Q: What kind of logic puzzle would you like? A: Yes

Question

A recent post and Stiv's answer provided inspiration for a new puzzle. I posted a study for this puzzle earlier; this is the bigger effort envisioned. I hope you enjoy!

This puzzle consists of four different logic puzzles: a Nonogram, a Nurikabe, a Kakurasu, and a Slitherlink, which need to be solved simultaneously. The clues for the first three puzzles are given in the following diagram:

The red numbers, at the top and to the left, are the Nonogram clues. The yellow numbers in the grid itself are the Nurikabe clues. The blue numbers at the bottom and to the right are the Kakurasu clues. The Slitherlink clues are given in this diagram:

As you might guess with this presentation, there is something funny going on. Not all of the Slitherlink clues are valid. A Slitherlink clue is only valid if its color matches the combination of the corresponding squares in the other three puzzle grids that are shaded. So for example, a red Slitherlink clue is only valid if the corresponding square in the Nonogram grid is shaded, and the corresponding square is not shaded in the Nurikabe and Kakurasu grids. Specifically the combinations are:

• White - no shading in any grid
• Red - Shaded in Nonogram, not shaded in Nurikabe and Kakurasu
• Orange - Shaded in Nonogram and Nurikabe, not shaded in Kakurasu
• Yellow - Shaded in Nurikabe, not shaded in Nonogram and Kakurasu
• Green - Shaded in Nurikabe and Kakurasu, not shaded in Nonogram
• Blue - Shaded in Kakurasu, not shaded in Nonogram and Nurikabe
• Purple - Shaded in Nonogram and Kakurasu, not shaded in Nurikabe
• Black - Shaded in all three grids

In the Nurikabe grid, the squares containing the clues themselves are considered unshaded.

Shading is determined by the background color in the Slitherlink clue; the coloration of the numeral is for legibility only and has no significance for the puzzle. A box around the numeral is solely to highlight the white background, and has no significance for the puzzle.

If a Slitherlink clue is invalid, it provides no information about the Slitherlink. The number may match the ultimate path, or it may not.

An accepted solution will have the solution for all four puzzles, as well as a description of the logic used to derive the solution.

As a final note: the set of four puzzles with the given connections does have a unique solution, but that does not mean that each of the component puzzles does, absent the given connections. The puzzles are meant to be solved simultaneously, not in sequence.

Solver Helps

Grids

As I was going through the puzzle, I found it easier to work each individual puzzle in its own grid. These individual grids are provided here:

For the Colorblind

The CSV below has the colors of the Slitherlink clues, with Bl for blue and Bk for black:

W,R,O,Bl,Y,O,G,Bk,P
Bk,G,G,Y,Bk,G,W,O,W
P,R,Bk,O,P,W,O,Y,R
O,Bl,R,R,Y,W,Y,Bl,Y
Bk,W,Y,Y,R,O,G,R,W
O,G,R,Bk,Y,Bk,G,Y,Bk
R,Y,P,Y,G,Bl,Bl,Y,Y
G,O,Bl,Bl,G,O,Bk,P,Bl
P,P,Bl,P,P,Bl,G,Bl,Bk

Kakurasu

This type of puzzle has not appeared on PSE before, at least not by this name. The rules are simple. The columns, left to right; and rows, top to bottom; are labelled with the values 1 though 9. When the grid is shaded, a row (column) sum is the sum of the values associated with the columns (rows) of the shaded squares in that row (column). The goal is to shade the grid so that the row and columns sums, presented at the right and bottom, respectively, are simultaneously satisfied.

Answer 1

This took me several hours. I did have other things to do today, but ah well, this was much more fun. Thanks for the wonderful puzzles. :)

And the three puzzles:

In this grid, the leftmost part of each cell is the nonogram, center is the Nurikabe, and right is the kakurasu. For the nonogram, pink is fill and black is not, for the nurikabe green is fill and black is not, and for the kakurasu, orange is fill and brown is not. I swear there is a method to my madness.

*** Detailed Solution ***

The first step to this puzzle was to get as far into each puzzle as you could before you have to start going back and forth in between the original three puzzles and the slitherlink clues. Here's the state of each puzzle before you have to start comparing/guessing:

Nonogram:

Nurikabe:

Kakurasu:

Now, it's time to compare. Use the clues that you have to determine which few slitherlink clues can be determined definitively true or false. This is approximately a 50000 step puzzle, so I won't outline every single step, I'll just highlight some specific strategies that I used in each puzzle.

