Hi, tonight I am working on actually solving a 4D Rubik's cube. Idk how long to expect this to take.

I was inspired to do this tonight because the concept of solving these recently got a popularity boost and because speedrunning sucks.

The software for this is "Magic Cube 4D" and can be found here http://superliminal.com/cube/cube.htm
On the website you can also find the "Hall of Fame" of people who have completed it. Currently there are 234 since 1988.

So what is the theory behind solving this?

The program functions exactly how a real 3x3x3x3 hypercube would function. The image you're seeing, of course, is a 2D projection of a 3D projection of a 4D object, and so it can be quite confusing to understand what you're looking at. Let me explain the pieces on this hypercube that are analogous to the pieces on a 3D cube.

-First of all, this is probably the biggest thing to realize: The small individual cubes floating around are the "stickers". They are not necessarily entire pieces on their own! On a 3D cube, there are center pieces which have 1 sticker, edge pieces which have 2 stickers, and corner pieces which have 3 stickers. On the 4D version, the 1 sticker pieces are the ones in the center of the larger units of cubes (referred to as cubical cells). These are the "centers". The 2 sticker pieces, "face pieces", are the ones in a center of a face of a cubical cell. The 3 sticker pieces, "edge pieces", are the edges of cubical cells. Finally, there are now 4 sticker pieces, "corner pieces", which are the corners of cubical cells.

-The basic theory behind solving twisty puzzles is conjugates and commutators. A conjugate is a series of moves in the form A B A', where A' is the inverse of A. How is this useful? Let's say you know a sequence of moves that rotates two adjacent corners, and you have two corners which need to be rotated, but they aren't adjacent. If you let A be a sequence of moves that makes the corners adjacent, then let B be the algorithm that rotates them, A B A' will result in the two corners being rotated.
A commutator is in the form A B A' B'. Let's say you've found a way to correct a piece on the puzzle, but in the process, you've messed up other things. This would be A. Now, you would move the corrected piece to another location (often with just one move) in a way that does not affect what was messed up by A. This would be B. When you do A', everything that was messed up will be fixed again, and one piece will be affected in the opposite way that the original target piece was. B' completes the process.

-Thankfully, the software has a great system of using macros to make this process bearable. Once a set of moves has been set as a macro, it can be done over and over again with a single click. It can also be done in reverse, and you can nest macros in other macros! This is all very useful in setting up and applying conjugates and commutators. Macro files of all the algorithms you need to solve this puzzle are available online but I will be making them on my own.

That's all I can think of to say. If you have questions ask in chat.