**Rules:** Draw a snake into the grid that does not touch itself, not even diagonally. The snake cells are numbered; head, tail and perhaps more parts of the snake are given (in particular, the cell containing the number 1 and the cell containing the highest number must mark the two ends of the snake). The numbers outside the grid indicate how many snake cells appear in the respective row or column, including those already given.

**Example and solution:**

Snake is a popular puzzle type which has been around for quite a while. It is worth mentioning that there are several rule sets for Snakes; they are of course extremely similar, but some of the details vary. For example, sometimes the snake cells are not numbered (hence the length of the snake segments between the given parts is not known). Also, head and tail may not be given but most be located as part of the solve. Finally, there are Snake puzzles with full information outside the grid.

It is difficult to say which version of the rules is the most prevalent or which comes closest to a “standard”. Personally, I prefer the instructions as given above. In particular, as was the case for some other puzzle types (e.g. Magnets), incomplete information can be a good idea to distinguish certain solving steps.

The non-touching rule has been the topic of some discussion – well, not the rule itself, but the wording. Basically, it says that there cannot be any enclosed area of empty cells (every cell not part of the snake must be connected to the border of the grid). However, just in case you want to be a smartass, it is also supposed to mean that there cannot be any 2×2 blocks of snake cells. Every cell occupied by the snake except head and tail must have exactly two adjacent snake cells, and whenever two snake parts lie in neighboring cells, the difference of the respective numbers must be minimal (1 for orthogonally adjacent cells, 2 for diagonally touching cells).

This brings us to the matter of notation. It can be painfully slow to use the full notation (i.e. enter numbers for all the snake cells and also mark all other cells as empty). In practice, many solvers prefer to simply draw a line for the snake and check the unused cells only if they are somehow important for the solving logic. The notation of a simple line is typically accepted in live contests. As for the snake cell numbers, it may be sufficient to enter them for very few cells where the numbering is actually relevant.

The point is, Snake is a highly intuitive puzzle type which can often be solved incredibly fast. Top solvers will therefore avoid “slow” logic in the heat of battle. In fact, it is quite common that a solver studies the general global behavior of the snake, draws something which comes close to a solution and eventually tries to make some adjustments so that all the clues are satisfied. (This what I usually do myself.)

Let us discuss solving steps. Since these puzzles are so intuitive, it feels a little weird to go very deep with logical explanations; I will only sketch a couple of techniques here. In the end, I believe one most discover Snakes by oneself in order to become really good at them.

As with many other puzzle styles, the most informative clues (outside the grid) are either very large or very small ones. Intermediate clues typically do not yield anything interesting at the early solving stages. It should be pointed out that a clue of 0 virtually never occurs: Since snakes are by definition connected, a row or column with a zero clue would imply that the entire snake lies on one side of the clue. But then one obtains a whole bunch of zeroes, and the entire puzzle could simply be presented on a smaller grid right away.

There is nothing deep about large clues. When the clue value equals the grid size (as in the bottom row of Snake 1 from the PDF file), all the cells in the respective row/column must be part of the snake, and one can draw a long segment right away. By the way, this is already an instance where the full notation is useless – after all, we do not know the exact numbers for the snake cells at this point.

Note that the cells R6C2 through R6C6 must all be unused due to the non-touching rule (this always happens near the intermediate parts of a known snake segment). As a consequence, all the remaining cells in Column 2 must again be part of the snake. Such “clue reductions” following other steps are typical for Snakes, they serve to exploit intermediate clues at later stages.

The next point of interest is that all “open ends” of snake segments we have located must somehow be connected with each other, and there cannot be any dead ends. In particular, there is only one way to go from the snake cell R7C1, and likewise for the second corner cell R7C7. The cells R6C1 and R5C1 as well as R6C7 and R5C7 must therefore be part of the snake.

Let us briefly study small clues. A clue of 1 – as in Snake 2 from today’s puzzle sheet – means that the row/column in question can be crossed exactly once. Ergo, there is one part of the snake on one side of the clue, then something we might call a “passage” or a “transition”, and finally another part of the snake entirely on the other side. This does not give us as much as the large clues in Snake 1, but it is something worth keeping in mind. Among other things, it tells us that the snake cannot go left from the head (the cell marked with the number 1), hence this cell has to remain empty.

