Sunday, May 28, 2017

Pyraminx Crystal

Originally posted April 10-12, 2017 on the BudLCuber Google site.

Recently Roark mastered solving the Rubik's Cube. It is a method that uses The Move, aka Al Bob Charlie, aka Up Replace Down GoBack. He also learned the basic Up Replace Down 3-cycle for corners. And how to twist corners with a Double-ABC. And how to double swap corners with a Triple-ABC.

I got to thinking that he could use the skills he has learned to solve Pyraminx and Skewb puzzles. We haven't tried it yet.

Then I wondered about the Pyraminx Crystal. It was half solved already when I got it out of the closet. I finished it up but wondered what my whole solution involved. I found this note in my Solutions spreadsheet:

White corners; white edges; 4 of the next 5 edges; middle layer lower corners, then last of the edges of the bottom layer; middle layer upper corners; middle layer edges; last layer corners; last 10 edges.

That sounded very familiar. I tried it out. Yes. Very comfortable. As far as all the corners except the last 5 it is all a matter of just putting them in place, or using Up Replace Down to place them. All edges can be done with ABC (The Move). The last 5 corners can be 3-cycled using the Up Replace Down 3-cycle, or double-swapped using Triple-ABC. They can be twisted using Double-ABC. Using the ABC moves on the last layer corners scrambles a few of the middle layer edges, so if wanted, the middle layer edges can be left scrambled until the corners are all solved. Or, of course, my traditional methods for placing and orienting corners can be employed.

Solving it a couple times using the old strategy for the bottom part (white on bottom) and the revised method for the middle edges and last layer was educational. I didn't really like the part of solving 4 of the next 5 edges. I decided to try solving 4 of the 5 bottom middle layer corners instead. It was quite easy. Then using the working face I solved the corresponding edges that go below the corners. Then I solved the last of the 5 corners and inserted the edge. I liked it better for some reason than doing the edges before the corners.

My Current Favorite Method:
  1. White Corners
  2. White Edges
  3. Four of the five Corners in the bottom half of the middle layer
  4. The Edges that go below those Corners
  5. The last Corner and Edge that go in the bottom half of the middle layer
  6. The five Corners of the top half of the middle layer
  7. The five Corners of the top layer
  8. Remaining Edges

Thursday, March 30, 2017

The Al Bob Charlie Method of Solving a Rubik's Cube

It has been a long time since I had anything to say. It was a long break from puzzling. But my interest has been rekindled this last month. New challenging puzzles have not sucked me back into the puzzle realm although I watch one of Rupert's videos once in awhile. In fact this last month it has been the 3x3x3 cube that has recaptured my attention. I wanted to teach Roark a simple method for solving the cube that he could learn and remember. No algorithm memorizing necessary. It fell into place like this.

First master solving the white edges. I did not require that he do it like I do it. We did not even talk about how to do it. Second, master the middle layer edges. I may have shown him how I do it but for the most part it was up to him to work it out.

Here is where the lessons really began. After mastering the white edges and middle layer edges I introduced him to the working edge—one of the middle layer edge spots. In other words, after solving the white edges, only solve 3 of the middle layer edges. Then solve the final 5 edges together.

Start by using the working edge slot to solve any two of the yellow edges. Each one is done with 3 or 4 twists. In fact, to this point the 9 solved edges have all been done with intuitive moves, and little instruction.

Next I taught him the Edge Piece Series, moves like Ri F R Fi, or U Fi Ui F. Only we didn't write down algorithms, or call it the Edge Piece Series. We called it Al Bob Charlie. To cycle Al to Bob to Charlie where Charlie is the one that needs to flip, you take Al to Bob, Charlie to Al, go back, go back. We discussed the problem of having to flip all three instead of just one. Cycle them the wrong way on purpose, then cycle them home. And we briefly discussed what to do if all the edges were in place but two had to flip. Cycle them out of place then back home correctly. We learned that if after solving two yellow edges, if one of the last three edges was in place and the other two had to swap, then that meant the two solved yellow ones were attached to the yellow center incorrectly. The top had to be twisted 90 degrees and solved again using the Al Bob Charlie method. He got it. He mastered the edges on his own at home over the next few weeks.

