Knowledge and awareness are vague, and perhaps better called illusions.
Everyone lives within their own subjective interpretation.
-- Uchiha Itachi

The full scramble

Pressing Ctrl+f, we have this scramble state,

My goal is to solve it without knowing exactly what to do. In other words, I will try to develope the algorithms we need along the way. Hopefully this can be done in no more than a week.

Wait... we need more notations

I planned to start our solve, but I notice that I have to introduce necessary notations at this stage. Without these notations, it will be difficult to specify any of the part of the puzzle, not to mention how I would want to do with them.

cell

You may have notice, I will keep using cell instead of "face" like other hypercubers often do. I originally plan to use modifiers like the top or the right-front cell to indicate cells at non-center positions, but Why don't we just go further? The modifiers are annoying.

• the right-front cell: X+
• the left-back cell: X-
• the right-back cell: Y+
• the left-front cell: Y-
• the top cell: Z+
• the bottom cell: Z-
• the invisible cell: W+
• the center cell: W-

Also, when I mention a cell, I mean 8 stickers clustered as a cube. A sticker is a 3D color block.

I prefer a absolute coordinate system rather than a relative system like R/L/B/F/T/D in 3D space. So this set of notation is my own preference and can be largely different from all other hyper/cubers.

cubie

It is worth noting that, geometrical speaking, a 2x2x2x2 hypercube can be divided into 16 unit hypercubes, and a 2x2x2 cube can be divided into 8 unit cubes. Cubie is a physical term to indicate each unit cube (and the mechanical structures with it) in 3D puzzles, and I would like to keep this word for the unit hypercubes.

Note that, a hypercube has 8 cells for each of its boundary, just like a cube has 6 faces. Among the 8 cells of each cubie in a 2x2x2x2 puzzle, 4 of them are visible in MC4D as stickers, and the other 4 are shared with other cubies. If a 2x2x2x2 puzzle is physically done in some 4D universe, the space of the later 4 cells should serve as the inner mechanical structure.

Unlike a physical Rubik's Cube, you can see a corner cubie's three faces directly. Now in MC4D, we don't directly see a cubie. Instead, we see a cubie's four cells on different places as if they were loosely connected or had no relationship, but that's the illusion or a visual trick. A cubie is a thing, and the four stickers are attached to it. No matter how a cubie moves due to rotations and twists, a sticker on one cubie won't move to another cubie.

I expect, sometimes we will need to take some time reading the state of the puzzle, and find out information like, "where is the yellow-gray-green-blue cubie?" or "where is the last cubie that has orange and red?"

colors in a cubie

I tend to mention colors of a cubie in X-Y-Z-W order. For the above example, what I really mean is, "There is a cubie, whose sticker in X- or X+ is yellow, Y- or Y+ is green, ..." You know what I mean.

Locating a cubie in a cell: the octant system

The octant notation is a generalization of 2D quadrant system. I follow the gray code notation from the left form in the Wikipedia page. Following the notation I introduce in the cell section, I have been and will always assume the left-back to right-front direction as X axis, the left-front to right-back as Y axis, and bottom to top as Z axis, from negative to positive. Therefore, the 8 octants of the center cell will be:

• Octant 0 (O0) is yellow, belonging to the gray-orange-pink-yellow cubie.
• Octant 1 (O1) is red, belonging to the orange-blue-turquoise-red cubie.
• Octant 2 (O2) is blue, belonging to the green-pink-gray-blue cubie.
• Octant 3 (O3) is yellow, belonging to the green-gray-turquoise-yellow cubie.
• Octant 4 (O4) is gray (the one below Octant 3), belonging to the orange-blue-turquoise-gray cubie.
• Octant 5 (O5) is yellow (the one below Octant 2), belonging to the red-pink-green-yellow cubie.
• Octant 6 (O6) is turquoise (the one below Octant 1), belonging to the orange-yellow-gray-turquoise cubie.
• Octant 7 (O7) is green (the one below Octant 0), belonging to the yellow-gray-pink-green cubie.

I will say something like, "I want to move away the W-O0 sticker" or "to get that cubie back to X+O7, we have to ...".

Conclusion

I introduce a bit more notations, not only for the readers to follow my terms in the rest of PART II, but also to share my mental model on these interesting yet abstract objects.

小結

多介紹了一些用語，其中最重要的是導入卦限表示法。