**Rules:** Enter numbers from 1 to 6 (1 to 4 in the example) so that each number appears exactly once in every row, every column and every outlined region. A clue outside the grid indicates which numbers are adjacent to the entry 1 in the respective row/column. (In that case all numbers are given, i.e. a single number clue indicates that there is only one number adjacent to 1.)

**Example and solution:**

This is the first variant in the Sudoku Compilation featuring clues outside the grid which restrict the contents of a row or column. Some others of the same category are X-Sums Sudoku or Sandwich Sudoku. Those, however, are much harder to handle on a 6×6 grid, since they are based on arithmetic techniques which are severely limited with fewer numbers available.

Before creating this set, I barely had any experience with the variant, and I actually struggled a lot to come up with six interesting puzzles. (You will note, for example, that the first two puzzles have very similar break-ins.) Maybe it has to do with my reluctance to use any givens; as with a lot of variants, I prefer to work without givens if they are not needed for uniqueness.

All in all, I felt that the rules have very little potential for exciting solving steps, and that most of what I did here was repetitive, to say the least. And still, I was warming up to the variant while I kept going. You will find (I hope) that the last few puzzles contain some beautiful tricky steps; even if the hard Sudokus in our set can once again be solved much faster using bifurcation, it is worth exploring the logic in them. Below you will find a hint for the sixth puzzle on the PDF sheet.

**Hint:** Study the possible locations of the entry 1 in Column 6, and in particular the Candidates for the cells R3C6 and R4C6. After that, examine which cells in Column 4 can accommodate the number 6.

By the way: When you encounter today’s variant on a 9×9 grid in an event, it is often called “Next To Nine Sudoku” instead (with the instructions adapted accordingly). The problem is that, if the clues indicated which entries are adjacent to the largest number, the puzzle designation would have to be adjusted every time the grid size changes. I believe it is desirable to have a uniform name for all possible sizes, which is why I prefer to use the “Next To One” version here. Anyway, enjoy the puzzles!

]]>**Rules:** Enter numbers from 1 to 6 (1 to 4 in the example) so that each number appears exactly once in every row, every column and every outlined region. Cells with grey squares must contain even numbers. Cells with grey circles must contain odd numbers.

**Example and solution:**

To be quite frank, Even/Odd Sudokus are not what I would call an exciting variant. That is because Even/Odd clues do not contribute any additional element to Sudoku. This variant belongs to the category of Candidate Sudokus, i.e. the clues only limit the set of candidates for the respective cell, a concept which is already present in Standard Sudokus. As such, they do not bring forth any new solving techniques.

Nevertheless, they appear to be somewhat popular; I guess this has to do with the simplicity of the rules. By the way, when I designed Even/Odd Skyscrapers a long time ago, I used to denote clue cells with small letters E and O (or G and U for the German designations). I prefer the layout with grey squares and circles, which is highly intuitive and also much easier to survey, unless this variant is combined with another which also uses grey symbols.

Regarding puzzle theory, I would like to briefly mention that N-2 givens are needed in this variant (where N denotes the grid size). That is because the different entry values are interchangeable only as long as the parity is preserved. Therefore, in each group – Even and Odd – all numbers but one must be given from the start for the solution to be unique.

In practice, easy Even/Odd Sudokus tend to be solvable using Naked and Hidden Singles only (similar to Standard Sudokus, although the clues may be necessary for the Singles to emerge). More difficult specimens may require the corresponding harder techniques. Since the candidate structures are mostly based on the Even/Odd clues, so are the solving steps.

It is not uncommon that the first part of the solving part consists of entering only further squares and circles, covering a significant part of the grid, before even a single number can be located. In that case, the last part of the solve typically breaks down into two rather independent pieces, namely solving the Even half of the Sudoku and solving the Odd half.

As usual, I have tried to design the six Sudokus from our set so that the difficulty is increasing. The first few puzzles contain nothing deep, it may just be a little harder at certain occasions to spot the next step. If you need any help with the last puzzle, feel free to highlight the following paragraph to obtain a small hint.

**Hint:** The two cells R6C1 and R6C2 must contain the same entries as R5C4 and R5C6. All three locations of even numbers in Column 4 are thus known, and the parity of several more entries can be marked. In fact, it is possible (although not strictly necessary) to take it from there and divide the entire grid into Even and Odd cells before entering any numbers.

As I said earlier, this set is not really thrilling. But I hope you enjoy the puzzles nonetheless, and stay tuned – the next installment of our series is supposed to be coming soon.

