# Volcanic 3D masyu

This is a three-dimensional masyu puzzle. The six squares depict the layers of a $$6\times6\times6$$ cube. The goal is to make a single loop in 3D space which fulfills the following properties:

• The line passes through centres of cells and makes 90-degree turns only.
• The line cannot cross itself or branch off in multiple directions.
• The line passes through every white and black circle.
• When passing through a white circle, the line must go straight through the circled cell and make a turn directly before or after that cell (or both).
• When passing through a black circle, the line must make a turn in the circled cell and continue straight for at least one cell on both sides.
• Red squares are lava. No line passes through lava.

Note: Feel free to use any notation you feel the most comfortable with, as long as it's properly explained. My way is to use diagonal lines to show movement between layers.

The solution is

Explanation:

We can go a long way looking at just the white circles. The ones in a corner must go in/out, and the ones near an edge must go parallel to that edge if they cannot go in/out (e.g. if they are on layer #1 or #6).
Once we have those filled out we can use what we've filled out to deduce even more white circles, including some places where the line must turn:

At this point we'll start looking at the black circles. Since these require at least 1 straight piece in each of its two directions, we can deduce a lot of them especially on layer #2 and #5. Some of them we know exactly (when they only have 2 valid directions) or we might know one of their directions (when they have 3 valid directions, 2 of which are on an axis, meaning we are certain of the third):

As we keep working on these we find a few places where the paths that a line could take are blocked off by the lines above/below it. Namely we can solve (almost) all of layer #1:

At this point it's mostly a matter of working our way down layer by layer, solving as many circles as we can. One notably different piece of logic is that we can deduce the top-left corner of layer #2, since it cannot go in and is entirely enclosed on layers #2 and #1 - this means that it must go out onto layer #1 and immediately back into layer #2:

Nothing much of interest happens until we've finished solving the cube. It's simply a matter of repeatedly finding a circle that we can deduce the solution for, and then using the cells that are solved by this to further deduce new circles, or deduce the path of open lines.
Eventually all layers are solved and we have the completed cube. A full list of steps can be found at this album

• This is correct. Nice work!
– Jafe
Aug 5, 2019 at 12:42
• @jafe This was fun to solve :) I have no idea how you make these puzzles so they're both uniquely solveable and require multiple kind of deductions, but you somehow do. Great puzzle! Aug 5, 2019 at 12:54
• Glad you liked it! Often the difficult part is trying to not blow a fuse when I'm almost done, realize I left a contradiction in step 1 and have to start over.
– Jafe
Aug 5, 2019 at 13:25