Most of us are familiar with Type 3 Unique Rectangles. Let me define them as being in the four cells:

A|C

B|D

which contain the pattern:

12|123

12|124

And, there are no conjugate constraints on <1> or <2> in C and D.

(Conjugate constraints lead to Type 4 reductions on the same pattern.)

<1> and <2> are the "deadly possibilities": At least one of C or D must not be one of these.

Now, the usual Type 3 reduction is this: One of C or D must be <3> or <4>. If there is another cell, "E", in the house(s) shared by C and D that has the possibilities <34>, then <3> and <4> can be eliminated as possibilities from all other cells in the house(s).

E is said to form a naked pair with the non-deadly possibilities in C and D.

There is an obvious extension to naked subsets when C and / or D contain extra possibilities. Also, in the above example, if we found two cells in the same house containing, say, <35> and <45>, we also have a naked subset, albeit involving a possibility that is not in A or B.

But, is this all?

I have found a few discussion threads that mention Type 3 reductions involving "hidden subsets", but they do not seem to come down to a clear definition of what this is, and what to do about it.

I do, though, have the following example from a real puzzle:

12|123

12|124

and there is another cell "E", in the same house as C and D, with possibilities E = <13>. It is immediately apparent that <1> can be excluded from D!

(OK, I confess: It is immediately apparent to me after thinking about it for a week. The solutions for {C,D,E} are {1,4,3}, {2,4,13}, or {3,2,1}.)

Sudoku Susser does not identify this pattern or reduction.

Questions:

1. Has someone figured this out? (Surely, someone must have?)

2. To me, this is Type 3, for it does not involve conjugate constraints on the deadly candidates. It is different, in that it eliminates a possibility in the UR corners, rather than their buddy cells. Can anyone send me to a thread on this? (Other than the UR threads in Mike Barker's sticky.)

3. Are you guys going to classify this as Type 19C-a?

Best wishes,

Keith

And, here is the example: Go for it!

- Code: Select all

. . .|. . 1|. . 8

. 8 2|. . .|. . .

. . .|9 3 .|. . .

-----------------

. . .|8 9 7|. . .

. . .|1 . .|. 3 .

. . 5|. . .|2 . .

-----------------

3 . .|6 . .|. 1 .

9 . .|2 . .|. . 4

6 . 7|3 . .|. . .