Wednesday, May 31, 2017

Skewb Pyraminx Family

Scrambled Skewbs

April 9, 2017
So I have 6 or 7 puzzles that are in the Pyraminx/Skewb family. I call it a family because their guts are the same. At least I think they are. Haven't had them all apart or done extensive research online. I have written posts about some or all of these before and even included videos, but that is all old news. In fact, Solution 1 and Solution 2 below are from years ago. My new interest is in light of a new interest in solving the cube using mainly Up Replace Down, and The Move, aka The Edge Piece Series, aka Al Bob Charlie aka Up Replace Down GoBack. So I wanted to see if I could do the same with Skewbs. The answer: pretty much. The puzzles are pictured and numbered above. 

1. The Pyraminx is easiest. Just some simple twists followed by a few ABCs. (ABC = Al Bob Charlie, my current name for the Edge Piece Series, or The Move)

2. Then comes the Meier-Halpern Pyramid (Meffert's version is popular and known as Jing's Pyraminx). It is like a Pyraminx with centers so there are 4 more pieces to deal with at the end but they are one color each so a Triple-ABC does a double swap if necessary and it is done. This assumes you start by solving it like a Pyraminx, ignoring the centers until the end.

3. Next is the Skewb. It is a cube shaped Meier-Halpern Pyramid. It has 8 corners 4 of which are attached to the core like a pyraminx, which can similarly be solved with a few simple turns. The 6 square center pieces can be solved with ABC just like the edges of a Pyraminx. The catch is that you have to twist the solved corners, not the other ones, while solving the squares, otherwise the solved corners will get scrambled. To solve the last 4 corners if they are not already in the right places use a Triple-ABC to double swap them. Then a couple strategic Double-ABCs can be used to twist them if necessary. The exact strategy came by thinking about the Double-ABC. Which way does it twist which corners? How could the cube be manipulated so that two corners only could be twisted while the others were untwisted? Double-ABC leading with the right hand. Roll the cube so the top corner in back rolls to the bottom left and the bottom right rolls to the top back. Then Double-ABC leading with the left hand. Two of the four twisted corners get untwisted by the left-handed move. And the squares that moved about by the right-handed moves get put back by the left.

4. The Skewb Ultimate is a Skewb in the shape of a dodecahedron. That is a 12-sided Skewb. Four of the small corners can be easily solved like the first four corners of the Skewb. Then the big pieces can all be put into place using The Move. It matters not whether they are oriented correctly at this point since when solving the last 4 corners the big pieces can get flipped. So after solving the last 4 corners flip the big pieces.

5. The Squished Skewb has to be solved like the Skewb Ultimate because its odd shaped center pieces can get flipped. It is also tough to solve because of its large size. And it shape-shifts because of its squishedness. But at least the pieces resemble those of a normal Skewb.

6. The Skewb Curvy Rhombohedron is by far the quirkiest of the Skewbs I own. First I want to say that it is not a rhombohedron. Maybe kite-o-hedron would be a better name, but whoever heard of one of those! At any rate there are six faces all of which are kite shaped. There are 3 different shaped pieces. Six triangles in the center of the kites, two small corners, and six large corners. The six large corners correspond to the six squares of the Skewb. The two small corners and six triangles correspond to the eight corners of the Skewb. Oh, did I mention it shape-shifts? Like the Skewb Ultimate the big corners can flip while solving the last 4 pieces. But then there is something else I have encountered that I did not on any of the other skewbs. Instead of a double swap I have had to twist a face at the end to get the last 3 corners solved. Then I have to resolve the "edges" that shift around. Very quirky indeed. Fun. The reason this happens is because the triangle center/corners don't have obvious orientation like the other skewbish puzzles have. If you make sure all 8 "corners" are solved in the beginning then that won't happen at the end.

7. There is one more puzzle with the word Skewb in the name, but it is not very skewbish in my mind. The F-Skewb. It is cut in such a way that it can be scrambled and solved exactly like a Skewb, but there are additional cuts that allow for non-skewbish scrambling. Four little corners can be solved easily like the first four of the Skewb. Each center Skewb square is elongated and cut in half so there are 2 squares per side. I think they can all be solved using The Move. More testing is required. Then the final four corners are each cut into thirds. I can solve them after all else is solved using a commutator I devised when solving the Face Turning Octahedron. The commutator is very much like the one used to 3-cycle corners of a cube.