I resisted the temptation to try to teach him an algorithm to swap two edges, or to teach him a popular algorithm for 3-cycling yellow edges. We did it all with the simple Al Bob Charlie method. And over the last month or so that this was happening I have used this method myself to solve the cube many many times. I like it. It is fun. It feels good. It makes sense. I even bought a couple more cubes. I had a stickerless speed cube already, but I wanted a genuine Rubik's Cube so bought one on for $8. One with tiles. I like it. Then I bought a Dreampark cube with black carbon-fiber stickers. I love it. So much so that I got their Pyraminx with black carbon-fiber stickers as well. It is by far the best pyraminx I've ever had. More on the Pyraminx later. Back to the cube.

It was time to learn the corners. At first I tried to teach him to use the same exact movements that I use to whip through some of the aspects of solving the corners, but eventually I decided it was best to let him work out his own ways to twist and turn.

At this point we transitioned from an edges first method to a working corner method. I showed him how to easily solve 3 white corners after solving the first 7 edges. Intuitive Up Replace Down moves were needed to accomplish this. Of course the bottom had to be turned to line up the bottom corners under the working edge, and sometimes we had to deal with a white corner on top that had white on top, but those were not difficult to deal with. Before long he had mastered all the edges plus 3 of the 8 corners. Only 5 corners to go.

Three skills to learn. How to 3-cycle corners. How to double swap corners. And how to twist corners. 3-cycling uses the familiar Up Replace Down move. It cycles two bottom corners and one top corner. The bottom corner that stays on the bottom is the one you move first. It goes Up. Replace it with the corner you are cycling that is on top. Move it Down by undoing the twist used to move the other one Up. Now twist the bottom to replace the corner that moved Down with the other corner that is being cycled. Move it Up with the same move used to move the first piece up. Replace it with the first piece. Move it Down. Move the bottom back. Done. I like this 3-cycle for two reasons. It builds on what is already used—Up Replace Down. And it is flexible. The pieces do not have to be in any certain spots as long as two are on bottom and one on top. You can do it either right-handed or left-handed. And with practice and keen observation you can discover how to cycle corners to avoid having to twist them later.

Twisting them isn't a problem though. It also builds on something already used. Al Bob Charlie. If you Al Bob Charlie the top right edge to the right front edge to the top front edge and do it again it twists the bottom right front corner counter clockwise. The key is that is the only change on the bottom. So then you can turn the bottom layer to replace the twisted corner with another corner that needs to twist. This time Al Bob Charlie the front top edge to the front right to the top right. That twists the second bottom corner clockwise and restores everything else that got scrambled twisting the first corner. Finish by putting the bottom back where it started.

Finally we will cover the double swap. It too is another application of the Al Bob Charlie move. Let's say we need to swap the top right and bottom right corners on the front layer. And we need to swap the back left and back right corners on the top layer. Al Bob Charlie the front right edge to the right top edge to the top back edge. Do those four twists 3 times in a row. That's it!

One other thing. This method relies on two things. Up Replace Down and Al Bob Charlie. Question: How are they related? The double swap explained in the previous paragraph can be thought of like this. Do Up Replace Down GoBack three times in a row. GoBack is simply twisting the top layer the opposite of the way you did on the Replace twist. So instead of calling this the Al Bob Charlie Method we could call it the Up Replace Down GoBack Method.

Friday, August 21, 2015

2x3x4 Camouflage Cuboid Reduced to 2x2x4

After working out a solution heavily relying on commutators and having one part that was totally baffling and that although I always found a way through I never really fully understood—anyway it dawned on me that the puzzle could be partially scrambled as a 2x2x4 and that made the solution much easier. So, although I am not real keen on reduction methods for most puzzles I thought that reducing the scrambled 2x3x4 to a 2x2x4 would make it much easier to solve. After a few such solves I was feeling confident enough and curious enough to watch Twisty Puzzling's tutorial. He reduces all the way to a 2x2x2. I suspected that would be his approach. :)

In trying to refine my method of reducing to the 2x2x4 I have run into a situation that is almost as bothersome as my first method was. Let me show you.
In pic 1 you see the scrambled state for this solve. My strategy in reducing to a 2x2x4 is to attach yellow and white edges to the 4 orange corners. The red corners stand alone due to the axis of rotation that is between the red and orange layers. The other skinny edges make up the middle layers of the reduced 2x2x4. Look at pic 7 to help visualize this if necessary.