]]>A few notes regarding the puzzles (primarily for those who have either seen them already or who have access to them): I had some reservations because, as it has been the case so often, the instructions for various innovative puzzle styles were overwhelming and appeared overloaded at times. But it turned out that many of my concerns were unfounded. I have often criticized puzzles, so it seems only fair that I also bring out the good things. The puzzles in Round 2 (“Die wilden Zwanziger”) were surprisingly accessible, and so were the Masyu variants and most of the puzzles in round 7 (Number systems) as well – at least those which I attempted to solve.

The Assorted Round covered much more stuff than one could handle in the given time, which is as it should be. I broke Round 6 (“Bewegung bitte”) entirely, which was of course my own fault (and which cost me pretty much all chances to reach a spot in the A-team). The last of the preliminary rounds, “Fishing for Complements” was hard to manage and overdid things a little in my opinion, but not by much, and in any case it is nice to have a manipulative round – of sorts – in the event.

All in all, it was a tough Championship (and it was a little depressing to see that, after missing the playoff spots, I managed to solve the playoff puzzles faster than three of the four finalists). But then, most of the puzzles were very nice, and there were no major disruptive incidents, which is the most important thing. Congratulations to Philipp and the runner-ups, and many thanks to all the organizers and helpers.

The secondary topic I would like to talk about is balance – not in terms of puzzle categories, which is something I have focused on frequently in the past, but regarding the scoring. There were several rounds which contained easy and hard puzzles, but no intermediate ones. As I see it, this is in general an undesirable round design.

For instance, one of the rounds contained two hard puzzles (70 and 75 points, respectively) and three easy ones with point values between 10 and 25. This gap in difficulty will usually produce a gap in the scoring table. Unless there is reason to believe that is impossible to finish with more than half the points anyway, the strong performers will likely start with the hard puzzles, because each of them is worth more than the entire rest. On the other hand, inexperienced participants may not attempt to solve them at all, since it costs too much time and risks ending up with no points at all.

There were two more rounds with similar traits; each time I started with the expensive puzzles, which was a successful strategy (at least in my case). I do not have the scoring tables available at this point, but my guess is that the results for these rounds broke down into two or three groups each, rather than a continuous spectrum of scores, according to the number of hard puzzles each participant solved.

The round design I prefer is based on a large number of easy puzzles and a gentle slope of increasing difficulty ranges with a decreasing number of puzzles for each range. Imagine, for simplicity: four puzzles worth 10 points, three puzzles worth 20 points, two puzzles worth 30 points and finally one puzzle worth 40 points. (One might call this the “Battleships model”.) I believe such a design has the best chance to produce a continuous scoring spectrum between the minimum of zero and the maximum of 200 points.

Obviously, the numbers must be adapted depending on the available puzzles and the length of the round. Still, I think this is better than having equal numbers of puzzles for each difficulty level or even a gap in the middle.

In fairness to the authors, this is quite an ambitious target, and in practice there are several aspects which should be taken into account. First of all, you have to live with what you have. If there are not enough puzzles available to achieve a perfectly balanced round, the priority should still lie on accuracy instead of balance. (Some special rounds may even require that the difficulty level increases in large steps only.) It would be questionable to distort the point values too much just to obtain the slope I mentioned earlier.

Next, it is not clear if the continuity in the scoring spectrum is really so important. After all, there may be a truly significant disparity in the overall performance between different groups of participants. If memory serves, the difference between my own score and the cut for the playoffs was about 200 points; it would have been a lot more frustrating if the fifth place had been separated from the playoff spots only by a much small margin.

Finally, it is worth noting that even a considerable number of (scoring-wise) unbalanced rounds in an event can be compensated for by one or two large balanced rounds. The lack of intermediate puzzles in some parts would still be there, but the gaps in the overall standing would disappear.

Let me therefore conclude by saying that the unbalanced rounds are not such a big deal for me. In a contest such as a national Puzzle Championship which is comprised of many individual rounds, the lack of balance in a few of them does not play a major role; it is something to keep in mind primarily if one wants to design a single-round event.

]]>I just managed to finish another installment of our Sudoku Compilation. To be honest, I am not as happy with today’s six puzzles as I was with those from the first two groups in the series. But then, it so happens that I find Clone Sudokus generally less exciting than other variants. They belong to a category one might call “Equality Sudokus”, meaning that cells are denoted which must contain the same entry (Palindrome Sudokus are another somewhat popular representative of the same category).

As I see it, such variants do not have much to offer when it comes to additional solving techniques. Basically, they are using the same stuff that is already known from Standard Sudokus (Naked and Hidden Singles, followed by Pointing Pairs, X-Wings, etc.), just in more confusing ways. You are welcome to disagree with my assessment, of course. Anyway, here is the new set: Clone Sudoku

**Rules:** Enter numbers from 1 to 6 (1 to 4 in the example) so that each number appears exactly once in every row, every column and every outlined region. Shaded regions with the same shape and orientation must contain identical entries in corresponding positions.

