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This puzzle is part of the Monthly Topic Challenge #14: Think inside the (very small) box!.


Rules of regional Yajilin:

  • Shade some cells in the grid. Two shaded cells cannot share an edge, but can touch at a corner.
  • Draw a single loop, going horizontally and vertically through centres of cells, which visits every unshaded cell and none of the shaded cells. The loop cannot cross itself or branch out.
  • A number in the top-left corner of a cage marked with dotted lines indicates how many shaded cells are inside the cage.

Solve on Penpa+

Empty regional Yajilin grid

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  • $\begingroup$ "Draw a single loop, going horizontally and vertically through centres of cells, which visits every unshaded cell in the grid." - and only the unshaded cells? $\endgroup$ Sep 19, 2023 at 7:30
  • $\begingroup$ Shaded cells can only appear in the numbered regions? $\endgroup$
    – justhalf
    Sep 19, 2023 at 7:43
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    $\begingroup$ @user2357112 Good point, the loop can't pass through shaded cells. I've clarified the rules now. $\endgroup$
    – Jafe
    Sep 19, 2023 at 9:03
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    $\begingroup$ @justhalf Shaded cells can be anywhere, not necessarily inside a cage. $\endgroup$
    – Jafe
    Sep 19, 2023 at 9:04

2 Answers 2

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The only way to get a loop into and out of the 3-cage without its getting stuck is by darkening those three squares. Once that's done, the only way to get a loop into and out of the 2-cage without its getting stuck is by darkening those two squares.

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The completed grid:

enter image description here

The solve path:

First, look at the 3-cage:

  • Note that diagonally adjacent shaded cells would always leave a dead-end cell in row 5. So the shaded cells must be in columns 1, 3 and 5.
  • Then the shaded cells in columns 1 and 5 must be in row 5 to avoid dead-end cells in row 5.
  • And then the shaded cell in column 3 must be in row 4 to avoid more dead-end cells in row 5.
    And that forces all the shaded cells in the 3-cage.

  • Now for the 2-cage:

  • Similarly to the above, we cannot have diagonally adjacent shaded cells in rows 1 and 2.
  • So there must be a shaded cell in row 3, which can only be in column 2.
  • The other shaded cell must be in row 1, and it cannot be in column 2 (dead-end cell in R1C1), so must be R1C3.

  • That gives us all the shaded cells in the cages.
    From there the loop path is forced.

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