A clue of 2 can either be used for a single transition – which means the snake remains for one step in the critical row/column – or for two passages. Have another look at the same grid: The snake must visit the rightmost column (because of the clue outside the grid), and then it has to return to the left part. It follows that the clue of 2 above Column 5 must be used for two transitions.

Moving on. There is an interesting observation which turns out to be the starting point for a great many Snake puzzles. Suppose a snake enters and leaves a boundary row/column, then it must occupy at least three of its cells (try to understand the essence of this statement). This implies, for example, that such a row/column with a clue of 3, 4 or 5 must have a single contiguous segment there. The situation occurs several times in the puzzles from today’s set.

The possibilities for the respective segment are often limited, and they may share common cells which must be part of the snake in any case. For instance, the 5 in the bottom row of Snake 2 must occupy the cells R7C3, R7C4 and R7C5. (This inference can be coupled with the implications from the clue of 1 above Column 3.)

Please note that the previous argument regarding clues in boundary rows and columns only works when the end points of the snake are known to lie outside the row/column in question. In the first puzzle we see a snake with a tail in a corner, hence the implications for the rightmost column are different: We know from earlier that there is a segment of length 3 in the bottom-right part, and the fourth cell in Column 7 is already given. Thus no other cell in that column can be part of the snake.

By the way, if the puzzle rules are such that head and/or tail of the Snake are a priori unknown, then this kind of step collapses entirely. Incidentally, this is one major reason why I do not like the alternative rule set: Many logical steps lose their power (at least some of it), and the solving path appears clumsy and sluggish.

There are some other interesting clue constellations, but I will only mention one more. When there are several large clues in adjacent rows/column, it may be necessary to pack the snake cells in this part of the grid as tightly as possible. It is virtually impossible to give a precise formula, so let us just look at Snake 4 for an example. The 5 and 6 near the eastern border of the grid leave very few options; you will note that a single long segment in Column 7 leads to a contradiction. Therefore, the snake must enter the rightmost column twice, and there is only one possible pathway in this region to satisfy both clues at the same time.

We finish our introduction to Snakes with a few thoughts concerning solving steps which either do not use the clues at all or which are heavily based on snake parts in the grid that are already known. One step of major importance is that certain open ends cannot be connected prematurely. This is where the numbered snake cells come into play.

In Snake 3, for example, the given snake cells R2C3 and R4C4 cannot be connected directly. As we have remarked earlier, a connection is already established via a diagonally touching cell. As a consequence, neither of the cells R3C3 and R3C4 can be used by the snake. A similar situation is present in Snake 4. This may not strike you as a big deal; the point is, this kind of argument occurs all the time, basically whenever a new piece of the snake has been located.

Such steps can occasionally be combined with the implications from clues. For instance, in Snake 2 we deduced from the clue of 1 that there must be a transition between Column 2 and Column 4, but the passage cannot lie in any of the three top rows. Also note that a single large clue can be affected by the presence of a known snake cell nearby, as in Snake 3: The cells R1C2, R1C3 and R1C4 cannot all be part of the snake at the same time, hence the other four cells in the top row must be used by the snake.

Finally, regarding the numbers of the snake cells: They can be used to decide whether snake parts which must be connected can be united directly or via a detour. For example, you will find that the cell R7C6 is part of the snake and carries the number 7 (or 15, but the latter is impossible). Now, the 1 and the 7 are six steps apart no matter what, so they must be brought together without any detour.

On the other hand, the cell R1C6 in the same grid will be occupied by the snake cell 17, and this is quite a long way from the 26 nearby. The two must eventually be linked, but since their grid distance is five steps, a detour covering four more snake cells must be found. Likewise, the top-left corner cell in Snake 3 (which we have seen to be part of the snake a few paragraphs earlier) must be linked to the 17, but not directly: once again, some extra steps are required.

I am aware that some of the last explanations were brief and vague. As I said before, I believe one must explore Snakes by oneself to some degree in order to get really familiar with them. Anyway, enjoy the puzzles.