**Example and solution:**

When it comes to the rules, it should be clarified what counts as a “region”. As far as I know, it is considered standard to interpret only that which is orthogonally connected as a single region; marked areas which touch each other diagonally are treated as separate regions. The latter situation is present several times in the PDF file, and in particular in our example as well.

Next, let me stress that numbers may repeat within a Clone region. This is something which cannot be seen in the example and should therefore be emphasized. (For example, the Clone constellation in today’s fifth puzzle is such that the puzzle would – even without the givens! – have no solution at all unless the shaded 3×2 block has one pair of repeating entries.)

It should also be noted that the Clone regions may appear more than twice within a puzzle. Once again, this is a situation I have omitted in the example due to the very limited space on a 4×4 grid, but the first 6×6 puzzle features such occurrences. Observe, however, that the different orientations of the domino-shaped regions in the same puzzle give rise to two separate groups of Clones.

There is, as I mentioned above, little to say about solving logic. It is usually a good idea to study the candidates for shaded cells, using the Clone rule. It may well turn out that certain positions in the Clone regions admit only one entry. The first Sudoku from the PDF sheet can be solved using hardly anything else. What we have here is the Clone equivalent of Naked Singles. Such puzzles can be quite boring, but in fairness, the same argument can be brought up for (easy) Standard Sudokus, too.

I have attempted to design the remaining puzzles slightly harder, so that at least the Clone equivalent of a Pointing Pair is occasionally required. In the end you should not expect too much, though; this variant is not what I would describe as “rich”.

Regarding the theory of Clone Sudokus, I would like to mention one more thing. Unlike the first two variants we have presented, Thermo Sudokus and Killer Sudokus, this variant has the property that all the number values are interchangeable. Neither the natural order of integers nor any arithmetic operations are used in the instructions. As a consequence, at least N-1 givens are required in an NxN Clone Sudoku to ensure a unique solution.

Several of today’s puzzle will do with five givens. In general, this has to do with the size and number of the Clone Regions. You see, more and larger regions provide more constraints, so that in extreme cases the entire grid can be broken down into N sets of N cells each (which must contain the same number). This is a little similar to Irregular Sudokus: The more geometry one adds, the less entries are needed.

Such extremes can quickly get boring once more. I think, as with other variants, Clone Sudoku should ideally have a moderate amount of shaded regions, so that the Clone rule is substantially required during the solve, but not so much that the Clone dynamics take over.

Difficulty-wise, I think none of the six puzzles in the PDF file is entirely trivial, but none of them is very hard either. As usual, you may find that a bifurcation is the quickest way to success on grids of this size. And as usual, I am urging you to try using only logic instead. Here is a small hint for the last puzzle:

**Hint:** You can at once determine the entry in the cell pair R2C4/R5C3. After that, you may want to look for Hidden Singles regarding the entry value of 1; there are several Sudoku regions with only one possible position.

Enjoy the puzzles!

]]>**Rules:** Enter numbers from 1 to 6 (1 to 4 in the example) so that each number appears exactly once in every row, every column and every outlined region. In each cage the numbers must add up to the sum indicated in the upper left corner. Numbers must not repeat within a cage.

**Example and solution:**

There are two issues on my mind regarding the nature of this Sudoku variant. The first is about the actual constraints for the killer regions (“cages”). Lately I have seen a lot of puzzles in the Portal which claim to be Killer Sudokus simply because the author has marked a couple of cages in the grid. The no-repeat rule is typically used, but sometimes there is no arithmetic constraint at all. It is my position that the designation as “Killer” should not apply in these cases.

My point is that the sum constraint should be considered the main ingredient in this puzzle variant, and the no-repeat part is only a supplement, in a manner of speaking. In fact, regions where numbers are not allowed to repeat have been given different layouts over the years. I consider it somewhat inappropriate to use a notation which is often associated with one meaning for something else and then lay down a claim on the puzzle variant’s name simply because of the distorted notation.

It makes sense to allow for killer sums to be somehow hidden, though (either coded or otherwise not explicitly given, as long as the sums still have some relevance). For example, there are puzzles where dots between cages are given and the sums must satisfy some Kropki-like condition. These are “variants of variants”, as Ulrich calls them, and I can live with the approach that they are labelled Killer Sudokus as well. I feel the same way about stuff like Product Killers (for each cage, the product of the numbers inside is given instead of their sum).

The second issue is about whether each grid cell should be part of a cage. In ancient puzzle history it was customary to dissect the entire grid into cages (as a consequence, there was a redundance of sum clues), and you had a hard time finding a Killer Sudoku without this feature. Nowadays puzzle authors are apparently trying to give as few cages as possible. It is debatable if this is a good idea or not; in any case, I believe Killer Sudoku should not require the cages to cover all the cells in the grid. Our example has been designed accordingly.

