Hi folks,

the Classics Collection continues! As usual, the idea is to present five popular puzzle types and then host another Beginners Contest on the Logic Masters Germany website. For today’s post I picked Hakyuu, also known as Ripple Effect, a puzzle style I used to be quite fond of in the past. In fact, I still like Hakyuu puzzles a lot, it is just that they have become quite rare over the years. Still, I feel it is a worthy addition to this puzzle series. Here is the PDF file: Hakyuu

**Rules:** Enter numbers into the grid so that each region of size N contains numbers from 1 to N, each number exactly once. If the same number appears multiple times within a row or column, the number itself indicates the minimum distance between these numbers: each pair of equal numbers K in the same row or column must be separated by at least K other cells between them.

**Example and solution:**

What I like most about Hakyuu is that, even though it is a Filling Puzzle type, it is so different from other Filling Puzzles. The stipulation which numbers must be entered does not come from rows or columns as in Latin Square based styles, but from the decomposition of the grid into irregular regions (of varying sizes). It is true that the rules include certain restrictions regarding rows/columns, but the “primary” issue – the number pool – arises from the regions. Also note that, unlike in Sudokus, the different number values are not exchangeable.

In case you are unfamiliar with the Hakyuu instructions (in particular the part about number repetitions within a row or column), have a look at the example puzzle. There are two entries of 1 in the leftmost column, and they have one cell between them; also, there are two 2’s in the same column, and two other cells lie between them. The distance is allowed to be larger (see the 1’s in the top row or the 2’s in the bottom row), but never smaller. This rule also applies to higher number values – the largest relevant cases in the example are the repetitions of 3’s in Row 4 and in Column 3. It is worth mentioning that rows/columns may have more than two entries of the same value if the grid size allows it, or no repetitions at all.

Another point of interest concerns potential givens. I have designed half the puzzles (of either size) with one or more given entries, and the other half with no givens. Both cases have occurred in past events; I am not sure if either way qualifies as “standard”. In any case, givens should not only be used to make the solution unique, but also to achieve a suitable difficulty level. For instance, the puzzle Hakyuu 8 in the PDF file is solvable without any givens at all, but its difficulty would then be way beyond a Beginner series.

Let us talk about solving techniques. Despite what I said regarding the differences between this puzzle type and other styles just a minute ago, the solving techniques for Hakyuu are not that special. The most important concepts are Candidates and Candidate Spots (possible locations for specific entries) – the same fundamental concepts as in Sudokus. Naked Singles and Hidden Singles may look different from those in Sudokus, but they still play a prominent role in Hakyuu puzzles.

A region of size 1 can only contain an entry of 1, and this number can be entered right away. A domino region (that is, of size 2) yields a pair of Candidates in each of its cells. Now, Candidates are eliminated in Hakyuu all the time, and the most obvious case is by other entries in the same region which are already known. For instance, the central domino in Hakyuu one already contains one number, so the other one – a 1 in R4C3 can be entered immediately.

Candidates are also eliminated by nearby entries of the same value. For example, the region in Column 2 of the same puzzle has Candidate triples, and those are reduced to two pairs by the given entry of 1 in R2C2. Of the remaining Candidates, a 2 in R3C2 is impossible due to the given 2 next to this cell, hence we can enter a 3 in R3C2 and a 2 in R4C2. Incidentally, the 3 in R4C4 would eliminate the 3 in R4C2, leading to the same outcome. Larger known entries have a bigger impact, but in general they are also more difficult to survey.

At this point, the region in the bottom row of Hakyuu 1 can be filled; the entries in Row 3 and Row 4 eliminate most of the entries in Row 5. After that, the region in Column 1 can be dealt with. And so on. This kind of progress is very typical for Hakyuu puzzles; completed regions usually resolve regions close to it. I guess this is where the name “Ripple Effect” comes from, although one could argue that similar expansions occur in many other styles as well.

So far we have mostly seen cases where a known entry in one region eliminates a single Candidate in another region. However, entries have the power to eliminate several Candidates in the same region at the same time; it is all a matter of the relative position. For example, once the 1 in R2C2 of Hakyuu 2 has been entered, it eliminates two Candidates in the corner region (even without the given 1 in R3C1). The 2 in R3C5 even eliminates three Candidates for the same value in the adjacent region. As before, larger entries can have a bigger impact in this regard.

In extreme cases, a (potential) entry can even “kill” an entire region. For example, the cell R4C2 in Hakyuu 2 cannot contain a 2. The point is, either R4C3 or R4C4 must accomodate an entry of 2, and a 2 in R4C2 would eliminate both possibilities. The insight is not very helpful at this solving stage, but you never know. Likewise, the cell R1C3 in Hakyuu cannot contain a 2. This kind of step works best with I-shaped regions (or constellations where all the remaining Candidate Spots for an entry lie in a single row/column); in particular, one should keep an eye on the cells above and below the ends of a domino region.

The analogous scenario for three or even more cells is rarer, but it still occurs every now and then. The cell R5C5 in Hakyuu 4 cannot contain a 4, since this would kill the entire region in Column 5. Following this step, the remaining Candidate Spots for the 4 in the L-shaped region in the bottom-right corner all lie in Column 4, hence R2C4 cannot contain a 4. Once again, it is not clear at this point what to do with the information, but it may come into play later.

It turns out that the search for possible locations for a fixed entry value is often more fruitful in Hakyuu than the study of Candidates in a specific cell. The critical question is: Which value? Well, there is no universal answer; it all depends on the sizes of the regions under consideration. It makes some sense to start with the small numbers, because those have the most appearances throughout the grid. Ideally – and especially for easy puzzles – one can start with the 1’s and simply work one’s way up.

Unfortunately, things are not always that easy. When it comes to more challenging Hakyuu puzzles, the above strategy – starting with the small entries – produces results only up to a certain point (and it would not be unusual to find that the remaining 1’s and 2’s can only be entered near the end). The smallest values may have the most appearances, but the larger numbers are more powerful. The Candidate Spots interact more strongly, and more regions can be considered at the same time. Quite often, there is kind of a “dominant” value which is most suitable for a global investigation of possible locations.

Take a look at the puzzle Hakyuu 6 from the PDF file. The linear approach will serve to locate half a dozen entries or thereabouts. After that, the search for more 1’s and 2’s is likely to lead nowhere. Instead, one observes that there are six larger regions (of size 3 or more), hence six 3’s most be located. Come to think about it, that is a lot for a 5×5 grid; the 3 is the dominant value here. Among the numbers one should have found in the first run is a 3 in R3C1. Now, four of the larger regions lie entirely in the top three rows. This means, the upper 3×5 section of the grid must accomodate four 3’s, and in particular two in the same row. One can verify that the only possible configuration uses another 3 in R3C5, and then one further entry of 3 in each of the two top rows.

Such “block arguments” are frequently necessary when there are a lot of bigger regions – of equal or similar size – in close proximity. Most of the puzzles in this set work primarily with regions of size 3 and 4 (plus a few even larger ones on occasion, as well as some small ones as entry points here and there). In my experience, this is a typical look for Hakyuu puzzles in contests, so the search for 3’s and 4’s has a good chance of success if nothing else works.

That is about all there is regarding solving techniques for Hakyuu puzzles. I mean, there are a couple of special constellations which could be considered independent techniques, but I am not aware of anything substantially different or deeper than what I wrote above. Larger Hakyuu puzzles are often more difficult simply due to the fact that one must search for the next step, not because that step is particularly hard. And in that regard, several of the 7×7 puzzles from today’s set are quite demanding. Have fun!