]]>To be honest, I do not have a lot of experience with Blackout Dominoes. I have solved them a handful of times, and I cannot recall ever designing one. This means, among other things, I am not sure if the puzzles you will find in the PDF file below are “typical” specimens of this style. Anyway, I hope you enjoy the puzzles. Here is today’s set: Blackout Dominoes

**Rules:** Shade some cells, divide the remaining unshaded cells into dominoes, and enter numbers into the unshaded cells, so that each domino from the given set appears exactly once. Horizontally or vertically adjacent cells that are part of different dominoes must contain equal numbers. Shaded cells cannot be horizontally or vertically adjacent to each other or to the border of the grid (they may touch each other or the border diagonally). Cells with numbers cannot be shaded.

**Example and solution:**

Blackout Dominoes is one of rather few puzzle types which regularly use an irregular grid, if you know what I mean. (Kakuro is another, but I do not think we have had any others with this property in the Classics Collection.) I have seen puzzle authors design them in a way that the outcome features some creative imagery. Personally, I prefer more mundane, compact shapes.

The next thing I would like to mention is that authors sometimes give shaded cells or walls inside the grid as clues. This is something I usually try to avoid, just as I prefer Skyscrapers with no given entries. Obviously one needs a few given numbers in Blackout Dominoes, since otherwise the solution cannot be unique, but you will find no other clues in today’s puzzles.

When it comes to solving techniques, it turns out that the solution virtually always breaks down into two parts. The first is about the dissection of the grid into dominoes (and shaded cells), the second is about entering the numbers. You will find that there is limited interaction between the two parts; most of the dissection typically follows from the grid shape and the givens. Only very few freedoms remain, especially in smaller grids.

The dissection usually starts at the border of the grid, near “humps”. Since border cells are not allowed to be shaded, a domino position can be inferred if a cell is connected to only one other. For example, in the puzzle BD1 (short for “Blackout Dominoes 1”) from the PDF file, the two leftmost cells instantly lead to dominoes. Such steps can easily lead to more humps, as with the cell containing a 1 in the bottom row.

Humps with a width of two cells are also interesting. The crucial observation is that two dominoes cannot lie together to form a 2×2 square, since they would have to contain the same pair of numbers. As a consequence, dominoes cannot go from such a hump inside the grid. Have a look at BD2: If one of the two leftmost cells was part of a horizontal domino, so would be the other, violating the above principle. Therefore, the only possibility for these two cells is a vertical domino.

Note that this has implications for the cells close to the hump. In our case, there cannot be a second vertical domino next to the first one, so the lower of the adjacent cells – which lies on the border again – must be part of a horizontal domino. This leads to more humps in the bottom row.

Another hump of width 2 can be seen in the top row, which requires a horizontal cut. This time, neither of the two adjacent cells lies on the border. It is easy to make a mistake at this point, trying to draw more dominoes here, but in fact one of them might be shaded. The only thing that is certain is that they cannot form another horizontal domino. I suggest drawing a wall between them, even though it is not clear yet if this wall separates two dominoes or a domino and a shaded cell.

There are other interesting constellations, but I will only mention one of them at this point. Take a look at the second row from the top in BD3, in particular the second cell from the left in said row. (I will refrain from using the RxCy notation here.) The cell in question cannot be connected to the cell above, because that would yield two vertical dominoes in a 2×2 block. Likewise, it cannot be connected to the cell to its left.

Thus we obtain a few more walls. In some cases, they do not seem to do us much good; every now and then they serve to locate a shaded cell (or a domino), though. For instance, the respective cell near the top-right corner of BD1 cannot be connected to any other cell, applying the same technique four times. In general, this is a ueful step for humps of size 3.

One more thing regarding the dissection part. The total number of unshaded cells is always known from the start. (As you can see from the list of available dominoes, exactly 20 cells are needed for the 0-3 set, 30 for the 0-4 set and 42 for the 0-5 set.) It might be a good idea to count the cells in a grid right away, because if can be helpful to know the exact number of shaded cells.

The point is that only the interior cells can be shaded as per the rules. And sometimes there is not much room for as many shaded cells. For example, you can determine that three cells must be shaded in BD1; using the early hump steps, only one possibility remains without any shaded cells being adjacent to each other or the border.