Pic 2 shows the Orange Green White block reduced. 

Pic 3 shows the Blue Yellow edges in place to slide over the Orange Blue Yellow corner, but the Blue Red edge is blocking it. It was a real struggle for me to get past this without losing track, but on a subsequent solve I think I figured out an easier way to deal with the situation. Namely, twist the corner made of the corner plus two edges so the Blue Red is horizontal. Then replace the Blue Red with an edge-center pair. Then twist it back so the Blue Yellow is back in place ready to slide over the Orange corner.

Twisting Corner Blocks. Using 2x2x2 moves do things like ( R U Ri Ui ) x 2 or ( R Ui Ri U ) x 2 or the left-handed versions, to orient corner blocks. Only one corner on the bottom layer gets twisted, so the bottom is the working layer where pieces are paired up. Three pieces in the top layer get twisted, but since they are corner blocks, already reduced corners do not get unreduced.

Pics 4 and 5 show two more reduced Orange corners. Somewhere during the hassle of getting the Orange Blue Yellow the 4th Orange corner got reduced too.

Pic 6—once the Orange corners are reduced it is ready to solve as a 2x2x4 Tower. For me that means solve the corners ignoring the middle layer edges and centers. Oh wait, I don't totally reduce it to a 2x2x4 since I ignore the centers until the end.

Pic 7—To solve the middle layer edges of a 2x2x4 I use the same technique as when solving the corners of a 3x3x2. With the 2x2x4 do not twist the top or bottom layers or they will get scrambled as the middle layers get solved. Just move the middle layers and the orange, red, blue, and green layers. When I say the orange layer I mean the reduced layer. Any color can be on the front when using the 3x3x2 corner cycling commutator. I use things like ( u L2 ui R2 ) x 2. 

Pic 8—The final step is to solve the centers. With Red on the right the following moves the square center on the Bottom Back to the Bottom Front and the Bottom Front to the Top Back.

( M2 U M2 Ui M2 ) Di ( M2 U M2 Ui M2 ) D

Sometimes this isn't what you need. If you need to cycle from Bottom Front to Bottom Back to Top Back, still make sure Red is on the right and do the following.

Di ( M2 U M2 Ui M2 ) D ( M2 U M2 Ui M2 )

Now that I have used the reduction to 2x2x4 method a couple times and am feeling comfortable with it I wonder if I could apply the corner twisting strategy to my first method in the step that gave me so much trouble to make it very easy to solve without any reduction. There is something about starting with the corners and filling in all the edges from there that really appeals to me.

(9/3/15) I pretty much like the reduce to 2x2x4 method. Another thing that is helpful at times is ( RU RiUi ) x 3 to invert a column. Also 3-cycles when they aren't blocked help out sometimes.

Wednesday, August 19, 2015

Ability, Knowledge, Understanding

Ability. I have the ability to scramble and solve this puzzle. But how well do I know what I know? Do I understand this puzzle?

I know that it can be scrambled and solved as if it is a 2x2x2 if you only do twists on the central axes. I've done this several times.

I used to have a 2x2x4 Tower. This puzzle can be scrambled and solved like a 2x2x4. It is much simpler than the 2x3x4. 

Wait a minute! What if I reduce the fully scrambled 2x3x4 to a 2x2x4? It would only involve pairing up 4 corners with 4 edges, so reduction wouldn't be as tedious and troublesome as I normally find it, yet it would greatly simplify the solve.

1. Pair up the Orange White Blue corner with the White Blue edge. The Orange White Green corner with the White Green edge. The Orange Yellow Blue corner with the Yellow Blue edge. The Orange Yellow Green corner with the Yellow Green edge.

2. Solve the corners using only 2x2x2 twists.

3. Solve the middle layer edges using the 3x3x2 Corner Commutator.

4. If necessary solve the centers.

Tuesday, August 18, 2015

Witeden 2x3x4 Camouflage Cuboid

I posted recently on the Facebook Puzzle Photography page:
From scrambled to almost cuboid took about a half hour. To go from scrambled to solved corners is easy. It took some fiddling and experimenting to get it to almost cuboid shape.

