4D 3x3

This cube is Physically Impossible, Meaning you have to defy physics to solve it.

## World Records

### Normal

15:57 by Mateusz Burnicki (using prepared macros)

The 3x3x3x3 (or the

3
4

{\displaystyle 3^4}

) is a Physically Impossible Cube only solvable Via computer.

## Info

The 34 has always been the main focus of the community. The first solve was done by Don Hatch in ~1988 using MC4D. After that, the Hall of Fame gradually began to increase in size. It eventually closed on December 7th, 2022 due to it reaching 500 solvers.

## Notation

### Cells

Each side of an n4 hypercube is called a *cell*, and consists of a cubic grid of n3stickers. Because each side is cubic, they can be rotated with cubic symmetry. A cube has 24 different possible orientations, meaning that each side of a 4D puzzle can be oriented in 24 different ways.

There are 8 cells, six of them using the same letters as that in the 33: `U` (up), `D`(down), `F` (front), `B` (back), `R` (right), `L` (left). The one that you can see on the inside is called `I` (inside), and the one opposite of that (which is hidden) is called `O` (outside). This means we use the letters `R` `L` `U` `D` `F` `B` `O` `I`.

### Pieces

Cell-turning hypercubes have 4 different types of pieces (with some slight variations for big hypercubes). These are the 1c (1-colored) centers, 2c ridges, 3c edges, and 4c corners. You can label each piece based on which cells it is on. The 3c piece that has stickers on the `I`nside, `R`ight, and `U`p cells would be called the `IUR` piece.

### Twists

Twists are notated based on which piece you click on to do the turn, with the first letter determining which sticker of that piece to click on. For example:

* `RO'` means to left click the R cell sticker of the RO piece.
* `IUR` means to left click the I sticker of the IUR piece (edge twists are symmetrical, so it doesn't matter if you left or right click).
* `FRUI` means to right click the F sticker of the FRUI piece.

For the slice layers, we use `M` `E` `S` `P`, where P is the slice parallel to I/O (although with notation for n5 puzzles beginning to be fleshed out, P may end up being used by the new 5D sides of Posterior/Anterior). Wide moves are tricky, as we are now using the letter `w` for the 4th dimension axis. One way to get around this is to write the numbers you hold down on the keyboard (in MC4D/MPU etc) while clicking on a piece. On a 3-layered puzzle, holding down 2 does a slice move, and holding down 1 and 2 will do a wide move.

* `{2}RO` means to hold `2` while doing an RO move (essentially an M move).
* `{2-3}UO'` means to hold `2` and `3` while doing a UO' move.
* `{2-4}IF` means to hold `2`, `3`, and `4`

### Rotations

3D notation for rotations doesn't really generalize to higher dimensions. For example, we call an `x` rotation x because it rotates the puzzle "around the x-axis". However, this is unhelpful because rotations don't actually happen around an axis. Instead it's better to think of rotations as happening within a 2D plane. When you're doing an `x`, the whole puzzle is really being rotated within the zy plane. And writing rotations like this generalizes to higher dimensions, so that is what we use.

First, we make a certain side from each axis be the "positive" side. There are standards for this in 3D (such as the right-handed rule). On the x-axis, going to the right is positive and going left is negative. This makes `R` the positive side from the x-axis. The same goes for the other axes: `U`, and `F` are the positive sides in 3D. In 4D we add two new sides, and have to decide which one is positive and which one is negative. Because of the projection, the side that we can't see is closer to the 4D camera, making `O` the positive w-axis cell, and `I` the negative.

To actually notate the rotations, write the letters of the 2 axes that form the plane that the puzzle is rotating in, in the order of which positive side on that axis goes to the positive side of the other axis. For example: rotating the positive y-axis side to the positive x-axis side (rotating U to R) would be written `yx` (and looks like `z` in 3D notation).

* `yw`: bring +y to +w (rotate U to O)
* `xz`: bring +x to +z (rotate R to F (this is called `y` in 3D rotation notation))

Note how we don't have to use the `'` prime symbol because you can just swap the letters (wy is the inverse of yw). You can also add a `2` to the end for double rotations, e.g. `wx2`, `yz2` etc.

This makes it a really nice system because it only ever uses 2 letters, no matter how many dimensions. You can also use it on n3 puzzles in order for more multi-dimensional consistency: `xy` means move x+ to y+ (R to U [z' rotation]).

### Turn metrics

There are many different ways to count the number of twists performed during a solve. All of these metrics (besides ETM) have their own Quarter Turn counterparts, where twists are broken up into 90° twists. For example, a 180° face twist takes two 90° twists, while a 120° edge twist takes three. Listed below are the metrics used by HSC. STM is used the most, but others are sometimes relevant.

## Methods:

* CFOP
* 3-Block
* Octachoroux (Roux)

## Trivia

* This is the first ever 4D Puzzle to be made.