The Bandage Cube

This section was posted on my Google Site. It says it was updated on September 3, 2011. Below this section is a post I made on WordPress even earlier.

Centers

Since each center is bandaged to an edge, orientation of centers matters.

Edges

Each edge is bandaged to either a center or a corner, so rather than thinking of solving edges, I think of it as solving pieces. One of the edges is bandaged to 2 centers, so this is the starting point of my solution strategy.

Corners

One corner is not bandaged to an edge.

Strategy

1. Orient the centers. Hold the cube so the orange is on the bottom and the white is on the back. Orient all the centers accordingly.
2. Get Blue-White (5). The BW move: F' U L F U L' U2. If BW is at GO or YW start with R U. Then do the BW move until BW is home. Then if you started with R U, do U' R'.
3. Get Green-Orange (6). The GO move: R U F R U' R2 F'. This is also a 5-cycle. If GO is at GY, it won't get cycled, because the 5-cycle I use doesn't hit that spot. So I use a double swapper to get the GO to the RB spot. Then the GO move takes it right home. The double swapper is (R U2 L' U) (R' U2 L U'). It swaps the piece at GY with RB, and the piece at RG with RY.
4. Get Yellow-White (8). The YW move: (R U R' F') (U L' U' L) (F U'). This is a 3-cycle that moves the pieces on top around. On a solved cube it would move RY to GY to YW. If the YW is not on top I simply move it to the top with the double swapper, then use the YW move once or twice as needed. The double swapper is (R U2 L' U) (R' U2 L U').
5. Get Red-Yellow (1). The RY move: (R U2 L' U) (R' U2 L U'). This is the double swapper used previously when necessary. If RY is at RG it takes it home. If RY is at GY it takes it to RB, where it can be moved to RG, then double swapped home. To move it from GY to RB use the Step 6 3-cycle.
6. Get last 3 with a 3-cycle (743). The GY-RB-RG 3-cycle: (U2) (L F' L' F) (U') (R' F R F') (U').

Bandage Cube Scrambles

Thanks to James the "thebackflipmaster" for the following list of scrambles.
02-----U F'L'U L U'F'L F U'F R2U2L'U2R'F'U F U'R'U F'L'U2 Good scramble. 
03-----R U2F'L'U L U'F'L F U'F R U L'U R'U F U F R'F L'U' Good scramble. 
04-----F'U L2F2R'F L'F2R F2L'F2R2U2L'U R'U'R'U2L F'L'U L Good scramble. 
06-----U F2R F'U2R'U F R'F'R U R'F2L F'R F'U'F'U'R U'L F2 Good scramble. 
09-----U F2R F2L'U2R'U2L F'L'U L U L U'F'L F L'U2F R F R' Good scramble. 
10-----F'U L2F2R'F L'F2R2U F2L'F U L'U'L F U L U'F2L'F U2 Good scramble. 
11-----U2L U2R'F R F'U'R U2L'U2R U R'F'U2L2F L'U R'U F U' Good scramble. 
13-----R U R U2L'U R'U L F2R'F'R U F'L'U R'U2L2F U'R'F'R' Good scramble. 
15-----U2L F U'R'F R F2L2U2R U R U'L U2R2F2L F L F'R F2L2 Good scramble. 
16-----F R2U R'F'U2L U2R'F'U'F R2U R'F'U F U L'U'F U'R'F' Good scramble. 
17-----F R2U R'F'U F'L'U L U'F'L F L'U2F R U'R'U2L2F2R'F2 Good scramble. 
18-----F R U F'L'U'R U'L U2R2F'U2L U'F'L F L'U'L F2R'F2L2 Good scramble. 
19-----R U2F'L'U L U'F'L F2U'R'U L'U2F'U L2F L'U2R U'L F2 Fell into place at the end but was a long time getting there.
21-----F R2U R'F2L F U'R'F'U F2R U F'L2U'L F U'F'U L2F2R2 Good scramble. 
23-----F'U L U L F'L'F U L'U'F U'R'F'L F'R F'U'F'U'R U F2 OK scramble. 743 was done, but I had to do a 743 earlier anyway. 
24-----F'U L2F'R F2L2U'F R U R'F'R'F'U2L U'F'L F L'U2F R 
25-----U2L F U'R'F2L'F R U2L'U'L F U2R'F R F'U'R U R'F2U 
26-----U2L F U'R'F R F'U'R U2F'L2U2R U'L U2R2F2L F'R F'U2 
27-----R U R'F'U2L U'F'L F L'U'F U'R'F R2U2F'L'F'L'U L U 
28-----U2L U'R'F2L F'R F2L2U'F R U'L U'F'L'U2F R U L'U2R' 
29-----R U'L U'F2L'F U L'U'L F L'U2R U'L U2R'U2L U'F'U L2 
30-----R U2F U'R'U'R'F R F'U'R U2F'L'U L2F U'R'F2L F2R'F2 
31-----F'U L F U'R'F R2U R'F'R U F'L F L'U'L F2R'F'R'F L2 
32-----R U R U2L'U R'U'R'U2L2F2R'F'R U F'L'U2R U'L U'F2L' 
33-----R U2F U'R'U F'L'U L U2R'F R2U F2L'U L U'F'L F2U'R' 
34-----U F2R F'U2R'U F R'F'R U R U F'U L U2R'F2L'F U L F' 
35-----R U R'F2L F2U F U'R'F R F'U'R U F'U L F'L2U2F R U 
36-----U F'L'U L U L U'F'L F L'U'L F'R F2L2U'F R'F'U F2R 
37-----R U'L U'F'L'U'F2R U'R2F'U F R'F'R U F'L'U'L F'R F 
38-----U L'U2R U R U'L U2R'F'U L F U L'U R'F R2U R'F'R'F' 
39-----F'U L2F L'U2R U R'F'U F'L'F2R U F U'R'U F'L'U'R U2 
40-----U L'U2R U'L2F U'R'F2U'F R2U R'F'U L F L'U'L'U'F2R' 
41-----F'U L U L F'L'F U L'U'L F L'U2F R U'R'U F R'F'R U' 
42-----F'U L2F'R F2L'F U'R'F'U'R U R'F'U2L F'L'F U L'U2F' 
43-----F'U L2F2R'F L'F2R2U F'L F L'U'L F2R'F L2U'F U F'L' 
44-----U2L F'L2U'L F2R'F'R U F'L F U2R'F R2U F2L'U'F R U2 
45-----F R2U R'F'R U2L'U R'U2L U2R'F'U L2F2R'F L'F2R2U F2 
46-----F'U L U'F R2U R'F'U L F L'U R'U2L F'L'U L F'L'F U' 
47-----F R U'R'U F R'F'R U2F'L'U L U2R'F R U F'L'U L U2R' 
48-----F R2U R'F'R U2L'U R'U L F U2R'U F R'F'R U2F'L'U2R' 
49-----F R2U R'F'U'R'U L'U2R U2L F U2R'U F R'F'R U F'U2R' 
50-----F R2U R'F'R U2L'U R'U L F U'R'F'U L'U2R U R U'L U'

