tried the one below and could not do it.


The one below, which I thought would be simpler, proved to be challenging as well. It wasn't until I posted these pictures and noticed the 2x2x3 block, that I thought the Petrus technique of building the 2x2x3 block then orienting edges might work. My first attempt worked, but I've been puzzling enough to know that might not mean it will always be so easy. Second attempt did not work out so well. So much for the Petrus method. I guess I got lucky the first time.

Another approach. Before solving the two edges make sure the corners are in a double swap situation.  Solve the two edges; keep checking the corners. When moving and twisting corners move the edge in danger to a safe spot before the move or twist. Sometimes it is a bit complicated, but it worked, and I got solve #2. Repeatable? Rememberable? We'll see.

First of all, if there are no added edges to worry about, the algorithm for swapping URF and DLF (2 and 4), and ULB and DRB (6 and 8), is quite simple: ( R' F' ) ( R U R' U' ) ( F R ). When you want to manipulate corners without disturbing the 2 placed edges, it gets more complicated, as the edges need to be moved out of the path of the corner swaps.

With red on the right and white on the front, ( R' F2 U F U ) ( R' F R F' ) ( U' F' U' F2 R ) does a double swap of corners. URF and DLF (2 and 4), and ULB and DRB (6 and 8). Same thing can be used with ( R' F R F' ) x 2 to twist 2 and 4 clockwise and 6 and 8 anti clockwise. Likewise x 4 twists 2 and 4 anti clockwise and 6 and 8 clockwise.

But what if red is on the front and white is on top and I want to swap 2 and 4? I could just use ( R' F' ) ( R U R' U' ) x 3 ( F R ). But that will not work for the twists that may be needed later.

3-cycle corners 3>4>2 then 3>6>2; then do ( R F' U2 F ) ( R U R' U' ) x 2 ( F' U2 F R' ); then cycle the corners back. It twists 2 and 4 anti, and 6 and 8 clock. There must be an easier way to protect those two edges! It isn't really difficult, but it is 48 twists.

Lastly, what if red is up and white on the right? ( R' F2 U2 F ) ( R U R' U' ) ( F' U2 F2 R )


The more edges that are added the more difficult it will be to use this method, as it relies on having 3 detiled edges that won't be affected by the EPS. I can't imagine how the fully tiled Fuse-3 can be solved. But for now let's get down what we have so far.

After solving the 3 2x1x1 pieces check to see if the corners are in a double swap OR fully permuted situation. It is important they get in one of these positions and remain so for the rest of the solve. The reason for this is that the 3-cycle of corners also moves edges. Perhaps it would be easy to find ways to hide the edges before the 3-cycle. Something else to think about.

Solving the 2x1x1 Pieces
Something that can help in some situations to get the 3 2x1x1 pieces is R ( U R U' R' ) or ( R U R' U' ) R'.

3-cycle of Corners
3>4>2 followed by 3>6>2 followed by 4>3>2 cycles 2>6>4 and returns the 3 2x1x1 pieces to their solved state.

Corner Swappers and Twisters
R' F' ( R U R' U' ) F R swaps 2 and 4, and 6 and 8.
R' F' ( R U R' U' ) x 2 F R twists 2 and 4 clock, and 6 and 8 anti.
R' F' ( R U R' U' ) x 4 F R twists 2 and 4 anti, and 6 and 8 clock.

F R ( F' U' F U ) R U R2 F' swaps only 2 and 4.

Thursday, July 10, 2014

Bandage Cube Kit—Detiled Corners Only Fuse-3

Progress on the Bandage Kit has been coming along nicely. I hit a road bump though with the Detiled Corners Only Fuse-3. I could get close but not quite get it. One time it was a single swap of corners that had me befuddled. Another time it was a 3-cycle, of all things. And then it was a 4-cycle. All I really knew how to do was a double-swap of corners.

It was time for some serious analysis and note-taking. Although some puzzlers do not enjoy this approach, for me it is quite enjoyable and satisfying. Although most of the Bandage Cubes so far haven't required this rigor, the Detiled Fuse-3 did.

Here is a picture showing a couple relaxed puzzlers with the Bandage Cube in the background.

And here is a picture showing some of my notes, the Detiled Fuse-3, not so detiled, but with only one edge tiled that was blank in the fully detiled one. Also featured are the guys we made with some of the leftover pieces.