Hi folks,

for the remaining five sets of our Classics Collection I have a somewhat clear schedule in mind. If nothing goes wrong, we will have today’s set and two more in 2021 and then the last two in early 2022, so that the fifth Beginners contest can take place in February or March. I have not yet decided what will come next on this blog after the CC; meanwhile, there is a WSPC substitute forthcoming, and I guess there will be new GP series (and other events) as well after the turn of the year.

The puzzle styles I am preparing for the CC groups 21-25 are perhaps not as exciting as the ones we already had. In fact, they appear not to be as rich in solving logic as some of the styles we have seen so far, which is why the new sets have only four puzzles of the smallest size (to demonstrate the basic solving techniques) and then two larger sizes of four more puzzles each. We are starting with Kropki; here is the next set: Kropki

**Rules:** Enter numbers from 1 to N into the grid, so that each number appears exactly once in each row and column. A white circle between two adjacent cells indicates that the numbers in these cells have a difference of 1. A black circle indicates that one of the numbers is twice the other. If there is no circle between two horizontally or vertically adjacent cells, then the numbers in these cells cannot fulfil either condition. A circle between 1 and 2 can be either white or black.

**Example and solution:**

Please note that the negative constraint – the stipulation that adjacent entries are not allowed to be consecutive or have a 1:2 ratio in the absence of a circle – is an integral part of the Kropki rule set. (Without it the example would not have a unique solution, as the 4’s and 5’s in the 2×2 block in the bottom-left corner could be switched.) Recently I have seen numerous Sudokus incorporating white and black circles with the same meaning as in Kropki, but the designation as “Kropki Sudoku” without the negative constraint is highly misleading in my opinion and should be avoided; I prefer the term “Kropki Pairs” for this variant.

The first thing to observe is that there are usually plenty of circles. Try plotting down a Latin Square at random (of reasonable contest size, i.e. somewhere between 6×6 and 9×9), and you will find that there are typically many places where pairs of consecutive numbers or numbers with a 1:2 ratio have ended up in adjacent squares. Assuming that there are no givens, it may be difficult to determine the best starting point.

It turns out that black circles are generally more informative. Every single digit can be next to a white circle, and there are N-1 pairs of consecutive entries comprised of numbers from 1 to N. On the other hand, the number of constellations which induce a black circle is smaller (it grows only logarithmically with the grid size, for the mathematically versed readers), and large odd numbers are impossible in a pair of cells linked by a black circle. Basically, it suffices to know the number combinations 1-2, 2-4, 3-6 and 4-8 (assuming the grid is large enough for the latter).

Still, a single black circle – just like a white circle – does not provide nearly enough information to enter any number with certainty. The best constellations to make progress are sequences of circles, ideally within a single row or column. Have a look at Row 2 of Kropki 1 in the PDF file: The only way to fill the first three cells such that the two black circles are satisfied is with the numbers 1-2-4 (in either orientation). The same applies to Column 3.

As a result, one obtains a bunch of Candidates for the respective cells. (I am under the impression that experienced solvers employ Candidates – a valid technique in virtually all Filling puzzles – less often in Kropki puzzles than in Sudokus or the like.) In particular, the cells R2C2 and R3C3 must contain 2’s, even if the other cells of the above sequences remain ambiguous. The cell R4C4 must also contain a 2, using the horizontal black circle in Row 4.

Most Kropki puzzles have a massive break-in of this sort and then many small logical steps to complete the rest of the puzzle. The point is that both circles and “non-circles” carry information, hence each entry creates new implications en masse. After getting started with a puzzle, the difficulty often lies in the question where to look next. In this case, the white circle between R3C4 and R4C4 tells us that the upper of the two cells must contain a 1 or a 3, but the former is impossible since it would force another 2 in R2C4, i.e. in the same column. Likewise, a 1 in R4C3 leads to a contradiction in R4C2, thus resolving the entire sequence of black circles.

It is not hard to follow up on the first steps, simply by searching in the immediate vicinity of the entries one has located so far. But it should not come as a big surprise that a large number of circles can cloud the issue. In many cases this is not a big deal because one logical path is just substituted for another – maybe not quite as clear or elegant as the first one, but workable nonetheless.

Still, I am not comfortable with Kropki puzzles that have little more substance than jumping from one circle to the next. In my opinion, the negative constraint should play a more prominent role during the solve. That is why I am usually trying to design the puzzles so that they have few circles, that is, fewer than one would expect for a puzzle of the respective size. Not only does it lead to a higher probability that the absence of circles in certain places must be used, it also makes it easier to identify the starting points.

As with the circles, a crucial part in the exploitation of non-circles is about where to look next. The best places are cells which are adjacent to already known entries (ideally several ones) which are “linked” – in terms of Kropki circles – to many other numbers, such as 2 or 4. For instance, a cell next to a 2 (without a circle) cannot contain a 1, 2, 3 or 4. The number 4 eliminates the candidates 2, 3, 4 and 5, and even 8 if the grid is large enough. Every once in a while, only one possible entry remains.

