Nurimisaki Puzzle: Introduction

Question

^{This is an entry for Fortnightly Topic Challenge #44: Introduce a new grid deduction genre to the community}

This is a Nurimisaki puzzle, an area-connection puzzle.

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Rules of Nurimisaki (taken from Nikoli):

• Shade each cell either white or black according to the rules below
• Cells with a circle remain white and must be a dead-end (Misaki). A white Misaki cell only has one cell adjacent to it remaining white with the rest being black (vertically and horizontally).
• Cells without a circle cannot be a dead-end.
• The numbers in the circles indicate how many white cells form a straight line from the Misaki cell. At the cells with empty circles, any number of white cells may form a straight line.
• White cells have to form a continuous network.
• Neither black and white cells may form a 2 x 2 square or larger.

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An example is shown below for greater clarity (taken from Nikoli):

Example puzzle:

Solution to example puzzle:

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Shown below is the actual puzzle you have to solve:

Good luck and have fun!

Answer 1

I think this is the solution

Explanation (known unshaded squares in yellow)

Step 1

The endpoint with the 5 can obviously not extend right or up and extending it left clashes with the white circle on that row. Therefore, the 5 must extend down and we can shade in some of the cells in this regard. Similarly, the 3 on the left of the grid cannot extend left or right.

Step 2

The path created by the 5 downwards runs adjacent to two other white circles so we can shade the cells around those. Similarly, we can start shading cells running down the right-hand side by iteratively using the no 2x2 box rule. We see that the white region described on the right must "get out" at the bottom here.

Step 3

The region extending from the 5 comes out adjacent to the white circle at the bottom right and we can shade the other cells around that. Seeing where this region emerges shows us that we cannot extend the 3, at the bottom, to the right (too many cells). Extending the 3 upwards leads to an issue, the white region on the right becomes closed off from the rest after shading. Hence, the 3 at the bottom must extend left.

Step 4

Now it's clear that the 2 just above the bottom 3 cannot extend right or down. If we allow it to extend left, we end up with a situation similar to the following, with a region closed off to the left (there is another way to continue the grid but it has the same results).

Hence the 2 must extend upwards.

Step 5

From here, the continuity of the white region allows us to shade in a lot of cells consecutively.

Step 6

Finally, extending the white circle north-east of the 3 upwards gives us the following

and, since all dead ends must have white circles, this guarantees a 2x2 shaded area. Hence, this circle extends to the left. Then, if we continue to use the no 2x2 region rule and extend the white region from the 3 around the top corner, we find there is just one way to complete the grid.

Answer 2

The answer from hexomino is valid, but here is my attempt of solving this puzzle without backtracking.

For this solution, a misaki cell is considered as neither black or white.

I use two additional rules (which is derivable from those 6 rules):

1. There is no sequence of black or misaki cells connected by edges or sides so that it either forms a loop or connects from side to side. (Because if it's all black, it effectively breaks a white cell network into two, and for every misaki cells, every possible placement of white cell near that misaki cells results in connecting the former cell and the latter cell with black cells)
2. A black cells cannot have 3 black cells as the corner neighborhood unless it shares a side with a misaki cell, as it ensures 2x2 black cells.
3. This configuration is impossible: (With _ is white and # is black or the edge of the level)

_
__#
_

Alternatively, you can leave out the last 2 rules but it has to be replaced with short backtracking.

Here is my solution:

Step 1.

The first step is similar to hexomino's solution, with stopping numbered misaki cells from expanding to a direction that has either has not enough cells or forces the numbered misaki cell to read more cells than what the number says.

Step 2.

The misaki cell with is labelled with number 5 is forced to go down, so we expand there.

Step 3.

Look at the red cells

This violates additional rule #3, so we color it back

I also expanded the white cells when there is only one way to expand the white cells network

Step 4

Look at the red lines

Those red lines are almost connected except for one cyan cell. These lines have to be broken as per additional rule #1

There is one extra white cell added just below the three white cells to avoid making the white cell above that added cell becoming a dead end. (Only misaki cells can be a dead end)

Step 5

If we want to extend the cell with label "5", we see that it's impossible to extend north or east, so we extend it west.

The result is this:

Step 6

This is similar to step 4, so no explanation is needed.

However, the next state changes drastically as the white expansion rule works most productively here.

Step 7

Look at the two red cells, the topmost one violates the additional rule #2 if it's colored black, so it has to be white. The lower one violates the additional rule #3 if it's colored white, so it has to be black.

Note since white cells are not allowed to be a deadend except for misaki cells, we have to do expansion on it.

Step 8

Probably you know this from playing other puzzle games like lights up. Basically, according to white expansion rule, at least one of the yellow cells has to be white. However, according to the misaki rule, there is only one white cell among the yellow and the red cells, so the red cells can't possibly be white, so we color it black

And an expansion rule as usual.

Step 9:

The last step is a straightforward additional rule #1 application.

After some expansion rule, the puzzle is finished.