This is the second collection of puzzles I have prepared for our new series on Sudoku variants. (It is likely that the pace of the series will slow down from now on.) I have picked Killer Sudoku, another highly prominent variant, as the topic for the second installment of the Sudoku Compilation. This group, like the previous one, consists of six puzzles of size 6×6. And here is today’s set: Killer 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. 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!