I have now had time to freshen up and take a better look. I think the right answers were posted even before my last hasty post.

Red Ed wrote:Hey, RW, why didn't you say? Your notion of "

closed system" seems to be exactly the same as Allan's "ISP". I like the terminology.

Back in 2006, RW wrote:Closed implies that no patterns outside this system could affect the numbers within the system, and no numbers within the system could affect any numbers outside the system.

I took a look at RWs old post. This is exactly the same concept that I tried to express in terms of constraint set arrangements. Closed system is perhaps the best term of all. Sheesh, 2006, half way back to the big Su-Bang.

Red Ed wrote:Right then

Allan, I'm convinced that your notion of "isolated sub-puzzle" coincides with Denis' definition (

green) below:

(warning: my attempt at paraphrasing him)

- Denis-ISP if and only if:
- it's a set of cells, C, in a puzzle-in-progress, P, such that
- for each candidate in C, there is none of the same value in the same unit (row, column or box) outside of C.

- Ed-ISP if and only if:
- it's a set of cells, C, in a puzzle-in-progress, P, such that
- for each unit (row, column or box) of the grid, there are as many different cells in C as there are candidate-values in C.

- RW-ISP if and only if:
- it's a set of multi-candidate cells, C, in a puzzle-in-progress, P (that has >0 solutions), such that
- assigning any individual candidate in C still leaves P with >0 solutions.

Of these 3 views I prefer Red Ed's as it is easy (for me) to see what makes these structures into closed systems. I.e., overlapping locked multiples that are locked in all their rows, columns, boxes. This is indeed a clear picture.

Red Ed wrote:Now the $6.67e21 question: so what?

If you were looking for a characterisation of "deadly patterns" or uniqueness tricks then, sadly, this ain't it.

Well, yes, I initially pointed out this property as I thought it might relate or lead to ways of testing/observing local uniqueness. This is precisely addressed in RWs 2006 thread. Perhaps you can remit the money, $6.67e21, to him.

Pure musing (I guess I have not learned my lesson)

RW wrote:denis_berthier wrote:- can we make a list of all the possible cases of absolutely isolated patterns?

This has been done up to a certain size of unavoidable sets. I guess there must exist a limited amount of possible sets, so theoretically they could all be listed. However, the amount is huge!

JPF wrote:Red Ed wrote:but I've seen at least one example of a minimal unavoidable set (and therefore "deadly pattern") that has all nine digits in it! It covered 55 cells.

even 60 cells

here...JPF

As the ISPs grow in size, would not a blank grid (no givens) be the ultimate closed system, described by the clear definitions given in this thread? Then, how many givens does it take to be able to prevent the formation of a closed system, or destroy the closed system? (We already know this to be 17?)

For the theoretically minded, perhaps using the clear definitions presented here, can it be shown that 16 candidates are not sufficient to prevent a closed system? Has this already been proved, or does it remain an observable fact?

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