# A Nurikabe-Kakurasu study

This puzzle is another study for a (hopefully) upcoming puzzle, which I think has an interesting solution path. This puzzle contains a Nurikabe and a Kakurasu (see a previous puzzle for details). The connection between the two is simple: for each clue in the Nurikabe, if the corresponding square in the Kakurasu is shaded, the Nurikabe clue should be increased by 1. Unclued squares in the Nurikabe are unclued, regardless of the Kakurasu. Here's the grid:

The different colors on the numbers in the grid are simply to differentiate the clue types, and do not have any bearing on the puzzle. You will need to solve both puzzles simultaneously; there is a unique solution for the pair of puzzles as connected. Hope you enjoy!

### Solver Notes

Text versions of the puzzles:

1  2  3  4  5  6  7
----------------------
1 |  |  |  |  |  |  |  | 18
----------------------
2 |  |  |  |  |  |  |  | 20
----------------------
3 |  |  |  |  |  |  |  | 11
----------------------
4 |  |  |  |  |  |  |  | 22
----------------------
5 |  |  |  |  |  |  |  | 15
----------------------
6 |  |  |  |  |  |  |  |  7
----------------------
7 |  |  |  |  |  |  |  | 10
----------------------
12 14 12 21 12 8  14
---------------
| |1| | |1| | |
---------------
| | | | | | | |
---------------
| |2| | | | |2|
---------------
| | | | | | | |
---------------
|2| | | | |1| |
---------------
| | |2| | | | |
---------------
| | | |2| | | |
---------------

Details:

First, I started by solving all I could in the Nurikabe without even looking at the Kakurasu. The keys here are remembering that the "oceans" must connect, and the islands cannot, and remembering that there can't be any 2x2 ocean tiles.

Also, given the fact that each Nurikabe can only expand by one, are there any squares that MUST be oceans given the possible reach of the islands? I filled those in, too. And if there were any points in which I knew the direction the islands needed to head, I filled those in too. So, on its own, the Nurikabe is already about halfway complete.

Now, what about the Kakurasu (gonna abbreviate the puzzles as N and K from now on)? The only spot we know for sure right now is row 4, col 7, because 7+6+5+4+3+2+1 = 28, so removing the 7 would leave us with only 21.

At this point, I got stuck. Where can I have another inroad with the K? All the other places are "okay, it has to be X or Y" but nothing definitive. My strategy at this point, and there's probably a better one, was to pick a "domino decision" - that is, one that I knew would have a lot of ramifications to a lot of other squares. If I followed perfectly logically from there, if it ended up not working, I could rewind and know the other possible decision was correct. The decision I chose was to make the 7th row, 4th column NOT highlighted. Which gives us this:

From there, several things fall into place. We know that (5r, 4c) must be an island to avoid a 2x2 ocean, and that (6r, 6c) must be an island for the same reason. This means that both of those island numbers must be shaded in the K. That kicks off some updates there, too.

Now the 21 column is filled, as is the 22 row, and that sets off some other progress in the K, too. And over in the N, we realize that the 2 in col 7 needs to go left one to ensure that there's no 2x2 ocean.

From there, it's just generally a process of checking the different ways of moving forward:

- Are there any 2x2 oceans at risk of being created? - Are there any ocean blocks at risk of being cut off? - Are there any numbers we can eliminate from each row or column given what numbers are already selected?

Doing those steps over and over again yields this grid, in which the three unsolved K columns each need 3, and the three unsolved rows each need 7. This is a good sign, because each row can be 5 + 2 or 7, and each col can be either 2 + 1 or 3. It all matches up. But what next?

We know that the bottom row needs at least one of the islands to expand by one to avoid the 2x2 ocean rule. In fact, because of the possible 2x2 oceans left, we know that actually 3 out of 4 of them must be expanded. This was another "domino decision" for me - I chose to pick 7 as the remaining number in row 3 to extend that island left by one. That forced the (r1, c5) island to the right one, and the other top-row island right one as well. After that, it all just falls into place by process of elimination.

• You got it! But then, you know you got it! Sep 22, 2020 at 16:44
• super fun puzzle. and now I must get back to work! :) Sep 22, 2020 at 17:33
• Well done! Thanks for the very detailed write up! Sep 22, 2020 at 17:45