How to union a slitherlink and a nonogram?

Question

I would like to union two puzzles in one. The first puzzle is a slitherlink, the second is a nonogram. The main idea is: the slitherlink's solution is the entrance for the rows of the nonogram.

In the slitherlink one can use only five labels: 0, 1, 2, 3 and None.

In my case I used 1, 2, 3 (left figure). But in the nonogram I must use natural numbers that are greater than 3 (for instance, '4' in right figure).

Question. I am looking for any ideas how to union these two puzzles? I don't want to use explicit labels ("4" in my case) in the slitherlink.

Answer 1

You could give yourself a little bit more wiggleroom if you create the slitherlink on a hexagonal grid. This preserves the neat rows of numbers that can be plugged straight into the next puzzle, while increasing the maximum "run" of any active cells in the nonogram from 3 to 5.

Hexagonal slitherlinks are a reasonably common existing form of the puzzle, one having been included in the 2016 LMI Puzzle Marathon (pdf warning), so the fundamental characteristics of the puzzle seem to be largely unaffected by the geometry:

Simon Tatham's Portable Puzzle Collection includes an impressive number of variants on "loopy" (slitherlink) grids, so if a range of 0-5 isn't sufficient for your purposes you may be able to find some more complex examples there. They're not quite as neat as a hexagonal grid, but they should still get the point across.

Answer 2

As someone who has created many puzzles based on combining two different puzzle types, here is one key piece of advice that you may not be hoping to hear...

Not all puzzle types can be paired up well in this way.

Here are a few issues with attempting to combine slitherlink and nonogram:

1. As you already point out, many nonograms require numbers greater than 3. It is important to note therefore that any nonogram requiring four or more consecutive shaded cells in a row will not equate directly to slitherlink clues.

2. With slitherlink puzzles, the numbers are already provided at the beginning. You are not trying to generate numbers, rather you are trying to create a line. If you wish to use the slitherlink keys for your nonogram row indicators, the solver would not even need to solve the slitherlink, making it a redundant puzzle feature - what you would have is just a set of numbers that could separately be both a slitherlink and the row indicators for a nonogram. This would just create a puzzle whose solution path spreads out in different directions rather than converging on a single answer.

3. Both of these puzzle types suffer from problems with ambiguities, making it very difficult to create such a puzzle while ensuring a unique solution exists for both. As a trivial example, in your example slitherlink above the entire bottom row of squares is redundant (the line does not need to dip two units below the keys provided). Infamously, nonograms requiring two isolated diagonal shaded squares cannot be uniquely defined:
   
   This point in itself does not mean it is impossible to create a unique solution for both, just that you will have to take extra care (and probably a lot of extra time) to ensure both solutions are unique, otherwise you will have a bad grid-deduction puzzle on your hands.

4. The key to producing a combination grid-deduction puzzle where the final step is a nonogram is to work backwards from the intended nonogram solution. After all, the reward of solving a nonogram is to find some kind of meaningful recognisable image, like a picture, a set of words, or a visual cipher (e.g. pigpen) - it just isn't enjoyable to solve a nonogram where the solution looks like static on a screen. You will therefore need to create your desired nonogram first, taking care to ensure no numbers greater than 3 are required in the row indicators. However, there is no guarantee that your target nonogram solution will generate numbers that also serve as keys to a uniquely solvable slitherlink.

5. Even if you do manage to create a compatible nonogram, the resulting slitherlink may be heavily over-clued, making it almost trivial to solve.

As these points hopefully make clear, combining these two puzzle types well is not going to be easy. Also, as the issues in Point 2 suggest, we're really going to need to find a way to produce numbers from the slitherlink that are not just its starting keys that are already provided from the off...

All of which leads me to a second (more reassuring) piece of advice:

Expand your thinking to other ways you can encode information in a specific puzzle type.

Just because a puzzle type relies on certain starting numbers doesn't mean those are the only numbers you can use for your target encoding.

Here's one idea: Use certain unclued squares in the slitherlink to be your nonogram row indicators. Shade these squares a different colour; instruct that the number of line segments around these squares yields the target number. The user must then actually solve the slitherlink to generate these.

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Trivial example: Let's say the target nonogram is this:

A suitable slitherlink for generating the row indicators would then be something like:

Note that this approach could even lend itself to generating numbers greater than 3 if you use colour-coding. Using a natural ordering like colours of the rainbow, you could say that in each row you need to find 'the total number of line segments surrounding individual squares of the same colour'. Ordering them in rainbow order (ROYGBIV) then gives you the specific ordered arrangement required for the nonogram.

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In conclusion, this will not be easy - but it may just about be doable and produce a meaningful combination of the two puzzle types if you use unclued slitherlink spaces for your targets.

Remember that not all nonogram solutions will lend themselves to an easy puzzle creation process. It is also important to note that the larger your nonogram, the larger your slitherlink will be! It might be best to aim to create a fairly small puzzle in this way before attempting anything larger, or you might find yourself stuck while creating it! Good luck :)

Answer 3

@Stiv's answer is excellent, but here's another idea: maybe have solvers work the nonogram first, and then finish with the slitherlink. Here's an example (notional to illustrate concept, please don't judge quality...I generated the slitherlink online):

Solving the nonogram first, you will shade a certain number of cells:

What remains are the clues to a slitherlink. The advantages of this approach, which address some of @Deusovi's very important concerns:

1. There are likely to be very few logical constraints on your slitherlink, allowing you to tailor the level of difficulty.
2. You will likely have a fair degree of freedom in setting up your nonogram. Even though the example grid is full, it does not have to be. In fact, for larger puzzles, it is likely that your biggest constraint will be the need to have white nonogram squares that do not provide slitherlink clues not have a number.
3. By having the nonogram done first, it gives you a bit of relief from the "interesting picture" problem Stiv mentions...getting the correct grid for the slitherlink is the payoff. This lets you make difficulty tradeoffs in the solution of the nonogram.

The good news is that there are a lot of potential variations, and you have here a good community of solvers who will give it try!