An extension of this technique can be seen by studying an entire row or column which has already several of its cells filled. Even without the benefit of known entries in a neighboring row/column, there may be very few possibilities to complete the row/column in question such that all the constraints from circles and non-circles are satisfied. In order to understand what I mean, let us have a look at Column 5 in Kropki 2.

As with the previous puzzle, the numbers 1-2-4 must occupy the cells covered by the sequence of black circles (in either direction). This leaves the numbers 3, 5 and 6 for the upper three cells in the same column. The white circle between R1C5 and R2C5 can only be satisfied by the consecutive pair 5-6, which means that the cell R3C5 must contain a 3. At this point, the rest of the column follows as well, since a 6 cannot be entered in R2C5 due to the lack of another black circle, and another white circle would be required if R4C5 contained an entry of 4.

Although it is not relevant at this moment, it is noteworthy that there would be no way to fill the upper three cells with the same numbers 3, 5 and 6 such that there are no further circles at all (apart from the two black circles in the lower half). The entry of 6 is linked – in the above sense – to both 3 and 5; whereever it goes, it must have a circle next to it. This would only work if the sequence 1-2-4 were in the middle of a row/column, the 6 would go on one side of it and the entries 3 and 5 on the other side. Incidentally, the same holds for a 7×7 grid (see Kropki 7), and even for an 8×8 grid if there are three black circles forcing a 1-2-4-8 sequence. Also, three consecutive numbers (e.g. 4, 5 and 6) can never be placed in three consecutive cells without at least one white circle between them.

In general, entire rows/columns can be decent starting points if they contain a “powerful” constellation of circles. See Column 4 in Kropki 3, for instance; it is almost full of circles and serves as a decent starting point. One could check the various options to enter numbers in certain positions (for example satisfying a particular circle or group of circles) and see which of them lead to a contradiction; this is certainly a reasonable strategy for contests, since one can surmise the extent of a full case differentiation without much difficulty. Sometimes, however, there is a more subtle solution.

In the present case, one could wonder which cell in said column contains a 5. Remember, a high odd number cannot lie next to a black circle, and the only cell without one is R6C4. Next, there is only one possible location for the 6, namely R1C4, since a 6 next to a white circle within the column would force another 5 as part of the consecutive pair. This leaves a 3 in R2C4, and so on. There is no playbook for such steps, one simply has to gather some experience with Kropki logic.

Kropki 4 features a useful sequence of white circles in Row 3. A priori there are three possibilities for the respective entries: 1-4, 2-5 or 3-6 (with two potential orientations in each case). Once again, the lack of a circle between the remaining cells R3C1 and R3C2 provides the decisive clue: those numbers can be neither 1 and 2 nor 5 and 6. Therefore, the four-cell sequence must contain the numbers 2-5.

Furthermore, the entry in R3C6 is linked to another cell via a black circle, which means the number 2 must lie at this end. It is generally advisable to look out for additional black circles connected to the sequence one is investigating; they can often reduce the number of available constellations considerably, sometimes even to a single possibility as in this case.

So far we have mostly studied sequences of circles within a single row/column, and there is a reason for it. Sequences which make turns are generally less significant because numbers can repeat along them. For example, two black circles connecting an L-shaped group of three cells could accommodate the numbers 1-2-4 or 4-2-1, but also 1-2-1, 2-1-2, 2-4-2 and 4-2-4, and let us not forget 3-6-3 or 6-3-6. Among other things, there is no longer any certainty about the entry in the middle.

However, there is one formation of particular interest, namely three circles around a 2×2 block. For example, consider the cells R5C2, R5C1, R6C1 and R6C2 in Kropki 3 from today’s file which are linked by three white circles. The cells R5C2 and R6C1 cannot contain the same number; otherwise it were impossible for the entry in R6C2 to have a white circle in the one direction but not the other. Likewise, R5C1 and R6C2 must contain different numbers.

Since we are dealing with pairs of consecutive numbers, the four entries must form a strictly monotonous sequence (and, by the way, the sequence 3-4-5-6 is impossible due to the lack of a black circle in the fourth position). A similar argument – ruling out repeating entries – applies if one of the “outer” circles in the same arrangement is black, and also if all three are black or one of the outer circles is white. Such constellations can be found in various places in our set. Note, however, that this step no longer works if the circle in the middle is of the opposite color, as in the bottom-right corner of Kropki 3 or the bottom-left corner of Kropki 4.

There are a great many interesting constellations we have not covered yet, but it makes little sense to list them here; in order to get better with Kropki puzzles, one has

to discover them on one’s own. Instead, it remains for me to point out that the last stage of a Kropki solve is typically unstructured because one can make progress all over the place. Kropki is a rather rigid puzzle style, and the abundance of information (circles and non-circles) means that the final deductions can often be made in virtually any order.

Let me stress once more that my puzzles rely quite heavily on the negative constraint, probably much more so than creations by other authors. Also, the initial techniques for some of today’s puzzles are sometimes rather deep – perhaps a bit harder than one could expect in a Beginners set, but I believe it is worth seeing such examples with non-trivial starting points. That’s all for now; enjoy the puzzles!