I used 3-cycles to move edges around. Made good progress fairly rapidly but then wandered around in circles near the end. It took an hour and 20 minutes to solve all the edges.

Getting the centers at the end wasn't too hard.

This was my third solve of this puzzle. It was by far the best scramble.

I've had this puzzle for almost a year and made an effort to analyze it months ago but couldn't make enough sense of it to scramble it. Eventually put it away and forgot about it. On the return trip from the CR I got back to it. Since being home I've made good progress in my analysis and have solved it maybe a half dozen times. Here is my approach so far.
  1. Solve the Corners like a 2x2x2.
  2. Try to get it to cuboid shape.
  3. Cycle all the Edges home.
  4. Cycle the Centers home.
Step 1 is self explanatory.
Step 3 has two parts. First get the whites and yellows. Use a commutator that goes from one white or yellow piece to a spot that should be to a spot on the side that shouldn't be but is. Or use a commutator that goes from one side edge to one that is white or yellow but shouldn't be, to its home. Use the 3x3x2 corner cycler to put all the non-white/yellow pieces home. Any face can be forward.
Step 4 can be accomplished with a commutator also. I hold 2 on the bottom and one on the top. The one on the top that is the wrong color is in back. The one on the bottom that is the right color is the first one that should be brought to the top. The commutator is a bit different than most:
( Twist D if necessary ) ( r2 U r2 Ui r2 ) ( Twist D one way or the other ) ( r2 U r2 Ui r2 ) ( Twist D if necessary ).

Steps 1, 3, and 4 are becoming somewhat routine now and feeling more and more like most other twisty puzzles in my collection. Step 2 is a big mystery to me at this point. I just keep fiddling with it until I get it, but there is usually some blockage to deal with and I'm not totally sure how I've done it.

Monday, August 3, 2015

Bandage Kit Cube

I wanted Annie to play with a corners only cube. She had ideas of her own. Pretty soon it was covered with tiles so big it wouldn't turn. We made some slight adjustments and two sides would turn but each blocked the other. After further modifications one layer has two positions that do not block the other turntable layer. And the other does no blocking as it is made up of all 1x1x1 cubies. 

It does not scramble very much. The Move and Sune can both be done so it is easy enough to solve the 5 edges. The Move X 3 can be used to do double swaps of edges to permutations the edges. It was challenging to come up with a corner twister for me and the ones I have come up with are pretty long. 

At first I could only find ways to twist four corners. I wanted to be able to twist two corners, so I didn't keep track of how to twist four. Now I can twist two, but usually during a solve four or five need to twist once the edges are solved and corners permuted. 

A discovery (rediscovery?) made last night is that the edge swapper routine I learned once upon a time done twice in a row does a double swap of corners. Combining that with The Move X 3 twists two corners. Specifically (U R U Ri U R U Ri U) (U R U Ri U R U Ri U) U (R U Ri Ui) X 3 Ui twists FLU anti and BLU clock.

Another way to twist two corners uses Sune. (R U Ri U R U2 Ri U) X 2 R U2 (Ui Ri U R) X 3 U2 Ri twists FRU anti and BLU clock.

Wednesday, July 29, 2015

Dayan Gem V Solution

In the fall of 2013 I participated in a discussion at the forum about how to solve the Dayan Gem V. The solution I came up with originally was OK, but by the end of the discussion I had come up with a better way. I don't think it incorporated anything from the discussion. The discussion simply spurred me on to find a better way.

Here is the complete method as presented on the forum.

1. Get all the centers—small triangles, large triangles, and if you have stickerless v1 as I do, the thin strips on the square faces. All 8 triangles can be solved with either 0, 1, or 4 moves. I do not mean each! I mean that after either 0, 1, or 4 twists all 8 triangles are solved!

The 6 thin strips take me up to 3 Double-EPS, or 24 twists. I use Double-EPS to maintain centers. Rarely it takes three of them. Rarely are they already solved. Usually it takes two Double-EPS to solve all 6 pieces.

2. Place all the small edges (corners) from the bottom up. Uses an Up-Replace-Down technique. Some people call this truncated EPS, since it is the first 3 twists of the EPS. 

3. Use [3,1] commutators to solve the large edges.

And in an earlier post I said:
doctor twist wrote:
I am left with 3 unsolved center pieces. How you can deal with that?