Nonogram:

Nonograms are my favorite. The important thing here is being able to identify which cells in a given row or column you are CERTAIN about. Sometimes that might only be one cell out of a three-cell block, because if the block is all the way left, or all the way right, the cell that is always covered is the middle. Does that make sense? This was the first puzzle I solved in its entirety.

Nurikabe:

The key for this puzzle was that the entire "sea" needs to connect. Without that piece of information you can't solve the puzzle without guessing I don't think. There's also a lot of "What if X goes here, what will happen to Y?" I even occasionally chose an inflection point in the puzzle, made a copy of my puzzle at that point, and then picked one of two possible directions. If that was wrong, then I went back to my "save" and started again.

Kakurasu:

This one was the hardest, and the one that took the longest. The key here was to start with the biggest numbers. 9+8+7+6+5+4+3+2+1 = 45, so for 40, each of 9, 8, 7, 6 HAD to be shaded. For 39, 9, 8, and 7. The rest of this is just going back and forth with the slitherlink clues, as there's not much you can do beyond guess and check without additional information.

Slitherlink clues:

While a clue being invalid doesn't mean that it's incorrect, the inverse is true. If the lines you're creating render a clue invalid, that's an extremely useful bit of information. If you have two out of three puzzles complete for a given cell, and they both match with the clue, if you prove the clue false, then the third puzzle's cell must be the opposite of what the clue dictates. I used this strategy many times throughout the puzzle.

Final nonogram:

Final Nurikabe:

Final Kakurasu:

From there, it's a matter of simply solving the slitherlink, which gets a bit tricky in the top-left but is ultimately the easiest part of the puzzle.

Answer 2

With one teensy correction to the final slitherlink (EDIT: Now corrected!), @kristinalustig has already solved this puzzle. However, I thought it would be beneficial to provide something a little more step-by-step to supplement her excellent answer, so that anybody following along and getting stuck would have a resource to guide them through...

Notation: Throughout this explanation I will use grey fill to indicate shaded cells and pale blue fill to indicate confirmed unshaded cells. For the slitherlink, correct line segments will appear in black; clues will have a small tick beneath them if confirmed correct by the other grid puzzles, a red cross through them if confirmed incorrect by the other grid puzzles, or a red circle around them if the unfolding slitherlink logic shows them to be incorrect and thus useful for resolving the other grid puzzles.

Step 1:

First solve as much of the nonogram, nurikabe and kakurasu as possible until you get stuck without any further logic. Simultaneously augment the slitherlink to show which clues have been proved correct or incorrect, solve as much as you can and then circle the clues that the slitherlink logic shows to be incorrect - this will be used in the next step. At this point your grids should look as follows:

Step 2:

Consider the kakurasu...

Thanks to the two incorrect clues circled on the bottom row, we can make some deductions about the kakurasu that allow us to resolve:
- The entire bottom row (R9C7 must be unshaded; since R9C3 is already shaded we must also leave R9C1 and R9C2 unshaded to total 35 for the row; shade the rest),
- R8C4 (unshaded, since 8+9>16, exceeding the column total),
- The rest of row 8 then follows (R8C2 must be unshaded, the rest shaded to total 39 for the row),
- R6C8 (unshaded, since the column total would exceed the target of 29 if shaded),
- Much of column 1 (anything contributing a value of more than 3 to the column total would exceed the target of 11).

We can also confirm the bottom right-most corner of the nurikabe to be unshaded, for what it's worth!

Importantly, we can also make some progress with the slitherlink by indicating which clues are confirmed correct or incorrect, reaching a contradiction in R9C5.

Step 3:

Now let's turn our attention to the nonogram...

The circled slitherlink clue at the end of the last step must be incorrect, meaning that R9C5 must be unshaded in the nonogram. This one piece of information now lets us resolve the entire nonogram!

(Via the following: forced logic in the bottom left section, column 1, R2C5, thus all of row 2, some of row 1, some in row 5, all of column 7, all of row 6, R4C6 must be shaded, R5C7 must be unshaded, all of row 5, column 6, row 1, column 5, and the rest follows!)

