Composite Puzzles

Composite Puzzles: Skyscrapers with Matching Cells

This is another post about “Composite Puzzles”. It features a Skyscrapers variant I call “Matching Cells”. At this point I should probably clarify that the series is not supposed to be about Skyscrapers only, but since the substance of the variant can be viewed as a generalization of the one from the previous article, I am taking the liberty of publishing a few more puzzles of this style which used to be my favorite for many years.

Here is the new set: Matching Cells. The PDF file does not contain an example, but you will find one on my Skyscrapers website. Anyway, the instructions are quite simple: Apart from standard Skyscrapers rules which apply to the individual grids, the linking condition is that the solutions must coincide on the marked cells (and no others).

When I first designed a Matching variant of the multi-grid type for Skyscrapers, the requirement was that certain entry values should occur in exactly the same spots in both grids (I rarely tried it with more than two individual puzzles). I selected the small numbers, 1’s or 2’s, because they are typically located only at a late solving stage, which means that the interaction between the grids lasted as long as possible.

I was never quite happy with the nature of this link, though. There were only minor interactions before the “breaking point” (let us use this term to describe the moment when the last link has been exploited) and obviously none after it. Up to the breaking point it was possible to come up with some solving steps which made active use of the link, but they never appeared to be quite rich.

This is why I invented the “Matching Cells” variant. I immediately got the impression that it had far more potential for interactive techniques, because they can basically occur with any number values at any point during the solution. In fact, the puzzle author seems to have a great degree of control over the solving path, because he can mark the cells more or less as he sees fit.

It is noteworthy, however, that this comes at a price. Unlike the highly flexible puzzle styles I presented in the Classics Collection (e.g. Nurikabe or Slitherlink), Latin Squares still have a complex structure. The result of the linking condition is a higher level of rigidity near the end of the solution; when one designs such a puzzle, there comes a point when the link seems to take over, and it becomes surprisingly hard to fill the remaining cells in accordance with the rules.

The point is, cells which are not marked still serve as constraints. In fact, the rule that all cells where the individual solutions coincide must be marked yields a NxN matrix of clues. “Negative” clues are weaker than positive ones, but if there are many marked cells (or if the grid is rather small), it may be far from trivial to satisfy them all at the same time.

Of course, as an author one can overcome such problems by marking a few more cells as needed. (This is what I did in the second puzzle from the set; the original shaded shape did not work well with the intended solution path, hence I made a couple of adjustments.) But one should keep in mind that changing the links may have a devaluating impact on the early solving steps.

The first puzzle is very likely the easiest one for a neutral solver. However, I think that the third puzzle best demonstrates the effects that rows/columns with many marked cells can have. It is quite difficult until one grasps the dynamics of the link in such situations. (By the way, it should be possible to apply this variant to other filling puzzles as well.)

In the end, I like the “Matching Cells” variant a lot since it utilizes the composite aspect of the puzzles very well. I am particularly fond of the fifth and sixth puzzle; they are exceptionally hard, but on the other hand they have a very sophisticated solving path which uses virtually every trick in the book, and they also feature a perfect clue symmetry.

I had published these two puzzles briefly on my old blog after the WSPC 2014 in the UK (call it an homage), but I felt that hardly anyone noticed them because soon after I stopped posting there. This seems like a good time to dig them out once more. Enjoy the puzzles!

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