Pavel's Puzzles

Adalogical Ænigma #10

General advice for solving the current ænigma

1. As with so many ænigmas, it is ever so much easier to proceed when one has as much contextual information as possible. In this case, you will naturally record each square that you know to be darkened, but perhaps equally important is to record is every square that you know not to be darkened! (On the example ænigma, I notate such squares with a small, central dot.) Pray do not hesitate to mark every such square immediately, as soon as you discover it.
2. Right at the beginning, for example, you may find regions labelled with a “0”; you know, a priori that all of the squares within such regions may not be darkened, so I urge you to mark those squares with dots as described above.
   Remember that the rules forbid darkened squares from being horizontally or vertically adjacent to one another. Thus, as soon as you darken any square, you may also mark the adjacent squares with dots and, again, I earnestly advise you to do so.
3. That same rule barring adjacent darkened squares will oftentimes create situations in which a numbered region may only be completed in a few possible ways. Indeed, sometimes there will only be one or two ways in which such the darkened squares in such a region may legally be assigned. (Please note that, although this may not be true of a particular initially, it may later become true, as you determine that some of that regions squares must not be darkened.)
   Pray do be ever vigilant for such situations. If there remains only one legal way to darken squares (for example, in a 1×3 region labelled with a “2”), then of course you must immediately fill it in accordingly. If there are but two such legal fillings-in (e.g., in a 2×2 region labelled with a “2”), then it will often prove productive to perform an analysis of the two cases, as described below.
4. When examining two cases as described above, it may happen that one of the two may immediately be rejected, as leading to a violation of some rule (most frequently, the rule requiring that all un-darkened squares be connected to one another, horizontally and/or vertically). This is, naturally, a delightful outcome, as it means that you may then reliably conclude that the other case must prevail.
5. In other case analyses, you will not be quite so fortunate as to be able to rule out either case at once: both will appear plausible, given your then-current understanding. Nonetheless, you will often find that you can make productive use of the analysis. Specifically, you must look for commonalities between the cases: conclusions about other nearby squares that follow identically from both cases.
   Sometimes, this will take the form of nearby squares that must not be darkened, regardless of which case is eventually found to be the truth. In other situations, the common conclusion will be that two already-darkened squares are linked (by a chain of diagonal adjacencies of darkened squares); such information can be invaluable in determining whether or not an undarkened area is in danger of being “cut off” from the remainder of its undarkened kin.
6. The strangest rule in this ænigma is, likely, the one forbidding any horizontal or vertical sequence of undarkened squares from intersecting more than two regions. (One may equivalently comprehend this rule as forbidding such a sequence from crossing more than one region boundary.) Strange though this rule may be, its application will actually drive much of your deductive process.
   If you follow my advice to mark all squares that you know not to be darkened (as I once again strenuously urge you to do), it will make it much easier to see opportunities where you may apply this strange rule. Those opportunities will come mostly in two guises.
   First, look for sequences of known-undarkened squares that cross one region boundary and extend right up to a second boundary, followed by an as-yet undetermined square (i.e., one whose darkened or undarkened status you do not yet know). If the sequence was to continue into the undetermined square, it would cross two region boundaries (or, equivalently, intersect more than two regions), thereby violating the rule. You may thus conclude that said undetermined square must, in fact, be darkened.
   Second, and much more subtly, look for a sequence of known-undarkened squares that is interrupted by a single undetermined square, where the interrupted sequence crosses two (or more) region boundaries. Again, such a situation allows you to deduce that the previously undetermined square must actually be darkened.
7. In my ænigmas, the individual deductions themselves are often quite straightforward, once identified. The difficulty lies in locating the opportunities for those deductions around the grid. Frequently, I myself must attempt many such deductions, at many sites in the grid, before finally hitting upon one that yields success. I fear that these notes, walking through the solution of the example, may give a false impression, lacking as they do any signs of that search for deductive weaknesses in the conceptual armour of the ænigma. I pray you will not allow yourself to be thus misled.
8. As always with my ænigmas, progress toward the solution here will come almost entirely in the form of small, incremental steps, often adding but a single darkened or known-undarkened square to the grid. Do not endeavour to rush ahead, attempting to guess or otherwise ascertain the darkenings within an entire region in one fell swoop. The whole of your solution will come as the sum of many steps; “patience” and “perseverence” must be your watchwords.