Another four 3D masyu

This puzzle is inspired and descended from Brain-breaking 3D masyu
Here the exact copy of rules, for convenience:

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).
Note: in this particular puzzle those not present, this rule here just to emphasize compatibility with Brain-breaking 3D masyu and general masyu rules
• 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.

Here are four additional grids. Each is an independent puzzle. Provide a solution, or show there is no solution.

After solving this, you can try

Dramatic follow-up 3D masyu problem

• This may sound dumb, but is it intended that no white circles are present? Commented Apr 19 at 17:36
• @Fluorine yes, I'll add a hint in rules, thank you! Commented Apr 19 at 18:35
• Re: "Note: in this particular puzzle those [white circles are] not present". Does that mean that all non-blob cells are 'white circles'? Else, if they are 'not present' why are they mentioned? What is unclear, is whether the lack of white circles voids the 'must turn' rule or voids the 'must pass through every' rule too. Commented Apr 19 at 22:52
• I managed rot13(gb svaq n fbyhgvba sbe gur svefg gjb chmmyrf, vf gur nafjre vagraqrq gb or bar nqqerffvat nyy 4 ceboyrzf ng bapr)? If not I'll be happy to post an answer for those I solved. Commented Apr 19 at 23:37
• @Fluorine, As you wish, I don' mind if you or someone else post partial answer, using spoilers with >! . But, A and B supposed to be warming up tasks before for C, D, rot13(naq cebonoyl, pregnva sbyybj-hc ceboyrz) Commented Apr 20 at 10:42

The 2nd and 4th layers are omitted for simplicity. The edges that connect different layers are not explicitly drawn, except the ones that connect the top and bottom layers which are indicated with a green X in the middle layer.

Variant A

A solution

exists:

If you rotate the entire 3D grid 90 degrees, you can put two faces with five dots on the top and bottom layers, and just two dots on the center layer. I fiddled with a partial route that goes through all five dots without escaping the layer, and got a nice simple solution above.

Variant B

A solution

exists:

I tried various partial paths connecting the four dots on one layer. Connecting all four dots with one path didn't seem to help, so I tried using two paths, and got the one on the first layer in the above solution. Then I spotted a possible symmetry with the cross-layer edges. Putting a 180-degree turn on the opposing layer gave two loops, and putting a reflection instead worked.

Variant C

A solution

does not exist.

There are 14 black dots and 13 small diamonds. Each black dot adds 2 to the number of edges connected to small diamonds, which makes 28 in total. By pigeonhole principle, some diamond has at least 3 edges going into it, which violates the rule of Masyu.

Variant D

A solution

does not exist.

The logic for this one is more involved and probably not perfect, but I'll try anyway.

The center dot has to connect to two of the diamonds that are not opposite to each other. Let's rotate the cube so both edges are oriented as in the image above.

There are 13 black dots and 14 diamonds. By the same logic ans Variant C, 13 or 14 diamonds must be used. This means that all 5 diamonds are used on the top or bottom layer, and at least 4 are used on the opposite one.

If the top layer uses 5 diamonds, at least 6 edges must come out of the layer. If it uses 4 diamonds, at least 4 must come out. It can be the case that all 14 diamonds are used but two of them have only one edge coming from black dots; a layer containing one of them gives at least 5 edges coming out of the layer. In any case, at least 10 edges in total are coming into the middle layer.

On the other hand, there can be at most one connection that connects the top and bottom layers, since such connections cannot go through dots, the diamonds on the east and south are unavailable, and one of the north and west diamonds must connect to the northwest dot. The center dot cannot accept any cross-layer edges, so the center layer can only accept nine such edges in total, leading to contradiction.

• JNQE, V qba'g guvax lbhe ybtvp sbe inevnag P vf nvegvtug. Gur ehyrf fnl gung gur qenja yvar frtzragf zhfg or ng avargl-qrterr natyrf gb bar nabgure, abg gb gur tevq. Fb rnpu oynpx qbg qbrf abg arprffnevyl nqq gjb gb gur ahzore bs vagrefrpgvbaf bs frtzragf jvgu qvnzbaqf. Commented Apr 22 at 6:03
• @msh210 Hmm, I guess the first rule can be interpreted differently (it is unambiguous in 2D but in 3D it can be seen to allow diagonal lines, as long as the lines meet at 90 degrees). In the spirit of original Masyu puzzle, my interpretation is that the lines are constrained to one of three cardinal directions. Commented Apr 22 at 6:30
• Bubbler, great analisys! Now it's time for drammatic follow-up problem. @msh210, Bubbler: cyrnfr, qba'g fcbvy zl qenzngvp sbybyybj-hc ceboyrz urer. Npghnyyl, zl chmmyr vf vafcverq ol guvf tnc va gur ehyrf. msh210, vs lbh jvyy cbfg na nafjre sbe sbyybj-hc ceboyrz, V jvyy pbafvqre vg nf zbfg rneyvre. Commented Apr 22 at 10:45

Here I'll show my own proof for 'D' variant:

In this diagram, there are 6 stars, corresponds to faces of the cube. Note, that every circle rests in the middle of 12 cube edges, and obligation to turn at black circle, forces every circle be connected to some star, so every star location is exhausted. This means, central location appears isolated.

Dramatic follow-up 3D masyu problem