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Composite Puzzles

Composite Puzzles: Overlapping Skyscrapers

Hi folks,
this is the first part of a new series I have labeled “Composite Puzzles”. Unlike the Classics Collection, which is primarily intended to attract inexperienced solvers, this puzzle series is devoted to more complex puzzle concepts and, as a consequence, contains somewhat harder puzzles. It is kind of a follow-up on some articles from an earlier incarnation of this blog, which also dealt with multi-grid puzzles here and there.

First of all, I have no clear definition of “composite puzzles” in mind. As with many things, I expect to know it when I see it, and in a sense the term “composite” speaks for itself. If I were to give an explanation, though, I would start by saying that there is a large variety of “basic” puzzle styles in the puzzle community. These puzzles are not necessarily what we would call “standard” puzzles in the sense of popularity; many of them may be relatively unknown. What I mean instead is that they use a single grid (at this point I will limit myself to grid-based logical puzzles) and a single set of rules throughout that grid.

Composite puzzles can emerge from basic puzzles in various ways, for example by connecting initially separate basic puzzles (either of the same style or of different styles) through certain linking conditions, by adding an additional task of matching rule sets to different grids (Matchmaker puzzles), or even by introducing different rule sets in a single grid in order to arrive at some kind of hybrids, as in the Permaculture round of the WPC 2019.

There may be many more ways to create composite puzzles, which is actually why I am refraining from any attempts to draft a precise definition. In any case, the last example I gave above is why I will avoid the designation multi-grid puzzles which I used before. It is certainly true that a great number of composite puzzle concepts rest on the presence of multiple grids within a single puzzle construction, but I would rather not view this as a crucial requirement.

Composite puzzles typically make use of interactions between the “components” and their respective solving techniques for the global solving path. However, I am once again reluctant to treat this as a part of the definition. In a very early German Puzzle Championship – 2006, if memory serves – we designed a “Medley” round which was essentially a linear sequence of basic puzzles, and the solution of each puzzle provided a couple of clues for the next one. The interaction was thus quite limited and in particular one-way; I will defer the discussion of such sequential concepts and what one can do with them to a later post.

Today’s article will be about a very specific kind of composite puzzles, namely overlapping grids. Actually, the puzzle set is even more restrictive, since it deals only with overlapping puzzles of the same type. However, some of the abstract thoughts that follow apply to overlapping puzzles of different types as well.

As I see it, the appeal of overlapping puzzles mainly lies in the possibility to design solving paths which require to jump between the individual grids, because (logical) progress can sometimes be made in one grid and sometimes in another, but not always in the same place. If the puzzle was such that one could solve one grid right away and then simply move on to the next grid using the full information from the completed overlapping region, I would probably consider it much less attractive. Of course, my “dearest readers” – an expression I am borrowing from Tom Collyer’s blog – are welcome to disagree with my taste.

I once tried to classify solving steps which constitute a transition from one grid to the next. Looking back, I think this was a misguided attempt, which is why I will not replicate that classification here. Instead, I will simply say that the transition can come from either explicit information in the overlapping area (an entry which has been found in one puzzle can be used for the next step in the other puzzle) or from implicit information, e.g. a candidate that has been eliminated.

I am particularly thrilled if the implicit technique is of a more complex kind. Sudokus clearly have a huge potential to include such steps (perhaps using Pointing Pairs), and I admit I am slightly disappointed when I have a go at an overlapping Sudoku which essentially comes down to just a sequential solve. Many other Filling Puzzles are no less suitable for transitional steps, including Skyscrapers, my favorite puzzle type in the past. I have therefore put together a set of overlapping Skyscraper puzzles which, I believe, make good use of these techniques.

Of course, one should not expect too much in terms of interaction, because the overlapping region will at some point be complete, and after that one only has to solve a bunch of independend puzzles; this is in the very nature of overlapping puzzles. Since I enjoy the interactive part most, at least I try to design the puzzles in a way that parts of the overlapping area will remain unknown until very late in the solving path. Also, the complex interactions typically require more than an overlap of just a few cells. In my experience, an overlapping region with a size of up to one-fourth of each grid (or thereabouts) will work out nicely.

The set in the PDF file below contains a few puzzles which are not new; I believe I have published four out of the six puzzles either in my old blog or someplace else. As you can see, the sizes of the grids and the overlapping part vary, from minimal intersections to large overlaps in two-grid or even three-grid constructions. For one of the Skyscraper contests I have held, I even designed a puzzle where complete rows of one grid were part of the other. You will not find such a layout here, so it is probably fair to say that there still lies some potential in the “Overlapping” category of composite puzzles.

Regarding the difficulty, I believe the last puzzle is the hardest even though it has only two components, not three. That is because it rests heavily on some of the harder “implicit information” steps I mentioned before. As for the others, the third puzzle is less elegant; I had a hard time coming up with something that resembles a logical solving path (I guess bifurcations will work much better), and I have only included it because of the perfect symmetry of the clues.

One last remark about the rules: In the past I have defined the clues in corner spots (positions which are adjacent to more than one grid) in different ways. My preference is with sum clues, because they add to the interactive character of overlapping puzzles. Obviously, though, corner clues are not essential for this type of puzzle; they are merely a bonus. Two of the puzzles in the set (2 and 5) feature an alternative treatment in corners, but a moment’s reflection will make it clear that those are just “normal” Skyscraper clues, depicted in a slightly unnusual way because of the other grid nearby.

Here is the PDF file: Overlapping Skyscrapers. By the way, the new series has been added to the PuzzleCheck tool, in case you want to verify your solutions. Anyway, enjoy the puzzles!

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