In my solution outlined directly above, I deal with all 8 of the triangle centers, and all 6 of the thin strip centers of the square faces in step 1, so that I never end up in the situation you describe of being left with 3 unsolved center pieces. My motto: Deal with parity as early as possible so you don't have to unsolve and re-solve at the end.

So, here is how I deal with the centers in a little more detail. Starting with a scrambled puzzle,

1. I hold the puzzle so there is a square face on top, and the large red triangle is on the upper half of the front, and the large purple triangle is on the upper half of the back. This is totally arbitrary, but it is what I am used to looking for and it defines for me the color scheme of the whole puzzle. (It is sort of like when I am solving a Rubik's Cube I start by holding it with the white center on the bottom.)

2. With the Red Triangle in front, I can see 3 small triangles. By looking at these 3 I can tell whether they are solved, or whether they need a double swap, or whether I need to do one twist that will solve them. If the one twist is required I do it. If the double swap is required I do it. (So I deal with the "3 unsolved center pieces" before anything else on the puzzle is solved

3. Since I have the stickerless version, there are colorful thin strips on the square faces. I place them next using Double-EPS moves. I use Double-EPS instead of just EPS so that the 8 triangle centers don't scramble during this phase. So if I need to move thin strip A to thin strip B to thin strip C, I go the opposite way twice! C to B to A twice is the same as A to B to C, and with each EPS there is a double swap of center triangles, so doing it twice leaves them solved. :D

So that is how I deal with it. But what if someone handed me a puzzle with everything solved except for 3 triangle centers? Then what would I do? I would do a single twist that solved them, then re-solve the puzzle from there. What a pain! That is why I like to insure at the very beginning that this will not happen.

As I read through the forum topic today some interesting things came out. themathkid originally solved the Dayan Gem V like he solves the Vulcano, Konrad solves it like the F-Skewb, and I originally thought of it like the Jing's Pyraminx. 

The thing that got me into this puzzle today is that I left it downstairs where Alex could reach it and Annie brought it to me in pieces. Three pieces had fallen out. She brought the puzzle and the pieces to me upstairs. It popped back together easy enough but I'm not sure the pieces are oriented correctly so I have to solve it to find out. I've solved it a few times already this summer using the abbreviated guide in my spreadsheet, but something told me I had recorded more about it so I went looking and found it at the forum.

Monday, July 27, 2015

F-Skewb Squares

I do not think of this as a Skewb at all. The only thing skewb-like about it to me are the four little corners. They are quick and easy to solve requiring only a twist or two. Maybe 3 or 4. No more. Other than that it remings me of the Face Turning Octahedron. The squares are like the central triangle pieces and the edges are like the edges. So this summer I think when I first solved it I used the same 8-twist commutators that I developed for the FTO. I don't recall if I've ever watched anyone else solve it (rline) or not or run across any other solutions, but the last few days at least I've figured out that The Move can be used to get the squares, which is easier than the 8-twist sequence. Now that I think about it, I think rline did introduce me to this concept only on the Master Skewb. I no longer have the Master Skewb. It was stolen. And I no longer play with the FTO because the stickers are so rough.

Monday, July 20, 2015

Český Krumlov

We went to the castle yesterday. We had been at least twice last year—this year was the best yet, except I forgot to enjoy the cliff face from below yesterday! I can't believe it!

This year we started at the top—the castle gardens. We hadn't been there at all in 2014.
(Below) it amazes me how they built the castle around the rock.
(Below) the kiddos seemed to enjoy looking at the river and the boats from the lookout up above. I zoomed in to see what the boaters were doing. Apparently they stopped to take pictures of one another.
(Below) the Bears are always a treat.

(Below) and of course trdelník is always a treat too! This time we ate it at Zapa Cafe, where we ordered drinks, and let the kiddos play in the playground. David had a ball.
(Below) Most of the time the streets were filled with people walking, but sometimes we had to yield to a car or truck.

(Below) Raymond and I enjoyed a made-in-the-USA Dr. Pepper he found in this little store. Each of the children got to ride in the stroller at some point.
(Below) On the way home we stopped for lunch. It was great! I had fried cheese. Lunch cost less than the local McDonald's and was a beautiful, peaceful, quiet restaurant.

