Diagonal pattern for 4x4x4 Rubik Cube with all colors on each side?

Question

I love creating patterns on my 4x4x4 cube, so I came up with the idea to create the following pattern:

• All sides have a diagonal pattern, meaning something like:

1234
2345
3456
4561

• All sides contain all colors

I was able to create such a pattern with a lot of 'heavy thinking' by first creating a draft on a sheet of paper - hoping the pattern I chose was doable - and then solving the cube step by step (2x2 centers first, then edges, then apply 3x3x3 solving):

Note 1: On my first attempt I drafted an unsolvable pattern, but I was able to resolve this by swapping the 4 corners that are not part of the main diagonals.

Note 2: Since each face has 7 diagonal lines, obviously 1 color has to appear in 2 lines. I soon found out that it is not possible to create the diagonals in a way that the opposite corners which are not part of the main diagonal line always have the same color. So I decided to always reuse the main diagonal line on one of the corners... (e.g. View 1, Top Face, Yellow main diagonal, with 1 white and 1 yellow other corner)

My question now is:

Is there an easier way to create this pattern (or a similar one that fulfills the requirements) with an algorithm? Or the other way around: Is there an easy way to solve this pattern back to normal? And finally: Is my note above (with the 2 corners) correct?

Answer 1

Is there an easier way to create this pattern (or a similar one that fulfills the requirements) with an algorithm? Or the other way around: Is there an easy way to solve this pattern back to normal?

I've put your current configuration in this 4x4x4 solver, which gave the following 49-move algorithm (with the white diagonal at the front and orange diagonal at the top - your second picture):

Bw' D' Rw' b' Uw' U' D2 L R2 Rw F' B Lw' R2 F' R2 B' R2 Uw2 Lw2 F' Lw2 Bw2 U D2 R2 U Lw2 U' F L2 B2 U R' B' D' R F2 L F2 L2 D' R2 D' B2 D L2 D2 F2

Or with pictures for anyone unfamiliar with the move notations:

Reversing this algorithm with a solved Rubik's Reverse 4x4x4 Cube will result in your pattern:

F2 D2 L2 D' B2 D R2 D L2 F2 L' F2 R' D B R U' B2 L2 F' U Lw2 U' R2 D2 U' Bw2 Lw2 F Lw2 Uw2 R2 B R2 F R2 Lw B' F Rw' R2 L' D2 U Uw b Rw D Bw

Unfortunately it's not the most intuitive algorithm, but it's faster than trying to solve it into that pattern manually.

---

And finally: Is my note above (with the 2 corners) correct?

It's indeed not possible to have diagonals of corners on each face. Ignoring all the other pieces and only looking at the corners, we basically have a 2x2x2 Cube.

1) Let's say we use the pretty popular checkerboard algorithm:

U R F2 U R F2 R U F' R

We do end up with diagonals on three sides, but have a single incorrect corner at the other sides:

As you may know, it's not possible to orient a single corner on an otherwise solved 2x2x2 cube. The same applies to the single rotated corner of the pattern above.

2) Using different diagonals also won't work. If we for example try to put two opposite colors as diagonals on a single face, e.g. orange and red, we have no way to arrange the other pieces:

2a) In the picture below we've used all four blue stickers, but the orange-blue-white corner at the bottom-front-right position has its blue sticker at the correct diagonal on the right side, but its white sticker NOT on the correct diagonal, since it's on the bottom face instead of front:

2b) Replacing this red-white-blue corner with red-green-white will put two white stickers next to each other on the front face; and replacing this red-white-blue corner with red-blue-yellow will put two blue stickers next to each other on the right face.

2c) Replacing it with the only other corner left with a red sticker (red-yellow-green) might look promising at first:

But it's unfortunately also a dead end, because: to prevent what occurred for 2b, we can only put the red-white-blue corner we've removed at the top-back-left position. But doing so, we won't have any valid corner to put at the front-right-bottom position (marked dark gray above).

So whether we try to create diagonals using adjacent or opposite colors on one of the faces, we can't finish the corner-diagonals on all faces.