# Masyu keep making those 3D puzzles?

This is a three-dimensional Masyu puzzle. The five squares depict the layers of a $$5\times5\times5$$ 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 through at least one cell on both sides.

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.

• May I suggest "through" rather than "for" in the last bullet point? I think that makes it harder to misinterpret. Oct 14, 2019 at 13:02
• @GarethMcCaughan Edited. Thanks!
– Jafe
Oct 14, 2019 at 13:03
• A ray-traced solution would be awesome. Oct 14, 2019 at 13:09
• I've experimented a little with ray-tracing and it's less good than you'd think. Unless you also have the ability to spin things around in 3D etc., it's hard to see what's going on. Oct 14, 2019 at 15:23

Solved! I use X, Y, and Z as axes to describe my logic.

White circles along edges have only one possibility:

Then the line through white circles in the bottom right of #2 must be in Z; if they were in Y, the black circle in (5,4) of #1 would be X and Y, leaving the white circle in (4,5) of #1 with nowhere to go. This forces a turn in (5,5) of #4, which means that (5,5) of #5 does not turn Z. A few other simple deductions later:

The black circle in (2,1) of #1 can't be Y and Z, as then the white circle in (2,2) of #2 would have nowhere to go. It also can't be X and Y, as then there would be a Y line through (2,2) of #2, which would have nowhere to go from (2,1) of #2. Thus it's X and Z.

The black circle in (1,3) of #3 only has two legal directions left. This forces (1,1) of #5. We then note that the white circle in (2,3) of #2 must be in X. A few more simple deductions:

A line in Z from the black circle in (2,4) of #5 would end up stranded, so that black dot is in X and Y. The line must turn in (1,5) of #2 - it's gone through two white circles. A colour-coded image at this point, showing the paths so we don't get separate loops:

The most obvious deduction at this point is that (5,4) of #3 can't go down. That swiftly brings us to our conclusion!

• Looks correct. Nice job!
– Jafe
Oct 14, 2019 at 15:24