We are approaching the part where the numbers come into play. Before we get to the actual techniques, you may notice that the unshaded cells form “groups” which must be filled with identical numbers – basically every time the cells are separated by walls.

Imagine for a second that all numbers are removed from the solved example grid, and only the walls and the shaded cells remain. The two upper cells in the leftmost column must contain the same number, i.e. they form a group of size 2. The lowest cell in the same column forms another group of size 2 with the cell right next to it. In the second row from the right there is a group of size 3. And so on.

Please observe that, near the top hump, we even have a group of size 5! Larger groups occur every time one domino touches two others in a 2×2 block. Note that this constellation contains in particular a domino with twice the same number – this may not sound like a big revelation, but the study of locations for those specific dominoes can be useful because they need either a large group or two adjacent groups.

The notion of “groups” is pretty much the foundation for all the filling that follows in the second part of the solve. If one entry in such a group is given, the rest can be entered immediately. For instance, it will turn out that there is a large group near the bottom of BD1, with a 1 inside it. This yields a lot of 1’s in this part of the grid.

Although it is primarily relevant for authors, I would like to point out that Blackout Dominoes are surprisingly rigid when it comes to creating them. Each number most occur exactly the same number of times (e.g. five times with the 0-3 domino set). The grid must be designed such that the groups work out accordingly.

Back to the solver’s perspective. Suppose we already have located a number four times (using the 0-3 set again). This means the same entry has to come up exactly once more. This requires a “dead end”, usually in a hump of size 1, but keep in mind that dead ends can also occur inside the grid using shaded cells.

An entry with three known occurrences needs two more, often in a group of size 2. (Two dead ends would be possible, but this is rather rare; one fitting group can be encountered far more often.) In larger puzzles, there are various ways to reach the required number of occurrences. As we have already seen, one single large group is possible.

In practice, large groups are quite common, since it is hard to have a unique solution otherwise. However, the groups cannot be too big. In areas which are known from earlier steps to contain many unshaded cells, shaded cells nearby are often needed to limit the scope of the respective groups.

Look at the the second cell from the right in the second row from the bottom in BD2. The early hump techniques give us a couple of domino positions in this part of the grid. If the cell in question was unshaded, it would create a rather large cluster with more than five equal entries – just try it. (This argument often leads to minor bifurcations.)

Let us continue with a couple of practical steps. If two adjacent numbers are given (as in BD2 or BD4), they must belong to the same domino. Remember, number cells cannot be shaded, and if there was a wall separating them, it would in effect create a group of size at least 2 with two different entries.

Another constellation I like to use consists of two different givens in cells with a common neighbor (as in two diagonally touching cells). The situation can be seen in BD3; look at the cell adjacent to the 1 and the 2. Such a cell can be shaded, or it can be connected to one of the given numbers. However, it cannot be part of a domino in one of the other two directions, or else it would create a group of size 3 with different entries.

As a consequence, a cell adjacent to three different entries must be shaded; it simply cannot accommodate the various groups next to it. This constellation comes up several times in this set, in BD3 as well as some of the larger puzzles.

When you are solving a Blackout Dominoes puzzle, starting the dissection from the humps and working your way down to the center of the grid (exploiting the above number constellations in the process), you will often reach a stage where large portions of the grid are filled and some minor pieces remain. Typically there are a few groups of size 2 and 3, and maybe a couple of dead ends as well.

When that happens, you will need some smaller, less impressive techniques to complete the grid. There is no brilliant strategy for this part of the solve, it is just a fuzzy bunch of tiny steps. Basically, you will have to use the rule that each domino must occur exactly once in the solution.

Suppose a large group has been located and some of its neighbors are already known. That group often has a few more “exits” which must then contain other numbers. For example, you will find a large group of 3’s in the east side of the BD2 grid (including a 3-3 domino), and the 1-3 domino has already been used, too. The cells adjacent to the remaining 3’s must therefore contain a 0 and a 2.

