Pavel's Puzzles

Adalogical Ænigma #13

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 path segment that you know must be drawn, but perhaps equally important is to record every square edge that you know cannot contribute to the loop! (On the example ænigma, I notate such edges with a small “X” mark.) Pray do not hesitate thus to mark every such square edge immediately, as soon as you discover it.
2. Remember that the rules require you to draw a single loop that neither crosses nor even touches itself. Thus, any time you connect two one-unit segments of the path, you may also mark the other square edges that touch the connection point with X's and, again, I earnestly advise you to do so.
3. It will often occur that a segment of the loop can only continue in one of two directions. In many such cases, it will prove productive to perform an analysis of the two cases. It may happen that one of the two may immediately be rejected, as leading to a violation of the rules (most frequently, the rule requiring that a numbered square must contribute exactly that many sides to the loop). This is, naturally, a delightful outcome, as it means that you may then reliably conclude that the other case must prevail.
4. 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 square edges that follow identically from both cases.
   Sometimes, this will take the form of nearby edges that must not be part of the loop, regardless of which case is eventually found to be the truth. In other situations, the common conclusion will be that certain nearby edges must contribute to the path. Of course, whichever case obtains, this yields valuable information that I urge you immediately to record on your paper.
5. 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 my 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.
6. 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 path segment or X mark to the grid. Do not endeavour to rush ahead, attempting to guess or otherwise ascertain the path 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.