Saturday, July 18, 2015

Bud vs. The Fan

Yesterday I had a cup of coffee in one hand and a plate with a cinnamon roll on it in the other. There is a little table next to the couch that would make a nice place to set my coffee while I ate the roll. Only one thing stood in the way. A fan. There was enough space between the fan and the table that I could sit on the edge of the couch straddling the fan and enjoy my treat. But to get there I had to step over the fan. I'm not sure what went wrong but I didn't make it. Instead my foot came down near the top of the fan, slid down the side of it, and got impaled by a piece that was sticking up on the side.

I threw my foot up as I fell backwards onto my back on the couch in pain. By some miracle the plate and cup remained upright and nothing spilled.

Raymond and Marae pitched in to doctor my foot. As they worked on it they agreed it should probably have stitches. We opted to do the best we could at home and see if it would close up and stop bleeding without stitches. Shown also is what it looked like this morning.

Psalm 106:21-23

21 They forgot God, their Savior, who had done great things in Egypt, 
22 wondrous works in the land of Ham, and awesome deeds by the Red Sea. 
23 Therefore he said he would destroy them had not Moses, his chosen one, stood in the breach before him, to turn away his wrath from destroying them. 
Psalms 106:21-23

I haven't done any serious research into this but when I read it this morning it hit me that
  1. I, like the Israelites of old, too easily forget my Savior, who has done great things on my behalf to secure my salvation and a place in His family forever.
  2. The holy God, creator of the heavens and earth, to whom all respect and obedience and reverence and worship is due, has every right to destroy those who treat His salvation so lightly.
  3. Jesus, His chosen one who is greater than Moses, stands in the breach before Him to turn away His wrath from destroying us.
If I were a preacher...

Thursday, July 16, 2015

Gotta Love It

The Before pic shows how I left the puzzles in the morning when I left with Raymond. The After pic shows what state they were in when we got home. I don't think anyone over 4 years old touched them while we were gone.

Wednesday, July 15, 2015

The Gravel Pit

Ok, so actually it is a playground, and we did swing for a long time, but then...

Tuesday, July 14, 2015

To the Pond and Beyond!

(Above) Annie wanted to walk across the board. I said no.
The path beside the pond.
The pond.
Annie told me we could see the church (the Catholic Church in the village). I took a picture then zoomed in and took another.
Looking back on the path beside the pond.
Looking ahead beyond the pond.

(Below) Crossing the pasture beyond the pond. We eventually came to a road.

(Below) Beginning the walk back. The sidewalkless part of the trip on the road.

(Above) The winters are long and cold—you see lots of wood stacks like this.
Annie was leading the way as I had no clue where we were. I didn't remember this short road but later learned I had been there on a walk before.
The kiddos posed in front of these flowers for me. This is up the road not far from their house.
These flowers are in the neighbors' garden at the end of the driveway.

Hluboká Zoo 2015

I uploaded 71 photos to Flickr. Here are a few highlights:

Sunday, July 12, 2015

Fun at the Big Slide Playground after A Long Walk

After walking down the steps, and across the street, and up the road back to the Catholic Church, we stopped at the playground on our way home. It was fun. While Annie road the lion Alex checked our my Music library and chose a Steven Curtis Chapman song. They had fun on the slide together. Alex tried to climb the swing set pole while Annie was swinging. I brought three puzzles along to scramble—mission accomplished.

Winnie the Pooh and 4x4x4 too

Summer 2015 Catch Up

Blogging has been a way for me to document some of the things that are happening this summer, but I don't always get around to it. Maybe I can do some catching up with a hodge podge of photos that haven't been given their own post. They obviously are not in chronological order, but they are all from the last couple weeks.

I enjoyed watching Alex do a Jigsaw Puzzle one day.


They grow some interesting plants around here.


There are several cats that hang out around the driveway area in front of the building that houses the landlords' place and the kids' place. The black one is the friendliest.

I know there is a lot the same about the plant life here and at home, but I was surprised to see this thistle.


Puzzles. Gotta do puzzles.

Our flight from SF to London was on and hot and cramped but we survived. Then the flight from London to Prague was great! It was short and cool and roomy. I could revisit a fun puzzle, and I even ate some clotted cream!


Chillin' in the pool with a puzzle and the kiddos.