# Mixed-breeds are puzzles too!

This is a three-dimensional mixed-breed puzzle. The five squares depict the five layers of a $$5\times5\times5$$ cube. The goal is to solve the odd-numbered layers as yajilin and even-numbered layers as masyu.

Short rules:

• Shade some cells on odd-numbered layers. The numbered cells show how many shaded cells are in the direction of the arrow.
• Shaded cells cannot be adjacent to another shaded cell, but they can touch the numbered cells.
• Unshaded cells on each yajilin layer are all connected in 2D (i.e. each layer is treated separately).
• Make a single loop in 3D space which goes through every unshaded cell on odd-numbered layers, and every circle on even-numbered layers.
• A line passing through a white circle must continue straight through its cell and make a turn in the directly following or preceding cell (or both).
• A line passing through a black circle must make a turn inside that cell and continue straight for 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.

• Just a 3d Yajalin left to do :) (I was thinking about the arrows, you could use NESWUD)
– JMP
Aug 7, 2019 at 10:23
• @JonMarkPerry Maybe I'll try that next!
– Jafe
Aug 7, 2019 at 10:31
• Or even a 4d Masyu!!!
– JMP
Aug 7, 2019 at 10:46

A step-by-step solution follows. Click on pictures for a full-size version. Layer #1 is the "top", so moving towards it is going "Z-up".

Step 1:

Layer #1. The 0 clues allow many cells to be designated unshaded (I use blue for this purpose). Then, there is only one way to resolve the cluster of 1 clues and retain connectivity of unshaded cells. Finally, the dead ends must go Z-down.

Step 2:

Layer #2. The shading of several of the cells in layer #3's R1-2 can be determined by connectivity logic. Now layer #2 can use information from #1 and #3. R1C3's line cannot go Z-up, so must go left and right. R2C5 cannot go Z-down, so it goes left and Y-down. The dangling end in R1C1 must go Y-down.

Step 3:

Black circles. In layer #4, R2C4 cannot go Z-up, so it goes left and Y-down. R4C3 and then R5C3 can only go Z-up and left; in R5C3's case its line hits into a white circle and keeps going. Now layer #2's R5C4 can only go Z-down and Y-up.

Step 4:

Layer #3. R4-5 have their shading forced. Then R3C5 and R2C4 cannot go Z so they go left and Y-down. This forces the last shaded square in R2's 2 clue, which forces R1C2 to go right and Z-down. Finally, R4C2 must go Z-up and Y-up.

Step 5:

Maysu cleanup. Layer #2's R3C2 must go right and Y-up. The dangling end in R2C1 must go Y-down. Layer #4's R1C4, R3C5, and R1C5 cannot go Z, forcing connections. The dangling end in R5C1 cannot go Z and so goes Y-up

Step 6:

Layer #5. R5C5 is shaded because if not, it would be forced to go Z-up and violate the white circle in R4C5. Connectivity unshading above. The new dangling end must go Z-up, which then forces the dangling end in Layer #3's R4C5 to go Z-up.

Step 7:

Layer #1. Crosses are where Z movement is impossible. R5C5 must be shaded because any line put there would dead-end. Then several things are forced: the R4-5C3-4 curve, going straight through R2C4, and having R5C2 go Z-down and Y-up.

Step 8:

Layer #2. R4C3 goes left because Z-up would form a loop. R5C1-2 can't go Z, so they are forced around the corner. Additionally, a few more Xs for layer #1.

Step 9:

Layer #5. R3C1 cannot be shaded as then the C5 2 clue would isolate an unshaded cell. IT can't go Z, so it is forced to go right and Y-up. Connectivity & clues determine shading. Now the loop in this layer can be determined, with helper Xs to guide the forced lines.

Step 10:

Layers #4 & #3. R3C1 cannot go Y-up and so goes Z-up and right. The dangling end in R2C1 cannot go Y-up, so it goes right. The dangling end in R1C3 is forced Z-up. As a bonus, layer #3's loop parts can be finished; note the two helper Xs added in as guides.

Step 11:

Finish. First, both dangling ends in layer #2 are forced to go to layer #1. Then, with some more helper Xs, loop connectivity forces layer #1's lines, and its last shaded cell.

Solution (with helper Xs removed):

Observations

The Masyu layers behave like they do in the previous 3D variants.
The usual Yajilin rules of dead ends only apply when the layer above and below is blocked as well. The 2D-connectivity allows easy deductions.

Finished grid