Of course this technique works just as well with several smaller groups containing the same entry, as long as many of its neighboring cells are already filled. Basically, whenever there is a large number of identical entries, it may be a good idea to study which of its potential “partners” have been used so far. Ideally this yields new entries, but even sets of candidates may help.

If a group of cells – in the above sense – is empty, a few deductions about its neighbors are still possible. In particular, two cells touching the same group cannot contain the same number; otherwise it would give rise to two identical dominoes. In the simplest case, the “outer” cells of two touching dominoes must contain two different entries. This technique serves only to eliminate a candidate in a specific position, but it may still be useful to know this step.

Finally, I want to stress once more that dominoes containing the same number twice are a good tool when it comes to the final filling stage. The point is that these dominoes cannot occur in isolation, they need more space. In BD1, for example, you will reach a position with several empty cells in the top-right part of the grid. For dominoes like 0-2 or 0-3 there will be several possible locations left, so you may want to study locations of the dominoes 0-0, 2-2 and 3-3 instead.

So much about solving techniques. Have fun!

]]>I have designed a small bunch of puzzles on other grids – hexagonal ones, to be precise. This is still not quite the full spectrum, but it feels like a decent start. The puzzle instructions are the same as for the respective styles on square cell grids (or reasonable approximations, whenever the geometry is relevant). And I have made an effort to design at least some of the puzzles suitable for beginners, in accordance with the CC spirit.

The only snag: you will have to find the puzzles first. There is a new section on my website labelled “Hex Maze”, and I have hidden them in there. (Yes, I could have made the puzzles available in a more straightforward manner, but I wanted to have some fun in the process.) The idea is to extend this project to other grid types at some point. I find it difficult to plan these things ahead, though, hence once again I will make no promises.

]]>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!

]]>I will be critical of certain aspects of the Convention, but before I get to the actual discussion, some acknowledgements are in order to the people who made this event possible. First and foremost, my thanks go to the puzzle authors, and also to the test solvers. The contests were hosted by Logic Masters India, so let me extend my thanks to the people who have been working on their website. Finally, I am sure there are many more organizers and others who deserve a mention and who I am missing right now, so let me just say they have my gratitude as well.

Regarding the course of events, let me start with the bad news. I found it very unfortunate that the fundamental announcements were made at such short notice. I do not care much about minor gaps in the instruction booklets; it just so happens that there was very little opportunity to generally prepare for the Convention schedule-wise. I do not know how others feel about this matter, but it had a major impact on my perception of the WS+PC.

If this had been a live event of the same magnitude, people would have taken a week off (making arrangments half a year earlier or so) and, even without a fully detailed schedule at hand, I am sure the WSPC week would pretty much have their undivided attention. Such is usually not the case for online events, even for those of the highest order. Obviously I can only speak for myself, but in essence I just lived my everyday life and added a few puzzle rounds each day.

Until a week before the Convention started, I had no idea about what other “gatherings” would be there, so I did not make time for any social (or pseudo-social) activities in the WS+PC context. As a result, I simply took part in the two main events and skipped all the Bonus Rounds and Time Trials. I briefly looked into two of the Panel chats some hours later, when they were available as YouTube videos, but did not join in on any of them in real time.

Do not get me wrong. I do not think that I would have taken a week off even if I had seen a schedule a month earlier, and I would probably not have taken part in all the activities beyond the 18 main rounds in any case. Still, I was reluctant (at least initially) to view the Convention as something special. My point is that I was in my usual 9-to-5 routine and did merely a little extra puzzle-solving. Perhaps this is as it should be after all.

Now, about the puzzles. I felt that the puzzle selection was – mostly – very good; the Convention featured a wide range of puzzles and variants, and the mix of puzzle categories was nicely balanced. Most of the rounds were of an assorted kind, not thematic ones. But then, there is no real justification to include a lot of thematic rounds (perhaps apart from traditional reasons). The above is an observation on the Puzzle contest in the first place, but it also applies to the Sudoku part.

It should not be surprising that I liked some of the puzzle styles more than others. The Dim Sum puzzle was the one I would consider the most dispensable. (While the other puzzles of the round were being printed, I had a brief glimpse at the graphics and then, without doing any actual calculations, tried the two which looked like the most abundant ones.) The Mahjong Mazes were also painful, but given the large number of participants from certain Asian countries, I guess it is somewhat fair to include some symbolism that Western solvers are unaccustomed with every now and then.

