Wednesday, June 17, 2015

Square-1—How I Got Started

I got a shipment of twisty puzzles from Cubefans.com (Cube4You) on Monday, August 4, 2008. I’m not sure if I even touched the Square-1 on Monday, or if I waited until Tuesday. I think I started with the Megaminx. Sometime Monday or Tuesday I checked out the Square-1. Before putting on the stickers I scrambled it. It was quite a challenge to get it back into cube shape. I wasn’t sure how I did it, but it involved getting symmetrical shapes on each side. This was probably Tuesday.
I guess I put the stickers on it on Wednesday. After putting stickers on I wasn’t going to scramble it right away. I was going to do just a few moves at a time and then undo them to see what I could learn. That didn’t last more than a couple minutes. Before long it was thoroughly scrambled. No amount of fiddling and twiddling could even get it back into cube shape. I worked on it off and on for a couple days.

Finally on Friday I got out a pad of paper and started systematically recording starting positions, and results. After writing down 65 of them, I wanted to quit being so systematic and start following different trails as I came to them, and pretty soon I had 6 corners on one side which made a 6-pointed star, and on the other side an eye-shape, which was 1 corner, 3 edges, 1 corner, 3 edges. I had a feeling this had some potential. It was late though, so I went to bed.

Saturday morning it all quickly fell into place. The 6 + 1313 became 44 + 44 became 2 tops + 2 bottoms became 2 halves + 2 halves became a cube. Now all I had to do was figure out how to get the colors solved!

First I got the edges. I don’t know exactly how I did it and I wonder if it will always be that easy. Then I was able to get 3 of the 8 corners fairly easily. Somewhere along the line I realized that I was moving corners around without messing up the edges, so I looked up a notation for doing Square-1 moves and used it to write down some moves that move corners around without messing up edges. I also wrote down where the corners were both before and after, so I would be able to do something with it. What it does it swap the 2 front corners on top and the 2 corners on the bottom right. I used my corner swapping algorithm with setup moves to swap a blue on the bottom with a green on the top, so that I had 3 of the 4 blue corners on top, and, of course, 3 of the 4 green corners on the bottom. And they were all 6 in the correct places. So I just had to swap a top corner with a bottom corner to have the cube solved. I got all the blue corners on top and all the greens on bottom, and I had the bottom solved, but 2 blue corners needed to swap. I hadn’t had a situation in which I needed to do a 3-cycle, but was sure that I could use the double-swap algorithm twice with setup moves to do a 3-cycle. So to swap a pair of corners I could turn the top 90˚ which would make exactly 1 corner right and then I could use my algorithm to 3-cycle the other corners into place. The only problem would be, my 4 blue edges would all be out of place. And I didn’t know how to move edges without messing up corners.

After much experimenting, that resulted in having to practically solve it over again several times, I ended up with all the edges solved, but 2 green corners swapped. Somewhere along the line I had all the corners solved, but 2 edges swapped. I figured out how to swap edges without messing up corners, but I couldn’t use it to fix the 2 swapped corners after all. I could just use it to switch from 2 swapped corners to 2 swapped edges.

With most of the cube solved it was relatively easy to write down moves and their results and to come up with ways to swap pairs of corners, and to swap pairs of edges. This gives me a way to 3-cycle corners, and to 3-cycle edges. In the midst of all the experimenting there were times when I would mess up and have to use the moves learned so far to get back to one pair of swapped something. Sometimes it was quick and easy, sometimes it was difficult. Try as I might I couldn’t find a way to swap 2 edges, the very thing I needed to do. I figured I was facing some kind of parity issue. I still don’t understand exactly what parity is, but I figured to fix it I would have to back out of the cube configuration and come back to it. Finally I went to Jaaps page and read just enough to get me through. Here is the link to the page that I look forward to studying in depth some day, after I am convinced I can solve the Square-1 on my own.

http://www.geocities.com/jaapsch/puzzles/square1.htm#oddperm

And here is his paragraph that helped me figure out what I had to do.
“The quickest way to perform an odd permutation is to go to the scallop-scallop shape (3 twists), swap three corners from one layer with three from the other (1 twist), and return to the cube shape (3 twists) which takes 7 twists in total.”

I did take a quick look through his diagrams to see what he meant by “scallop.” It is what I called the 44 in my analysis when I was trying to get it into a cube shape at the beginning.
All the above in this post was written in 2008 when I first got the Square-1. As it turns out I don't use any of the Corner Swappers and only 1 of the Edge Swappers routinely now.