Last time I announced the beginning of a new puzzle series on this blog, so here we go. The Sudoku Compilation features popular Sudoku variants, and for the time being each group is supposed to consist of six puzzles of size 6×6. I have selected Thermo Sudoku, one of my favorite variants, for the first issue. Here is today’s puzzle set: Thermo Sudoku

Before we move on, let me clarify the goals of our new series. I will assume from the start that you are familiar with Sudoku basics. The puzzles in each group will be of increasing difficulty, starting at a low level, so that they can serve as an introduction to the variant. In particular, I am trying to incorporate typical solving steps. However, the puzzles are not designed to demonstrate standard Sudoku techniques; those will just come along during the solving path.

The variants that will occur in our Sudoku Compilation are supposed to be simple and intuitive. As I tried to explain in the previous post, it is my view that puzzles do not tend to get better when they take up more space, and they certainly do not get better when their instructions take up more space. Ideally, the extra condition which defines each variant will fit into a single short sentence, so that no deeper efforts are required to comprehend the rules.

The puzzles themselves are not extraordinary, and they are not meant to be. On the limited space that is available on 6×6 grids, I am trying to create versatile puzzles with various different constellations of whatever type of clue the variant contributes. You should not mistake this for a claim of great difficulty, though. I believe elegance and simplicity can coexist peacefully (and do so on many occasions) in the world of 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. Inside each thermometer the numbers must be strictly increasing (starting from the bulb).

**Example and solution:**

Thermo Sudokus belong to the larger category of Greater Than Sudokus, i.e. Sudokus where the constraints from the clues are such that the entry in one given cell must be larger than the entry in another. Each thermometer is in essence a sequence of relation signs, just in a more pleasing visual style.

In such puzzles you will typically work with candidate sets which are intervals of integers, and that may not sound too exciting. However, quite interesting constellations can arise from the combinations of thermometers, even without any givens inside the grid (which is how I prefer Thermo Sudokus). The first few puzzles are designed to demonstrate the typical basic techniques for this type of constraint.

The fifth puzzle from the PDF file features branching and intersecting thermometers. As long as each thermometer segment has a well-defined orientation, this is not a big deal as far as the puzzle rules are concerned, since one can still view every pair of adjacent cells along a thermometer as a separate (comparison) condition. But overlapping thermometers can create stronger inferences, which is why I am including such a specimen here.

The sixth puzzle is considerably harder than the rest of the set. No doubt it can be completed quite quickly using bifurcations, given its size, but I think it is worth exploring the solving logic behind it. If you need a hint, highlight the following paragraph.

**Hint:** Study the 2×2 square consisting of the cells R4C4, R4C5, R5C4 and R5C5, their connection to the cells in the bottom row and in particular the role of the entry in R6C4.

I have decided that I will not explain the solving techniques in detail (as I did in the Classics Collection) in this series. I could make up some reason, but in the end it all comes down to the fact that I am too lazy / not in the mood. Anyway, enjoy the puzzles.

## 2 replies on “Sudoku Compilation: Thermo Sudoku”

I look forward to see where you get to with this series – your classic collection was really good.

I am on the fence as to whether you can do full justice to some variants using only 6×6 grid sizes: your hypothetical non-consecutive collection is going to be pretty dull for example. However I think you pulled it off pretty well with this Thermo set, and in particular the last puzzle I much preferred the logic being embedded in a smaller grid size. I tend to struggle and get bogged down in the harder thermo puzzles at the larger grid sizes, where the constraint doesn’t always play intuitively with classic sudoku solving.

Thanks Tom. You are right that some variants will not work well on 6×6 grids, such as Consecutive/Kropki constraints, but also some type of outside clues. Also, the 2×3 regions (instead of 3×3 blocks) will limit my options somewhat.

Still, I have decided to stick with 6×6 grids. I have prepared a list of Sudoku variants which I think will work reasonably well with this size. Perhaps my claim “if something cannot be done in size 6×6, it is usually not worth doing it at all” was too harsh; at least it provides a good challenge.