I came to enjoy several puzzle types and variants which I had not encountered before or which were otherwise unfamiliar (Norinori, Sashigane, Place by Product, Nanro Cave, Index Yajilin, Five Cells). Also, some Sudoku variants (Max Triplet, Coded Pairs, Search Nine, etc.) were more fun than I expected. In some cases I was held back by my lack of preparation; on other occasions I made stupid mistakes. Still, I want it on record that I liked quite a few puzzles which I might not have given a chance without the Convention.

The Skyscrapers round was a disappointment, I am afraid. It is not just the puzzles; I think the entire round concept was flawed. The multi-grid rule set with shared clues may work reasonably well for a small number of puzzles, but it is simply not rich enough to support nine individual grids of the same size in a single large pack. I would have preferred a slightly larger variety in this round.

This brings us to the matter of results. The Skyscrapers were the only round where I had a very clear ambition. Apart from that, I hoped to end up in the Top 20 in the overall table of the Puzzle Convention (i.e. the Top 10 outside of Japan), but it was not to be. In the Sudoku event I missed the Top 50, but since I never was particularly good at Sudokus, it is no big deal for me. All the results should be taken with a grain of salt anyway, considering that some excellent puzzle solvers did not participate at all.

Philipp and Martin ended up ahead of me, and it seems we have a new star in the German puzzle solving sky: Christian König (nickname CJK) worked his way up to one of the German top positions in what appears to be less than a year – at least I do not recall seeing him in earlier German championships. Among other things, it means I can barely hope to get a spot in the German B-team, let alone the A-team, should I eventually make another attempt to qualify for a future WPC.

In the international competition, the number of participants who finished ahead of me is not as scary as the margin by which the top solvers win these days. Ken, Walker and Freddie are truly a league of their own these days (I am pretty sure even Ulrich will agree with me on that). The Sudoku results are perhaps not quite as telling, although there is again a considerable distance between rank 3 and 4.

Finally, some words on my solving strategy. When I restarted this blog, I wrote about contest strategies and “comfort lists”. Well, I barely did any preparation beforehand at all, and it was usually my approach to start with the expensive puzzles – except in cases when I really disliked the puzzle style in question. In general this seems to work well for me, unless I make errors and break the puzzles (which, I regret to say, happens far too often these days).

There is something that bothers me, though. As mentioned above, there are three participants who are so far ahead that they can usually complete all the puzzles no matter what (and I am not in a position to judge strategic matters in their case). Among the mere mortals, I cannot help but feeling that the latest contests seem to reward bifurcation, intuition and – occasionally – sheer guesswork over the application of cold logic. At least it appears to me that every hard puzzle I attempt can eventually be cracked faster by means other than logical solving techniques.

An extreme example, to give you an idea what I mean: In my very first round of the Convention, when I was still trying to solve them logically, I broke four Sudokus in a row; this took more than half the round and cost me 150 points or thereabouts. On the other hand, when I looked at the Big Tents puzzle (from the Triple Jump round of the Puzzle Convention), I decided to just place a dozen of tents where they fitted best with the clues and did not obstruct each other. This took no more than 90 seconds, including printing, and was rewarded with an instant 80 points. A week later, I still do not know what the logical solving path looks like.

This is regrettable, but let me clarify. My perception is not limited to the WS+PC; it seems to apply to most contests I have taken part in lately. Also, it may be just my own approach, and others may feel completely different about it. And still, I am confident that I am not alone with this. Not too long ago, Ulrich has expressed a similar sentiment: that it is a pity he cannot fully appreciate puzzles he solves under time pressure, because the most successful route in a contest is usually not the most beautiful or elegant one.

Anyway, the Convention confirmed that I am no longer a match for the top solvers out there, unless I have a very lucky streak. (It may even be that I am making a virtue out of necessity, playing for my strengths when it comes to intuitive solving. Chess players who have seen my more recent games will know what I mean.) Perhaps I will get another chance to mess with the best; in particular, I would really love to see Toronto. So long.

]]>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!

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