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Firstly, something that has been playing on my mind a bit recently was the announcement of a Cracking the Cryptic sudoku book via a kickstarter campaign. This does look very exciting, and I've paid up my $19 USD to secure a copy for whenever it's ready. The money they've raised to date is phenomenal and it's exciting to see exactly where Mark and Simon can take Sudoku, particularly when you compare that with what the WPF has been able to achieve. All the best to them!

And yet when I look at the list of the contributors, it's a bit like seeing all the cool kids running off to play their cool kid game, without being invited. Don't get me wrong, it's their book and their channel and they can do whatever they like, I have no problem with that. And I do get that the puzzles I create aren't really the sorts of puzzles that get their viewers' pulses racing, so there's no real reason why I would be included. And the authors on that list certainly include a few of my favourite authors so absolutely no complaints there. I suppose the crux of the matter is simply that I'm kidding myself a bit to look at that list as a "Who's Who" of the best sudoku authors. From that point of view I can console myself with the company of many more of my absolute favourite sudoku authors who didn't make the cut.

I'll leave things there - I do feel a bit better for simply expressing my feelings on the matter to you, dearest reader - I suppose all I'm really saying is that it would have been nice to have been asked. As presumptuous as I'm sure that sounds to even my most dearest of readers.

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Moving on, I've been thinking a little more on some of the recent posts and discussions on this blog, particularly to do with intersection theory. The news thing from that front is that I've created this resource to help any of my interested dearest readers to easily visualise the addition and subtraction of rows, columns and 3x3 boxes.

https://docs.google.com/spreadsheets/d/1dTAJNB-U-5d8bxwFsWwyVHEU9xAGiBf3mjMmUboHzFI/edit?usp=sharing

I've also started fleshing out a list of references which might become the basis of a more permanent article I might put together as and when I get my head around this stuff even more.

Fred's recent comment gives me some hope that there might be some kind of multiplication we can work into this algebra of constraints, but I haven't really given much more thought to it than that. If anyone else gets there before I do I'd be most interested to hear about it!

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The final item of business is simply to reflect that all this discussion holds my interest because it's fundamentally about linking different areas of a sudoku grid to each other in interesting ways. We had started off by thinking about these as partitions of cells, but I did want to discuss something more along the lines of a partition of digits.

At the danger of stating the complete obvious, hidden pairs/triples/etc using a subset of the digits 1-9 are exactly the same thing as naked septuplets/sextuplets/etc, and vice versa. The way I thought about it was like this:

- Suppose you have a naked triple of 3 cells in a row whose possible candidate placements are some of 1,2,3. That means there a 6 cells left in the row, and 6 digits which we haven't been able to place - i.e. we have the very definition of a hidden sextuplet.
- Now suppose you have a hidden triple of digits 1,2,3 - each of which can only go in some or all of 3 different cells in a particular row. This means that for each of the 6 remaining cells, we only have 6 possible digits from which we can choose from - i.e. we have the very definition of a naked sextuplet.

The conclusion is that which ever way you look at things, hidden or naked, you end up making identical deductions.

My next thought is that the naked point of view (i.e. looking at which digits are possibilities for given cells) I've always - perhaps mistakenly - viewed as the basis for some of the more advanced techniques when compared to the hidden point of view (i.e. looking at which cells are possibilities for given digits).

I say perhaps mistakenly because I think I realise now this is probably a misconception I've picked up trying to follow discussions of advanced sudoku solving techniques, where computer assistance is de rigueur and grids are very conveniently displayed with all the possible candidates automatically displayed - an approach to sudoku solving which generally leaves me very cold. When I tend to solve, the more cells in the grid I label with pencilmarks, the less I generally enjoy solving a puzzle.

Anyhow, the link between naked and hidden and partitions and so on reared its head to me again when I came across Bob Hanson's website, discussing:

- Bent naked subsets (in particular Y-wings)
- Intersection ideas briefly
- Intersection ideas in more detail

As I'm sure you can pick up from the somewhat dated web-design, in some sense all of this way of thinking is not particularly new - indeed it's been around nearly as long as sudoku mania (dating back to 2004) itself has. My interest remains in thinking about these things from different perspectives.

There's a bit more there for me to get my head around, particularly with a link to almost locked sets (ALS) and what he describes as almost locked ranges, but I want to finish with the two rules that Bob outlines with relations to bent subsets:

- If a bent naked subset contains one and only one candidate k that is present in both of its nonintersection subdomains, k can be eliminated as a candidate in any cell that sees all the possibilities for k in the subset.
- When there is no common value k in the two nonintersecting regions of a bent naked subset, the subset behaves as a standard naked subset. That is, candidate k can be eliminated from any cell that can "see" all cells of the subset containing k.

What I'd like to try and work out and see if there's a "hidden" point of view for "naked" subsets that span intersections of a row/column with a box, in the same kind of pleasing way that can alternatively view so-called Schroedinger's cells via intersections as a kind of semi-Phistomefel arrangement. Again if anyone else gets there before I do, I'd be most interested to hear about it!