# Brain-breaking 3D masyu

This is a three-dimensional masyu puzzle.1 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 for at least one cell on both sides.

Note: Feel free to use any notation you feel most comfortable with in your answer, as long as it's properly explained. As an example, I like to mark movement between levels with diagonal lines as shown below.

1 The concept of 3D masyu was previously introduced in 3D Masyu - A Die.
2 Paraphrased from the original rules on Nikoli.

• are you sure all steps can be deduced through pure logic and no guesswork is needed? thanks and +1! Commented Jul 25, 2019 at 14:26

## 1 Answer

Differences in 3D

White circles: Good to start with, only 3 possible ways for the line to go through (along all 3 axes), if two of those are blocked on one side, then the line has to go along the missing axis.

Black circles: To get anything at all, we need both sides along one axis blocked (reducing to 2D case) and then one side on another axis to deduce at least one of the legs.

Starting with the white circles and slowly expanding to all other circles gives

Now we have to check for possible ways to connect the segments

A2 can not connect to D2 because F2 would be a dead end, so D2 has to connect to F2. This leaves the following possible connections:
A1-B1-C1-E1-F1-E2-A2-D1-C2-B2-A1

Which leads to the solutions

Red using the connections A1-B1-B2-C2-C1-E1-E2-F1-F2-D2-D1-A2-A1
Blue using the connections A1-B2-B1-C1-C2-D1-D2-F2-F1-E1-E2-A2-A1

• Looks like my answer agrees with your blue version, but I didn't get a second answer... I'm not sure why. Commented Jul 25, 2019 at 16:34
• This seems to be right, I can't find anything wrong with either version (I probably assumed something wrong at some point). Good Work! Commented Jul 25, 2019 at 17:02
• Yeah, looks correct. My solution was the same as Dark Thunder's, and I hadn't realized there was a second possible solution either.
– Jafe
Commented Jul 26, 2019 at 6:19