blue wrote:Mathimagics wrote:Are minimal UA's of higher degree just unions of degree 1 UA's?

I'm assuming (...) that by "minimal" and "of higher degree", you mean that removing any cell from U (or equivalently, adding any clue from U, to P), would reduce its degree ... reducing it by one (and no more), necessarily.

The answer is "yes", and in fact, U would be a union of

minimal UA's of degree 1.

Assuming the definition of "minimal" from above, it follows from the fact that for a puzzle, Q, to have S as its unique solution, it is necessary and sufficient for Q to "hit" every minimal UA for S.

With that in mind, if U is any unavoidable set of S, and P is S - U, as before, then since P already hits the unavoidable sets that are

not subsets of U, the "degree of U" would be the size of the smallest hitting set for the minimal UA's that are wholy contained in U.

If U is "minimal", in the sense that removing any cell from U, reduces its degree, then every cell in U, must "hit" at least one minimal UA that is contained in U -- or else the size of the smallest hitting set for those UA, would go unchanged, and the (correspondingly) smaller U, would have the same degree as the original.

In the end then, if U is minimal in the sense above, then every cell of U is contained in at least one minimal UA that is a subset of U, and so U itself, must be a union of minimal UA's.

Cheers.

@Dobrichev: Hi. I vaguely remember the UA's thread.

A few friendly arguments, for a while, I recall ... but I don't remember much else.

It would/will(?) be nice to see what

Red Ed and

JPF settled on, for "final" definitions.

Someone else might be interested in the "Grids with minimal UA18s for all 2-row,2-col,2-digit UA sets".

I remember being a useless curiosity, in the end. No 16's in them, and a long time checking them for 17's