Sandwich/Cage Method Algorithms

[Boneless]

Hello, all! I have just gotten into 4x4, and I really enjoy the Sandwich Method, since it is intuitive and there is no parity. The problem is, I don't know how to solve the last four centers or the last four edges. I've only found one resource on the Sandwich Method, but the algorithms don't have descriptions or pictures, so I have no idea which ones to use. Also, many of them seem to mess up one or both of the solved layers. My question is, are there any algorithms that can permute the centers of a 4x4 without messing up the edges or solved sides, and are there any that can move the four middle edges? These algorithms would also technically be the same ones as the Cage Method.

I intuitively solve the corners using Ortega, then the 8 edges for the L and R layers, then the rest of the edges, but then I have no idea how to solve the remaining centers and put the edges in place. There are algorithms for solving the edges, but once I do that, they are still in the wrong place. For reference, here is the only thing I've found about the method: http://web.archive.org/web/20090423225534/http://rachmaninovian.webs.com/

I would much rather use Sandwich or Cage instead of a method that ends in having to solve like a 3x3, since I use Roux and doing M slices on a 4x4 is very awkward. Attached are pictures of the point to where I can currently get. The white and yellow faces/layers are completely solved. Thank you to anyone who has any information!

 

[xyzzy]

I used to use sandwich, although I did the steps a bit differently from how Nicholas did it (and he's faster than me at his method, obviously).

First off, you should hold the cube so that the white and yellow faces are on the left and right sides.

To solve the edges, you form two edge pairs with the same colour (you don't need all four edge pairs), and throw them in the D layer. The four remaining wing pieces are then solved in a single step. For example:

z' U2 m U2 // D edge pairs
2R' U2 2L F2 2L' F2 2R2 U2 2R U2 2R' U2 F2 2R2 F2 // U edge pairs (case 2 in the step 4b page)
m' // align

(Follow the link to see an animated view of what's going on.)

This way of solving the edges requires quite a few algs (and some of them are pretty ugly) for step 4b; I prefer to solve the ones on the left side first, followed by the right side. For example:

z' U2 2L 2R2 U2 2R' U2 2R2 U2 // left side
x y' R' 2U R' D2 R 2U' R' D2 R2 y // right side
L r R2 // align

(The alg used here is just an A perm, but with U moves replaced with u slices. If you get a diagonal-swap case, you can either use two A-perms to solve it, or learn another alg to solve it in one look.)

To solve the centres, the basic alg you need is just Niklas: 2L' U 2R U' 2L U 2R' U'. This moves the Ubl centre to the front face, and the Ful centre to the top face. Repeat this as many times as needed to finish solving the centres. There are much more efficient ways of doing this, but you can always learn those after you've familiarised yourself with the basics.

 

[Boneless]

Thank you for the information!

To solve the edges, you form two edge pairs with the same colour (you don't need all four edge pairs), and throw them in the D layer. The four remaining wing pieces are then solved in a single step.

So you can't solve the edges in any order and then put them in the right place? I know there's an opposite edge swapping parity algorithm, but it doesn't change the orientation. Is there an opposite edge pairs swapping and orienting algorithm? So in the picture I attached, I have to redo the edges? Even though the edges are paired, they're technically wrong, since they're in the wrong places. Or can I do U and M moves to get them correct once they're paired, sort of like the last part of Roux?

To solve the centres, the basic alg you need is just Niklas: 2L' U 2R U' 2L U 2R' U'. This moves the Ubl centre to the front face, and the Ful centre to the top face. Repeat this as many times as needed to finish solving the centres. There are much more efficient ways of doing this, but you can always learn those after you've familiarised yourself with the basics.

So the algorithm swaps Ubl and Ful? If so, then I would assume it's just a matter or doing setup moves, doing the algorithm, and then undoing the setup.

I also have a lot of trouble with step 3b. I can easily build the pair in the M slices, but then how do you get it into the last spot in the left or right face? I keep messing up the opposite face. Also, what if part of the last edge pair is in the spot that it will be in? How do you get it out to pair it up? Lastly, if the edge pair is already correct and in the slot, but upside down, would you just do the parity algorithm r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2 to flip the single edge?

 

[xyzzy]

So you can't solve the edges in any order and then put them in the right place? I know there's an opposite edge swapping parity algorithm, but it doesn't change the orientation. Is there an opposite edge pairs swapping and orienting algorithm? So in the picture I attached, I have to redo the edges? Even though the edges are paired, they're technically wrong, since they're in the wrong places. Or can I do U and M moves to get them correct once they're paired, sort of like the last part of Roux?

If you're doing it as on the step 4a page, you don't need to pair all the edges. You just need two pairs. If you do have all four edges paired, just pick two with matching orientations and place them together using M and U moves (as in Roux 4c). Occasionally, you can't get them to match in orientation (but this is rare), so just use any edge-flipping alg as needed to make the orientations match. (E.g. (M' U)4 (U M')4.)

So the algorithm swaps Ubl and Ful? If so, then I would assume it's just a matter or doing setup moves, doing the algorithm, and then undoing the setup.

If the setup includes only U and F moves, yes. It's not only a swap of Ubl and Ful; it's actually a 3-cycle Ful → Ubl → Fur. This only allows you to move pieces between adjacent faces, so you'll have to get creative for dealing with swapping pieces on opposite faces. (Or use the algs on the step 5 page.)

If you know how commutators work, you can cycle pieces from three different faces at once, e.g. 2L U 2R U' 2L' U 2R' U' or 2R U 2R' 2U2 2R U' 2R' 2U2.

I also have a lot of trouble with step 3b. I can easily build the pair in the M slices, but then how do you get it into the last spot in the left or right face? I keep messing up the opposite face.

You can do it the same way as you insert UL and UR in Roux once you have the last LR edge paired.

Also, what if part of the last edge pair is in the spot that it will be in? How do you get it out to pair it up?

Insert an edge that doesn't have white/yellow on it in that location. Set up some dummy pieces (without white or yellow) onto the DF location, then do U M' U2 M U (if the stuck piece is in UR) or U' M' U2 M U' (if the stuck piece is in UL).

Lastly, if the edge pair is already correct and in the slot, but upside down, would you just do the parity algorithm r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2 to flip the single edge?

(M' U')4 if UR is flipped, (M' U)4 if UL is flipped.

 

[Boneless]

If you're doing it as on the step 4a page, you don't need to pair all the edges. You just need two pairs. If you do have all four edges paired, just pick two with matching orientations and place them together using M and U moves (as in Roux 4c). Occasionally, you can't get them to match in orientation (but this is rare), so just use any edge-flipping alg as needed to make the orientations match. (E.g. (M' U)4 (U M')4.)



If the setup includes only U and F moves, yes. It's not only a swap of Ubl and Ful; it's actually a 3-cycle Ful → Ubl → Fur. This only allows you to move pieces between adjacent faces, so you'll have to get creative for dealing with swapping pieces on opposite faces. (Or use the algs on the step 5 page.)

If you know how commutators work, you can cycle pieces from three different faces at once, e.g. 2L U 2R U' 2L' U 2R' U' or 2R U 2R' 2U2 2R U' 2R' 2U2.



You can do it the same way as you insert UL and UR in Roux once you have the last LR edge paired.



Insert an edge that doesn't have white/yellow on it in that location. Set up some dummy pieces (without white or yellow) onto the DF location, then do U M' U2 M U (if the stuck piece is in UR) or U' M' U2 M U' (if the stuck piece is in UL).



(M' U')4 if UR is flipped, (M' U)4 if UL is flipped.

Thank you!