Message #3569

Subject: Re: MagicTile few colors
Date: Fri, 18 Nov 2016 17:49:25 -0800

I also find 3-cycles on complex puzzles, although I am generally not very strict about what I try to find. The cycles do not need to be pure. I also don’t try to find ways to orient puzzles, since I can usually either solve those with appropriate setup moves or by just doing two 3-cycles. Sometimes I find some unusual parity or orientation issues which I will figure out when they come up. Also, depending on the puzzle, I might solve a lot of pieces by intuitively solving them into place, or I might reduce the puzzle into something simpler (which I did with the cube/octahedron Super Chops, where my big first step was to build groups of pieces so I didn’t need to use the edge twists anymore).

Here’s an example of the nonpure cycle thing. Let’s look at Dodecahedron E0:1:0.11, where we have corners, corner centers, edge centers, tiny centers, and middle centers. I’d solve the corners first, intuitively, since a move is a 2-cycle. Then you can make an 8-move 3-cycle of corner centers that affects everything except corners and middle centers, but only corners are solved now, so it’s OK. Then there’s a 4-move 3-cycle of edge centers that also affects tiny centers and middle centers, but we haven’t solved those yet so that’s OK too. Now I have a 3-cycle of middle centers that affects tiny centers (again OK) and then the only pure 3-cycle algorithm, of tiny centers - that one is 14 moves.

For the puzzle you mentioned, {3,4} 4-Color Orbifold B F0:0:0.8 V0.8:0:0 on the program, I do puzzles of that type by doing the middle edges (very quick) and then solving each vertex along with the surrounding 8 pieces. There are only two distinct vertices here. You can think of each vertex’s group as a "center" (corner) plus four "edges" (edge2 pieces) and four "corners" (centers). So then it’s somewhat like 5x5x5 supercube centers. I built most of it intuitively, for instance making "blocks" of two solved "edges" with a solved "corner" in between. By about 32 moves it looks like I had everything solved except two "corners", although we can consider it a 3-cycle because we have some identically colored "corners". Then I solved that 3-cycle with an algorithm that roughly corresponds to the 3x3x3 algorithm: [R2 U’ R2 U’ R2 U2 R2, D’]. Align the vertices and it’s done.