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Place each of the twelve pentominoes into the grid once, with rotations and reflections allowed. No two pentominoes can overlap or be orthogonally adjacent, and all cells not occupied by the pentominoes must be traversed by a single, closed loop (connecting cells horizontally or vertically). The black circles indicate some cells that must be covered by pentominoes, white circles indicate spaces that may not be contained in pentominoes and must be part of the loop.

enter image description here

Source

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  • 5
    $\begingroup$ This is a nice puzzle, but I've downvoted it. I don't think just directly copying a puzzle from GMPuzzles deserves an upvote, and I'm not sure it can even be here due to copyright issues. I would have much preferred it if the puzzle were original, even if it was much easier or less elegantly constructed. $\endgroup$ – Deusovi Oct 4 '18 at 15:51
  • $\begingroup$ Just to make sure I understand correctly, can the loop go over one spot twice or does it have to be one path that doesn't overlap or go back on itself? $\endgroup$ – gabbo1092 Oct 4 '18 at 15:52
  • $\begingroup$ @Deusovi I think it's a really interesting variation of a puzzle I haven't seen before. It also have a pretty good difficulty. The solution isn't obvious, and it takes some time to work through, but I never got stuck for very long. I really enjoyed it, and wanted to share it, and thought it was unique enough for that. I also made sure I was allowed to post it here beforehand. $\endgroup$ – Kruga Oct 4 '18 at 20:28
  • $\begingroup$ @gabbo1092 The loop cannot touch or cross itself $\endgroup$ – Kruga Oct 4 '18 at 20:31
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Unless I made a mistake somewhere, this solution is unique:

enter image description here

To get there, I used a couple of rules of thumb:

1. Never make a separated area of empty squares
2. Never make a "dead end" for the empty square path
2a. The "U" piece comes with a dead end, so that piece must hug a knoll on the edge
3. Each piece must go somewhere.
4. If an empty square has only two (possibly) empty neighbours, the path must connect them.

Apart from those, there were a couple of slightly mind-boggly deductions required, but all in all, everything seemed extremely well designed, and no guesswork was needed at any point.

Progress, part 1:

enter image description here Starting with the easiest bits, I make a (non-critical) mistake right off the bat. The red pieces all directly follow from the rules of thumb above, as I keep plotting the only possible path from the lower left corner. The mistake is that the blue bit on the right isn't necessarily connected all the way up. (Fixed that in some later image, when I spotted my mistake.)

Progress, part 2:

enter image description here
There must be a path of empty surrounding the pentomino on the right, and the neighbouring pentomino forces that path so much that the border piece must be the N-pentomino. There is no room for another pentomino in the lower right corner, so it must be empty. There's only one path that can achieve that.

Progress, part 3:

enter image description here
Counting from the bottom right corner, the empty square at (4,4) forces the square below it to also be empty. If it weren't empty, there would be a 3-way junction of the empty path on the right.

Also, the black dots right above the (4,4) must be connected: if they were not, there would be a dead end (or a disconnected empty space) between them.

Progress, part 4:

enter image description here
This bit is very tricky. On the bottom side, the two rightmost black circles must be connected, because if they weren't, the piece in the middle lower side would have to be the P. No matter which way you try to put the P, there will always be a dead end somewhere. Since the empty path blocks the other options, the piece on the lower side must be the Y. This, interestingly, forces the empty path shape quite a lot, even to the point where the P piece's place becomes forced.

Progress, part 5:

enter image description here
The pentomino on the left lower side can only be T, P or N. Since the P and N are already elsewhere, it must be the T. This squeezes the empty path towards the left, so the pentomino on the left side has to be the V. Also, at this point, there is only one spot where the U can go, so that the dead end in its middle is outside the borders.

It's definitely a bit difficult to spot, but at this point, there is exactly one possible spot for the X piece.

After that, there weren't any major difficulties in placing the rest of the pieces and finding the only possible path for the empty squares. Here's one more picture from along the way:

enter image description here

(Apologies for the inconsistent scaling of the images, Imgur's auto scaling seems to have malfunctioned on some of the pictures.)

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  • $\begingroup$ Amazing amount of detail! +1 $\endgroup$ – Brendon Shaw Oct 4 '18 at 20:58

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