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. 

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