Below is not only a basic strategy, but also a story complete with dates.

My Basic Strategy

Hold the cube so orange is down and white is back. 

Piece 5 (Left Back Down)

• 14735 if piece 5 is at 3 or 7 (Fi U L F U Li) (U2)
• 37415 if piece 5 is at 1 or 4 (U2) (L Ui Fi Li Ui F)
• 34865 if piece 5 is at 6 or 8 (R U) (Fi U L F U Li) (U Ri)

Piece 6 (Right Back Down)

• Green up if 6 is at 1; 78416 (R U F) (R Ui R2 Fi)
• Red up if 6 is at 3; 87436 (R U F) (R Ui R2 Fi)
• Yellow up if 6 is at 4; 38146 (R U F) (R Ui R2 Fi)
• Red up if 6 is at 7; 87436 (R U F) (R Ui R2 Fi) three times
• Yellow up if 6 is at 8; 38146 (R U F) (R Ui R2 Fi) three times

Piece 8 (Left Back Up)

• 13/47 if piece 8 is at 3 or 4 (R U2 Li U) (Ri U2 L Ui). Now 8 is on top.
• 178 until 8 is home (R U Ri Fi) (U Li Ui L) (F Ui).

Piece 1 (Left Front Up)

• 13/47 if piece 1 is at 3 or 7 (R U2 Li U) (Ri U2 L Ui).
• 743 if piece 1 is at 4 (U2) (L Fi Li F) (Ui) (Ri F R Fi) (Ui). Now 4 is at 3. Do 13/47.

743 (The last three)

• 743 (U2) (L Fi Li F) (Ui) (Ri F R Fi) (Ui).

Extras

When I was first trying to solve the Bandage Cube early on it helped to categorize the pieces. Much like it is good to realize something about the three different kind of pieces in a normal cube, it helped to think about the different pieces of the bandage cube and how they worked.

The first challenge was simply in scrambling the thing. Because of the restricted movement caused by the “bandages” it is quite challenging to mix it up. But before long mixed up it was indeed. And as I just tried to solve one side it got scrambleder and scrambleder. I finally solved it a couple times but couldn’t really tell you how. I was trying to keep track and analyze it but it was tough. I experimented with different strategies that I thought would work. In the course of a week I was able to come up with a lot of algorithms through trial and error, being careful to record how to do what. I tried several approaches before finally coming up with the strategy above which is the easiest for me to remember without my notes. Below are some of the algorithms I figured out that are not part of the basic strategy, but may be useful in isolated cases.

