I just read this thread. Some interesting discussion and ideas although I haven't digested everything.

Viggo wrote:I have some difficulty in the terminology and the number of unavoidable sets. According to

this link you have 240000 unavoidable sets in the SF grid. At the same time it is stated, that it is sufficient to use about 50-300 of them - which ones?.

The smallest ones. But the question is, what are they sufficient for.....

That site is using them to search for puzzles with a given number of clues. To do this you really only need the smallest 50-300 unavoidable sets. This is what we mean.

Viggo wrote:A small subset of the unavoidable sets are the "templates", "the entwined pairs", "2-rookeries" or "[9-clue] Rookeries". There are at least 36 of them, and I think all these names have been used for the same subject.

A template is a set of 9 cells with one in each row and column and box. So if you look at the 9 cells in a completed grid where 1 is (or any digit) then this is a template. (Someone correct me if I'm wrong)

An entwined pair is an unavoidable set with two digits, I think. This could have 18 cells, or less.

A 2-rookery consists of all 18 cells containing two digits. This is an unavoidable set, but perhaps not a minimal unavoidable set. If it is minimal, we are calling the digits a fully entwined pair.

A template is a 1-rookery.

A 3-rookery consists of all 27 cells containing any given three digits.

Viggo wrote:I suppose, that when you have got a grid with:

21 2-permutable

6 4-permutable

1 8-permutable

and 8 16-permutable

then you have actually have got 21*1 + 6*2 + 1*3 + 8*4 = 68 unavoidable sets like this - am I right?

Yes.

You would have 68 unavoidable sets with only two digits in them.

You could say that 21 of the 36 2-rookeries are minimal unavoidable sets, and the other 15 are not minimal.

Viggo wrote:The SF grid and SFB grids are very different from the other "random" grids. Looking at these numbers alone, the SFB grid is the most special, and perhaps you would expect this puzzle to have the most 17 puzzles, but it IS the SF grid. So what is the difference in stucture between these two grids, that make this difference?

Good question. That is what we were trying to understand a while back, but we never did.