# Message #102

From: Matt Young <indigowombat@indigowombat.com>

Subject: My 3-color and 4-color sequences

Date: Mon, 30 Aug 2004 14:02:15 -0700

Still working on putting my thoughts into order re: avoiding situations with single pieces that have their orientations flipped, but I’ve at least had a chance to find an expression in Roice’s notation of my 16-twist sequence for cycling through three of the 4-color pieces. I’ve also found an expression of my own 8-twist sequence for cycling through three 3-color pieces, which differs from Roice’s published sequences.

I arrived at these sequences after studying the same 3^3 solution sequences by Phillip Marshall that Roice adapted his own sequences from. I took a slightly different tack in developing my own sequences, though. The key to their development was in treating one of the 3-D "faces" of the 3^4 puzzle exactly as though it were a standard 3^3 cube, and performing Marshall’s sequences on that "face" to cycle through its edges or corners. This works because of the highly symmetrical structure of Marshall’s sequences. I perform alternating twists of two 3-D "faces" in my sequences: one 3-D "face" that I treat as the standard 3-D puzzle, and one neighboring 3-D "face" that I use to perform twists on the 2-D faces of that 3-D puzzle. I’ll call these two 3-D faces "Cube" and "Neighbor" for the moment. I start by identifying which face of "Cube" I want to twist first in order to perform a particular Marshall sequence on "Cube." I set "Neighbor" as the 3-D "face" of the 3^4 puzzle that shares that 2-D face with "Cube." After beginning the Marshall sequence on "Cube" by twisting "Neighbor" in the appropriate direction to achieve that effect, I then twist "Cube" so that the next face of it that I wish to twist in that Marshall sequence is presented to "Neighbor." I continue through the Marshall sequence in this manner, each time using "Neighbor" to twist the appropriate face of "Cube", then perform a final twist to return "Cube" to its starting orientation in the sequence. Because of the symmetry of the Marshall sequences, "Neighbor" performs an equal number of clockwise and counterclockwise twists on "Cube" throughout this procedure, which allows this technique to work without disturbing the rest of the 4-D puzzle.

Okay, enough background explanations. Here are my 3-color and 4-color sequences in Roice’s notation. In both sequences, the face I designate as "Cube" is in the position Roice calls "Top". In my 3-color sequence, "Neighbor" is "Front"; in the 4-color sequence, "Neighbor" is "Upper". This is to preserve the sense of the Marshall sequences as presented on his site.

My 3-color sequence:

Front 5 Right

Top 11 Right

Front 5 Left

Top 11 Left

Front 5 Left

Top 11 Right

Front 5 Right

Top 11 Left

My 4-color sequence:

Upper 11 Right

Top 5 Left

Upper 11 Right

Top 5 Right

Upper 11 Left

Top 5 Right

Upper 11 Left

Top 5 Left

Upper 11 Right

Top 5 Left

Upper 11 Left

Top 5 Right

Upper 11 Left

Top 5 Right

Upper 11 Right

Top 5 Left

These can be reversed (right twists becoming left twists and vice-versa) to perform mirror images of the Marshall sequences, just as in Marshall’s original technique. You can treat any face you want as "Cube" and "Neighbor", of course; notating it as above was simply for convenience’s sake.

More hopefully to follow, on the tactical use of these sequences to solve the 3^4 puzzle without creating situations where a single puzzle piece is in the correct position but incorrect orientation. Time to rest the brain for a while.

–Matt Young