Pavel's Puzzles

Ada's General Advice on Solving Her Ænigmas

1. The single most important point I have for you, dear solver, is that my ænigmas are intended as purely deductive challenges. My hope, and indeed my expectation in constructing them, is that you will employ logic at all times in addressing them, and not yield to the tempting siren song of guessing at parts of the solution. There is a very important role for your intuition here, as discussed below, but it is not in making guesses and then checking whether or not they appear to “work out”. Instead, I urge you never to make a mark on the ænigma paper until you are quite certain that it is correct. Indeed, much of the solving process should consist of making progress only as you are logically forced to do so, when you can conclude that no other choice is possible. In so doing, you will effectively be proving that the solution you have written down is the only one even possible!
2. As any person engaged in mathematical or scientific endeavours will agree, the choice of a good notation is of paramount importance: a well-chosen notation makes critical information instantly accessible and greatly improves both the ease of making inferences and the likelihood of those inferences being accurate. As a starting point for the development of such a notation, I direct your attention to the example puzzle on each month's ænigma paper. The second illustration in the sequence shewn, in particular, displays a partially solved grid, employing thereon what I hope is a useful notation for your emulation.
3. It may seem that all of the additional markings on the grid, shewn in the example, are but useless noise, unnecessary steps that do not contribute to the solution. I implore you to lend this impression no credence whatsoever. In actuality, your swift progress in solving the ænigma utterly depends on your disciplined application of all such notational devices. To attempt solution without their aid is akin to a sculptor attempting to create great works with his right hand tied securely behind his back: it may, in theory, be possible for him to succeed, but only at great cost, and with much reduced likelihood.
4. Speaking of the example puzzle, I encourage you to work through the solution of that much simpler case before attacking the main ænigma. In so doing, not only will you come to understand the notation there employed, as discussed above, but also encounter simple variants of deductions you will need to make in the original ænigma, thereby better preparing you for the rigours to follow.
5. When solving, it is ever so much easier to proceed when one has as much contextual information as possible. In nearly every ænigma, there are a number of quite easy, even trivial deductions available right at the beginning. Pray do not hesitate to perform all of these deductions, writing in their results according to that ænigma's notation, right at the beginning of your labours. Doing so will often immediately grant you sufficient information to make a little further progress on several fronts, in addition to providing improved context for deductions later on in your solution.
6. Progress toward the solution to each ænigma will come almost entirely in the form of small, incremental steps, each adding but a single line segment, darkened square, number, or letter to the grid. Do not endeavour to rush ahead, attempting to guess or otherwise ascertain great swaths of solution in one fell swoop. The whole of your solution will come as the sum of many steps; “patience” and “perseverence” must be your watchwords.
7. The great majority of the deductions on the path to your solution will be relatively straightforward. The difficulty will come in determining where on the grid each deduction is to be found. Seek out locations where your options are most restricted, where progress on some entry or region or path can only be had in one of two directions. Quite often, there will be multiple such locations, but in at least one of them, a simple analysis of the cases will readily yield results.
8. In each such case analysis, use your intuition to guess which of the two possibilities is most likely. Then do something you might find rather surprising: concentrate your analysis on the other option, the one you concluded was less likely! If your intuition was correct, and you must trust that it quite likely was, then your analysis of the opposite choice will most probably yield a contradiction in very few steps indeed, thereby proving your intuition correct!