436: R 234 Ri 12
367: U2 L Ui R 234 Ri U Li U2 18
143: 743 13/47 19
173: 13/47 743 19
176: U L F Ui R U Fi Li U2 R U Ri Fi U F Ui Ri 17
678: 34786 743 18
146: (red-yellow switch) F U2 L U2 234 U2 Li U2 Fi 18

17/48: (R U Ri Fi U F) (Ui Ri Fi) (L Fi R) (F2 Li Ui) 15
35/78: (R Ui L) U2 (Ri U Li) U2 8
36/48: R U 13/47 Ui Ri 12

34865: R U Fi U L F U Li U Ri 10
68435: R Ui L Ui Fi Li Ui F Ui Ri 10
78635: R U2 L F Ui Ri Fi U Li U2 10
36875: U2 L Ui F R U Fi Li U2 Ri 10

The Story

December 10, 2009: What is a Bandage Cube? It is a 3×3×3 cube that has tiles applied that restricts movement of the layers. There are 14 1×1 tiles, and 20 2×1 tiles. There are 4 center-edge pieces. There are 7 corner-edge pieces. Then there are 2 unique pieces. One is a single corner. The other is a center-edge-center piece. At first I thought the challenge was to scramble it! After awhile it was barely scrambled and I tried to get the green side solved. It just kept getting scrambleder and scrambleder.

Perhaps the best strategy is to try to use the red-yellow-green corner piece strategically to move the orange-white 2×1 pieces to the big orange-white center-edge-center piece. If that is even possible. hmmm…

The strategy worked, to a point. This cube sort of reminds me of trying to solve the 3×3×3 cube using the Petrus Method, only not. I ended up with 3 pieces that needed to cycle. After an attempt to cycle them it ended up needing 3 different pieces cycled. And again. Then I got back to the original situation. Sad thing is, I’m not sure what I’m doing. So close. I’m afraid one of these times it is finally going to fall into place, but I won’t know exactly how I did it.

December 12, 2009: It took a couple days to finally solve it for the first time. I tried to figure out some algorithms. I have a few notes and a few videos, but I do not have it all figured out yet. Now that it is solved I want to think hard about it before scrambling it again. A strategy that might work:

1. Get the first two layers with orange down and yellow up. Mostly. For the left side get it so the greens are pointing down like all the other edges. Then at the end do Li to complete the solve.
2. Line up the yellow-blue center-edge so the blue is on the white side.
3. Cycle the 3 corner-edge pieces into place.
4. Twist the yellow layer so everything lines up on the sides.

Steps 2 and 4 are trivial. Step 1 I was able to do the first day I attempted it. I think I can solve the first two layers a piece at a time. It might be smart to look for a systematic way to do it. For Step 3 I need the yellows in the E formation. You hold the cube so yellow is on the left and the little cube is on the front top left. (Li Ui L) switch (R Ui Ri) D (R U Ri) switch (Li U L) Di

It moves the bottom-front-left to the bottom-back-left to the top-back-left.

December 13, 2009: Overall the strategy outlined yesterday may work. I’m not sure yet. But I should mention in Step 1 that it is probably best to hold white in front and get the front and side corners and edges before going for the back ones. In other words, fill in the 4 pieces to the sides of the white-orange foundation piece.

This puzzle is hard. It may take a couple more days to get Solve #2, and then I still might not understand!

December 14, 2009: Got it! Solve #2. But have I developed a strategy for systematically solving it? Not yet. In the meantime, another puzzle that has recently caught my attention is the 2×2×2. I’ve been attempting to solve it with as few looks as possible. I have been doing so for a few days. Some of my progress is documented in gmail. Basically I can usually get it in 2 or 3 looks. I would like to be able to study it for a minute or so, then solve it completely without looking.

December 20, 2009: I was practicing what I knew after Solve #2 and accidentally scrambled it. It was back to the beginning.

After much trying, failing, trying something else, failing, etc. something finally worked and everything was solved except the red layer. A new idea came to me that enabled me to solve the red layer with a bit more understanding. Since then I have solved it multiple times and recorded sequences of moves that do various double swaps, 3-cycles, and even a couple 5-cycles that may prove useful. A strategy that seems to work:

1. Hold the cube so orange is down and white is on the back.
2. Solve the centers.
3. Solve at least one piece by… uh… pulling and planting posts. First put the little cube over the visitor. Pull it up. Put the little cube over the owner. Pull it up. Take it home with either U or Ui and one of the L’s or R’s or F’s. If anything else is simple to move home, do it. If it isn’t scrambled in such a way that this is easy to do, skip right to the next step.
4. Write down the cycles needed to move the remaining owners home.
5. Figure out how to go about it using the following notes compiled through much experimentation.