Now, a few brief remarks on the essence of Killer Sudokus. Unlike the Thermo variant I have used for the first group in our series, Killers belong to a larger category of Sudoku variants which are based on arithmetic elements. Little Killers are pretty much the same, except that regions have a different geometry. Arrow Sudokus are also related to Killers, but note that it makes quite a difference if the sum must correspond to an entry or is simply given as a clue. I will probably get back to this variant in a later group of the Sudoku compilation.

There are rather few things for me to say regarding solving techniques. Obviously, a great many steps are about which collections of distinct entries add up to a given sum. Sometimes weaker steps come into play, such as estimates in one direction (a certain cage cannot contain a specific number because the sum would then be too small or too large) or occasional parity arguments. Plus, there are many Law of Leftover steps, i.e. the sum of one or several cages can be compared with the sum in rows/columns/regions.

As with the Thermo Sudokus, I have tried to design the puzzles such that the different techniques can be employed. The first two puzzles have the property that the cages cover the entire grid. These two specimens are thus fairly easy. I made no attempts to incorporate any particular step; because of the redundance I mentioned earlier, there are various logical solving paths in either of these two puzzles anyway.

The remaining puzzles in today’s set have cells which are not part of a cage, and I tried to give them a different character each time. The third puzzle has only a small umber of unused cells, so that essentially the same techniques as before could be applied (with minor adjustments). The fourth puzzle uses a very specific geometric formation of cages, and the fifth one has only very cages at all. They feel – at least to me – like substantially different puzzles.

The last puzzle of the set was supposed to be considerably harder, but I cannot really judge if I succeeded. You see, I had prepared this beautiful solving path; unfortunately, I did not get the final part to work, so I had to make a couple of changes which cancelled some of of the original logic. I felt the final version of this puzzle was still satisfactory, hence I included it as Killer Sudoku 6 in the PDF file. If you want a hint, just highlight the paragraph below.

**Hint:** You will be able to locate a couple of 1’s and 6’s using the aforementioned estimates, especially in the left half of the grid. I believe these steps will leave a bunch of candidate pairs, though. The decisive breakthrough will come from a study of the rightmost column in the grid (which yields one entry and helps resolve the ambiguity). However, the order of the steps can be changed, which means that not all the candidate considerations are actually required.

So much about today’s puzzles; I hope you appreciate how not only the solving path but the general character of a Killer Sudoku can be controlled and altered by the number and the arrangement of the cages, even in a small grid. Have fun!

]]>Chief editor of the magazine is Tawan Sunathvanichkul, one of Thailand’s best puzzle solvers, and it was he who contacted me at the time. (Apparently he was in Germany for the WSC 2019; unfortunately, I did not know him then.) Check out his old blog and his new website for more information. At this point I can only guess what the long-term prospects of this project are, but it sure is worth a look or two.

To be frank, I do not know how puzzle magazines in general are doing these days. More to the point, I am wondering if magazines are still a viable format at all. My guess is that puzzle publications which would have appeared in print ten oder twenty years ago can only survive now if they are available in some electronic form. Since PULZE can be obtained as a PDF file, this should not be a concern.

Some facts about PULZE and its premier issue. The PDF version I got spans 84 pages front to back, including the cover, table of contents, solution pages, etc. About 50 pages contain actual puzzles. Most of them – though not all – are WPC style puzzles (and there is in particular a Sudoku section). The puzzle pages contain only two grids on average. On the plus side, virtually all the logical puzzles appear to have been hand-crafted by highly proficient authors.

The rest of the magazine is dedicated to puzzle-related articles, such as puzzle creation, solving strategies, and some other stuff. The first issue of PULZE contains an interview with Tantan Dai (and I was thrilled to learn what her favorite movie is). I am assuming more interviews with other prominent members of the puzzle community will follow.

I did not yet have time to go over all the contents, but I read some of the articles and worked some of the puzzles. As can be expected, I did not always agree with the given difficulty assessment. However, there are some real gems in the first issue. Also, Tawan has put a lot of time and energy into the final layout of the various sections.

So here is the big question: Should you buy PULZE? Is it worth the money? Ten dollars (or twelve dollars, for later issues) seems to be a lot. On the other hand, one must recognize that travel and lodging for a single puzzle event are often a greater expense than the pile of puzzle magazines one would acquire over several years. In the end, it obviously depends on what you expect from a puzzle magazine. (Let me say again, this is not purely a logical puzzle collection.) So here is what I suggest: Do not commit yourself to anything. Just give it a chance.

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