Classics Collection

Classics Collection: Four Winds

As announced, we are continuing with our beginners series. Here is the new set: Four Winds

Four Winds is another plain puzzle style (some would say “dull”), probably even more so than Kropki. Its variety of solving techniques is limited in the first place, and it is also hard to come up with puzzle layouts and clue constellations which require applying the few available techniques in new ways. For some reason, I have always had a soft spot for this particular style, even though it has little to offer.

Rules: Draw horizontal and vertical lines, starting in the cells with numbers, so that each empty cell is used by exactly one line. Aside from their starting cells, lines cannot enter cells with numbers. Each number indicates how many cells are covered by lines starting in this cell, not counting the number cell itself.

Example and solution:

I have been thinking about a reasonable solution code for this puzzle style. Our current choice (see the PDF file) is not entirely satisfying in several regards. For once, it is error-prone; it is easy to mix up several vertical lines, for instance. Also, the code does not carry as much information as in other styles, since a considerable number of lines are without alternative. Incorrect solutions will include many of the same lines, and – in extreme cases – the code might not be affected at all by an error.

Finally, the solution code can only be significant if there are many different numbers in the grid. A puzzle which uses the same clue value many times will therefore suffer (it is often considered inappropriate to alter a puzzle only to suit a specific code structure). The puzzle FW4 from today’s PDF file with only 3’s and 4’s is a good example, and I have even seen specimens with only one number value throughout the grid.

There are two fundamental solving techniques in Four Winds. The one is the search for cells which can only be reached from one clue cell. For example, the cell R4C1 in FW1 can only be covered by a line starting in the 3 above (the line may or may not continue further south). Likewise, the line covering the cell R4C2 must originate from the 4 in R5C2, and there are many more similar inferences. In general, rows or columns with no clues at all are a good starting point, but note that the same result can be obtained if there is a small number given in the same row/column sufficiently far away, as with the cell R3C5 in FW2.

The other key technique is based on large clues. Look at the 6 in R6C4 of FW1 (something similar applies to the corner cell R1C6 in FW2). There are only seven empty cells which the lines starting in this number cell can reach at all, and six of them must be covered. It follows that lines from R6C4 to R6C5 and to R2C4 can be drawn; only two possibilities for the last cell remain. Such steps often occur if the number of “freedoms” for a clue is close to the clue value itself, in particular if there are few directions from it which cover much ground each.

You may note that we have encountered the “duality” of the above two techniques before – grid parts without clues, which must somehow be reached on the one hand, and large clues with restricted space on the other. We made a similar observation in Shikaku puzzles, but also in a few other styles like Nurikabe. In order to give Four Winds puzzles a nice balance, it is thus desirable to avoid areas (in particular rows and columns) with no clues at all, but also to refrain from designing regions stuffed with clues so that the second technique becomes prevalent.

There are just a few more constellations I want to bring up. Look at the cell R6C2 in FW3. This cell can be reached from two different clues, namely the 5’s in the top and bottom row. Observe, however, what happens if a vertical line is drawn from R1C2 all the way down to the bottom of the grid. This line would cut off a piece of the grid with a clue of 3 inside it, but the clue value cannot cover all the empty cells in that part.

Cutting off grid parts is an important advanced technique, although suitable constellations are typically harder to spot. (It may sound silly to call such a step “advanced”, but to my knowledge there is not really much in Four Winds which goes any deeper than this.) Let us look at the clue of 4 in R2C4 of FW3 next. With eight freedoms distributed over three directions, there does not seem much to go on here, but you will notice that it is impossible to manage without drawing a line south from the clue. Otherwise, we would need lines covering the other four cells (to the left, up to the cell R2C1, and R1C4). This would once again cut off a region with an unfitting clue inside.

By the way, it is noteworthy that the cut-off argument can also work with more than one clue or none at all in the area in question. Sometimes one must be careful because the clue cell which does the cutting-off can contribute a line into the region which is otherwise separated from the rest. There is no formula for it, one simply has to check if the remaining clues are suitable.

Finally, there are inferences which investigate lines starting from more than one clue. I will give just a simple example (which is not actually needed), once again in FW1 from the PDF file. The cells R2C2 and R5C6 can both be reached from the same two clues, but neither of them can cover both specified cells at the same time. In this case, the 4 cannot cover the rest of Row 5 due to the cut-off argument anyway, but sometimes it makes a difference to know that the clues must somehow split up the cells in question among them.

It is conceivable to come up with clue constellations which a priori work only as a group, not individually. However, such steps are tremendously hard to incorporate in Four Winds puzzles, especially since there are usually other ways to make progress during the solve. The puzzles from today’s set do not need such elaborate techniques; in fact, with very few exceptions they require only the basic steps. I hope you enjoy them anyway. Have fun!

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