In order to help keep track of what I have figured out so far, I have numbered the corners thusly:

1. top front left
2. top front right
3. bottom front right
4. bottom front left
5. bottom back left
6. bottom back right
7. top back right
8. top back left

(At this point in my notes there was a long list of 3-cycles, swaps, and 5-cycles.)

One really cool thing about the 3-cycles is the extensive use of The Move. Yes, The Move that I discovered worked so well with the Jing’s Pyraminx. The Move that I ran across when figuring out the Skewb. I read somewhere that it can also be used to solve the edges of the Pyraminx. I hadn’t approached the Pyraminx that way before, but perhaps now I will. And the 3×3×3 cube—remember the Edge Piece Series of the Ultimate Solution by Philip Marshall? What he calls the Edge Piece Series, I have been calling The Move.

December 21, 2009:

1. Hold the cube so orange is down and white is on the back.
2. Solve the centers and get the top configured as shown in the photo.
3. Move piece 5 home using one of the algorithms given below.
4. Move piece 6 home using one of the algorithms given below.
5. Move the last 5 pieces home using one of the algorithms given below.

There is much flexibility in steps 3, 4, and 5. It may be better after step 2 to find both 5 and 6 and consider which order they should be moved in, or if they can both be moved home simultaneously using one of the extra 5-cycles. Or perhaps moving 6 first, then 5 makes for a more pleasing solution. For example, if 5 is at 4 and 6 is at 3, then doing 436 moves 6 home and moves 5 to 3. It is only 7 twists to get 5 home from 3 as opposed to 14 from 4.

December 28, 2009: Another strategy, not necessarily a better one. I like puzzle solutions that do not require a lot of memorizing or a lot of notes. I have taken a lot of notes during experimentation, but now would like to look for a way to simplify the strategy. It may require more twists and turns, but fewer different sequences.  So instead of 5 different sequences of moves to choose from to get piece 5 home, there is a way to use one of only two.

The disadvantage is that sometimes I’ll end up doing a lot more twisting and turning than necessary. For example, let’s say 5 and 6 are home. If 3 is at 1 and 4 is at 4, if I did 713, which is 18 moves, I’d be done with 3 and 4, but if I use this alternate strategy, I will do the 8 move swap to get 3 home, then have to do a 19 move cycle to get 4 home, for a total of 27 moves.

The advantage of this alternate strategy is that if I can cut back on the number of sequences I use to solve the cube, then maybe I’ll be able to internalize them and be able to do it without the notes.

The Basic Strategy

1. Hold the cube so orange is down and white is back.
2. Solve the centers and get the top configured as shown in the photo.
3. Move piece 5 home by repeated use of one of the following:

• 14735 if piece 5 is at 3 or 7 (Fi U L F U Li) (U2)
• 37415 if piece 5 is at 1 or 4 (U2) (L Ui Fi Li Ui F)
• 34865 if piece 5 is at 6 or 8 (R U) (Fi U L F U Li) (U Ri)

4. Move piece 6 home by using one of the following:

• Green up if 6 is at 1; 78416 (R U F) (R Ui R2 Fi)
• Red up if 6 is at 3; 87436 (R U F) (R Ui R2 Fi)
• Yellow up if 6 is at 4; 38146 (R U F) (R Ui R2 Fi)
• Green up if 6 is at 7; 14876 (F R2 U Ri) (Fi Ui Ri)
• Red up if 6 is at 8; 34786 (F R2 U Ri) (Fi Ui Ri)

5. Move piece 3 home by one of the following:

• 178 if piece 3 is at 8 (R U Ri Fi) (U Li Ui L) (F Ui). Now 3 is at 1.
• 13/47 if piece 3 is at 1 (R U2 Li U) (Ri U2 L Ui)
• 743 once or twice if piece 3 is at 4 or 7 (U2) (L Fi Li F) (Ui) (Ri F R Fi) (Ui)

6. Move piece 4 home by one of the following:

• 178 if piece 4 is at 1 (R U Ri Fi) (U Li Ui L) (F Ui). Now 4 is at 7.
• 874 once or twice (R U) (Fi L F Li) (Ui F Ri Fi) (U F R) (U Fi U2 Ri)

7. Finish up with one or two times through the following:

• 178 (R U Ri Fi) (U Li Ui L) (F Ui)

January 3, 2010: The problem I have run into with the Basic Strategy I worked out, is in step 6. The 874 move. It is too long for me to remember easily. What to do? Save 3, 4, and 7 for the last step. This makes sense because I already know the 743 cycle. But it only makes sense really if there is an easy way to get 8 and 1.

(At this point the solution strategy presented at the top was fully developed.)