Step 4:

Meanwhile we can also make more progress with the kakurasu...

To hit the total of 31 in row 6, everything except R6C5 needs to be shaded. We can now shade all remaining cells in column 9 and everything in column 7 with a value greater than 3. In column 4, the 1 must be shaded and the rest left unshaded to hit the total of 16, and all remaining cells in row 7 must be shaded to hit the total of 24.

All of this produces a single contradiction in the slitherlink clues, which we will use next (the yellow 3, circled)...

Step 5:

Now for the nurikabe, which we can solve in its entirety from this single slitherlink contradiction...

- R6C5 must be unshaded, which must be reached by the 6-shape. This has a knock-on effect as there are now other squares that the 6 (or any other number) cannot reach - shade these.
- Resolve the bottom of column 3 through forced logic.
- To link the bottom-left section of the grid to the rest of the path we need to extend the path right the way up the space in column 1. This in turn resolves the nearby 4-shape.
- R4C5 must be shaded for connectedness.
- R3C6 must be unshaded - we need to avoid making two complete 2x2 squares of path to the right of the 3 in column 7. If we did this by leaving both its 'north-west' and 'south-west' neighbours unshaded, we would break the connectedness of the path - the only solution is to 'unshade' the space to its left.
- To avoid isolating the path down the RHS we need to add some path by forced logic to the right of the grid, simultaneously helping us to resolve the shapes of the 6 and the remaining 4.
- The rest follows with the requirement that R2C7 must be unshaded to prevent forming a 2x2 square.

Augment the slitherlink with new knowledge about confirmed correct and incorrect clues, and follow the logic to complete the path in the bottom-left section of the grid.

Step 6:

Just the kakurasu remaining then...

- R5C2 and R5C3 must be shaded because of the logical contradictions circled in the slitherlink at the end of the last step.
- R5C6 must be shaded for a row total of 11.
- The remainder of column 6 must be unshaded, as the column total of 35 has now automatically been hit.
- R4C8 must be shaded to bring row 4's total to 13.
- R1C8 must be shaded for column 8 to total 29 (rest unshaded).
- R1C1 and R1C3 must be unshaded, as 1 and 3 cannot be used in combination with other available numbers to make up the remaining row total's difference.
- R3C1 must be shaded (and R2C1 unshaded) for a column total of 11.
- R2C3 must be shaded no matter what combination of numbers is used to make the row total - it's a vital component of both possible sums.
- R3C3 must be shaded to total 40 for the column.
- R4C5 must be shaded (and R4C2 unshaded) for a column total of 13.

We then have three 3x1 cell blocks that cannot yet be resolved, as each row requires an additional total of 7 (either 2+5 or 7 alone) and each column requires an additional total of 3 (either 1+2 or 3 alone) - we cannot yet tell which it will be. Thankfully resolving the slitherlink a little more throws up another contradiction (circled in row 1)...

Step 7:

Consider that slitherlink contradiction...

This means that R1C7 must be unshaded in the kakurasu. Which means that R1C2 and R1C5 must be shaded, which then allows us to deduce the whole of rows 2 and 3 as well!

Now we have a complete picture of which clues in the slitherlink are real and which are fake.

Step 8:

Time to resolve that slitherlink in its entirety!

- Use the logic of the green 3 in the top-left to resolve the whole top-left corner.
- The white 3 (R3C6) being diagonal to the white 2 permits some more deductions. In particular, the path must go up the left-hand-side of the white 2 to avoid producing an odd number of loose ends in the top right section (which would make it impossible to resolve).
- Orange 1 in row 1: Path segment must be on its right-hand-side. The path in the top-right section can now be fully resolved using the logic of the 1's.
- Black 1 in middle of row 2: Must be passed to the south. Then the adjacent yellow 1 likewise.
- The remainder of the path can be solved through forced logic and the puzzle is conquered at last!!

Concluding remarks:

This puzzle was EPIC! It took HOURS to solve in its entirety and shows such a huge amount of thought and craftsmanship in its design and execution. I am seriously impressed! Huge kudos to @kristinalustig for being first to get an answer. I encourage everybody interested in grid-deduction puzzles to give this one a try - it's one of the best combination-puzzles I have come across not just on this site but anywhere (and I mean that)...