Xjyuk

This is a three-dimensional Statue Park puzzle.¹ The five squares in the below image depict the five levels of a $5times5times5$ cube. The goal is to fit inside it the eight pieces shown in the picture, so that the resulting space satisfies the following conditions:

• Pieces can be rotated or reflected along any axis.


• No piece can be orthogonally adjacent to another piece.


• The cells not occupied by pieces must all be orthogonally connected.


• Cells marked with a black circle must be part of a piece, and cells marked with a white circle cannot be part of a piece.

Note: The pieces are drawn in 2D above, but all are 3D pieces with thickness 1. So the daggers are made of 6 mini-cubes and dashes are made of 3.

_{¹ The Statue Park puzzle type was invented by Palmer Mebane. The original rules can be found on his blog.}

For clarity: in this answer, "up", "down", "left", and "right" refer to the obvious directions. "In" and "out" refer to movement between layers, with "in" increasing the layer number and "out" decreasing it.

Step 1:

The two diagonally-adjacent black dots in the corner mean that the corner is empty. The cell in the top left of layer 1 cannot go downwards, and that means it must go left or inwards. And it must be part of a dash rather than a dagger, meaning the cell in L2R1C2 (layer 2, row 1, column 2) must be unshaded.

Step 2:

If the cell in L3R4C4 was shaded, it would block off the black dot in layer 4.


If the cell in L1R4C2 was unshaded, both of the clues next to it would be dashes going inwards -- but that means we would have three dashes, because the very top left dot is also part of a dash. So it is shaded, and we can get a large portion of layers 1 and 2.

Step 3:

The black dot in L2R5C3 must extend downwards, making the plus sign of a dagger. The extra part of this dagger can only go in layer 5, since all other possibilities for it are blocked off.

The black dot in layer 4 then must extend upwards, making another plus sign that must be extended outwards into layer 2.

Step 4:

Note that every cell in a shape either has two adjacent cells in opposite directions, or a two-cell extension in one direction. (Or both.)


The black dot in the center of layer 2 is blocked from extending down, left, or out. So it must extend at least two squares in some direction. That direction must be inwards, meaning it connects to the black dot in the center of layer 5 (forming the 4-block-long part of a dagger). The only place that the handle can go is in layer 3.

Step 5:

By similar logic to step 4, the remaining black dot in layer 3 can only be a dash, extending two squares outwards.


No remaining daggers can fit entirely on a layer (apart from the nearly-finished one on layer 1), so both will have to be placed between layers. We can also mark off some cells in layers 4 and 5 as being unusable for daggers.

Step 6:

If one of the remaining two daggers is placed with its 4-square-long part horizontally, that must be in layer 4. But then the other dagger cannot be placed: there's no room for it to be placed horizontally, and its handle cannot be in layer 2, 3, or 4. So both must point inwards or outwards.


The handle for the left unplaced dagger only fits on layer 4. Then, the upper right dagger's handle also only fits on layer 4. And that finally resolves the unfinished dash and dagger in layer 1, completing the puzzle.