If the topic is not deadly patterns, but unavoidable sets, and in particular,
Sukaku unavoidable sets, then I'm with
dobrichev ... the 7-cell patters *do* represent minimal unavoidable sets. Adding any other pencilmarks to his DP-like presentations, would destroy the minimality.
For Sukaku, if you consider that a "solution" is a true/false assignment for each of the 729 candidates, then an unavoidable set, should be a set of candidates where if the assignments were removed, the result would be a probelm with multiple solutions.
With some thought, one comes to the realization that a
minimal unavoidable set, should be a set true candidates (taken from cell values in the solution grid), and a set of false candidates for the same cells -- just enough "false" candidates to admit a second solution, and not so many "true" candidates, that the two solutions have values in common, in relevant set of cells.
In a "normal looking" pencilmark grid, clearing those true/false assignments, would produce a "BUG" pattern. The BUG pattern should have exactly two solutions. Any more, and the UA set wouldn't be "minimal".
I see two ways to represent such UA sets.
In
dobrichev's representation, whether by intention, or whether he was just imitating
Red Ed's presentation of deadly patterns: he has candidates for one solution on the left, and candidates for the other, on the right.
Another representation, would be as two 81 character lines, one for the values in the solution grid, and another for thier alternates.
7-cell #1:
- Code: Select all
..1...3....23..1....32...........................................................
..3...1....12..3....23...........................................................
7-cell #2:
- Code: Select all
..........1.2.....32.1.....13....................................................
..........2.1.....13.2.....31....................................................
For a canonical form, I would suggest combining the two lines, and minlexing the whole 162-character string.
That would be equivalent to minlexing the first line, applying the same transformation to second line, and in case the first line has more than one automotphism, applying the one (or one of the many) that give(s) the "minlex-smallest" result for the 2nd line.
It should be noted that reversing the order of the lines, produces a second UA set (in a different soluton grid) that is also minimal.
The two orderings don't necessarily have the same canonical form, though.
That isn't a problem, if you consider that the first line is supposed to represent *true* candidates in a solution grid, and the second line, is supposed to represent "false" candidates -- really not the same thing.
For the two size 7 sets, swapping the lines doesn't change the canonical form.
Here is an "size 9" example, where it does.
Original, in minlex form:
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...........................................................1..2..1..3.4...4..2.31
...........................................................2..1..4..1.3...1..3.42
Lines swapped, and re-minlexed.
- Code: Select all
...........................................................1..2..1.32..4..3.4...1
...........................................................2..1..3.41..2..1.3...4
These represent two "essentially distinct" (Sukaku) UA sets.
The deadly patterns produced from them, however, are "essentually equivalent".
See
this post for related facts, involving deadly patterns produced from minimal
Sudoku unavoidable sets.
They're (sudoku) DPs if you accept the generalised notion that a DP is any set of pencilmarks all of whose solutions is a UA / contains a minimal UA. Maybe they're MUGs (see Andries Brouwer's 