# 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. – Gareth McCaughan Oct 14 '19 at 13:02
• @GarethMcCaughan Edited. Thanks! – jafe Oct 14 '19 at 13:03
• A ray-traced solution would be awesome. – Daniel Mathias Oct 14 '19 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. – Gareth McCaughan Oct 14 '19 at 15:23

## 1 Answer

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 '19 at 15:24