And here they are solved.

Solved Skewbs

The following was written years ago when I had a different take on the whole thing. My current favorite solution starts out like Solution 1, but when it comes to the last 4 corners is much simpler than the following. There are not cases and no jumping about. Simply permute then orient the corners. I must admit that with the Curvy Kite Skewb I frequently have to 3-cycle corners at the end instead of doing a double-swap. I twist them into place, use the Move to put the three big pieces back and finish it up. As I think on it, I seem to recall a way to avoid this situation. After solving the first 4 pieces check to see if the last 4 need to double-swap or 3 cycle.

Skewb Solution 1: (This is based on the Crazy Tetrahedron Method I learned from rline.)

Centers: First 4 Corners

Get two opposite corners on one face. This takes at most two twists. Get the two skew corners on the opposite face. This takes at most two twists.

Edges: 6 Center Squares

Use EPS making sure you twist the centers solved in the Centers section, and not the Corners that will be solved last. In the Crazy Tetrahedron and the Jing's Pyraminx, it is easy to tell centers from corners, since centers look like centers and corners look like corners, but that is what makes the Skewb challenging. The centers and corners all look like corners and all are centers of rotation.

Corners: Last 4 Corners

Hold the puzzle with a Center forward on the bottom, and 2 corners on bottom with one on the right and one on the left. Another center is on the upper left, and one on the upper right. Call the one on the left L, and the one on the right R. ( L'RLR' ) x 3 will swap the two corners on the bottom with each other, and the two on the top with each other. The colors on the corners on the front faces on the bottom will move to the bottom after the swap, so it is preferable they are the same as the edge on the bottom. The colors on the top of the front top corner and back top corner will stay on top.
Case 1: All are in the right place but at least two are twisted. Hold two twisted corners on top, and do ( L'RLR' ) x 3. Go to Case 2.
Case 2: Double Swap needed.
Holding the two bottom corners and top front corner still, twist the back half of the puzzle so the edge on top matches the color of the top corner that is in front.
( L'RLR' ) x 3 If necessary, go to Case 1.
Case 3: One is in the correct place, and may or may not be twisted.
Twist the Half which will place all 4 corners in their correct spots. They may or may not be twisted. Use setup moves and EPS to re-solve the edges. Go to Case 2.

Solution 2 (This is the solution I came up with on my own originally.)

Notation: Hold the Skewb so one face is facing you, and so there is a corner pointing right, one pointing left, one pointing up, and one pointing down. We will name these corners Rf, Lf, Uf, and Df, respectively. The corners behind these corners are in the back so we will name them Rb, Lb, Ub, and Db. Twists are defined by the corner that is the center of rotation for the twist. Centers can be named Ru, Rd, Lu, Ld, F, and B.

First 4 Corners

  • Get two opposite corners on one face. This takes at most two twists. Get the two skew corners on the opposite face. This takes at most two twists.

Place the Last 4 Corners

  • Rb Lb Rb' Lb' swaps Rf with Lf and Db with Ub, and twists Lf, Ub, and Db counterclockwise.

Twist the Last 4 Corners

  • ( Rb Lb Rb' Lb' ) x 2 twists Rf and Lf anti, and Db and Ub clockwise.
  • ( Rb Ub' Rb' ) Df ( Rb Ub Rb' ) Df' twists Rf anti and Lf clockwise. This is a very easy to see what is happening algorithm. Move Rf up; Twist it; Move it down; Replace it with Lf; Move Lf up; Twist it; Move it down; Move it back.

Place the Centers

  • [ ( Rb Ub' Rb' ) Db ( Rb Ub Rb' ) Db' ] x 2 moves Lu > Ld > Rd. I hold my left thumb on F throughout. I go for 3 centers in a row, rather than 3 adjacent centers, but if I end up with 3 adjacent I can use a setup move when placing the last 3.
  • ( Rf' Lf Rf Lf' ) x 2 Df ( Lf Rf' Lf' Rf ) x 2 Df' moves F > Ld > Lu. That is, it moves 3 centers around a corner without scrambling anything else.

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