@François,
Thank you for your interest for this kind of solution.
DEFISE wrote:I couldn't find the definition of a remote triple (RT) anywhere.
I was in the same questioning
a few weeks ago.I guess it is a set of 3 cells C1,C2,C3 each containing exactly the same triplet {a,b,c} such that C1 and C2 are in the same house, C2 and C3 also, but not C3 and C1.
This definition is a bit restrictive, if "containing exactly" is meant as "containing no extra digit" to the triplet {a,b,c}. It is also restrictive to require that C1, C2, C3 be in two houses only. shye's
Graffiti puzzle is a nice example of a remote triple in which C1, C2, C3 are in three houses and in which one cell contains an extra digit !
So my definition would be a set of 3 cells C1,C2,C3 each containing the same triplet {a,b,c}, and having each a different solution within the triplet {a,b,c} .
The case that concerns us is RT {1,5,8} in r3c6, r7c6, r7c3.
I also guess that a RT has the property that the solution numbers of the 3 cells are all different, otherwise you could not have written this start of the chain: (x)r7c3 = r37c6
Right
Why, for example, could we not have:
r3c6=5, r7c6=8, r7c3=5 or: r3c6=8, r7c6=5, r7c3=8 ?
Such a solution is not in accordance with the requirement: "the solution numbers of the 3 cells are all different" Above, you are considering a case where two cells have the same solution digit.
The remote triple leads to assert: 1 is present in one of the 3 cells r3c6, r7c3, r7c6 & 5 is present in one of the 3 cells & 8 is present in one of the 3 cells.
Hence the three RT-derived strong links: (1)r7c3 = r37c6, (5)r7c3 = r37c6, (8)r7c3 = r37c6, but also other three (1)r7c36 = r3c6, (5)r7c36 = r3c6, (8)r7c36 = r3c6, if these would have been useful.
These derived strong links chained with the Empty rectangle in box 5 form three chains demonstrating, digit per digit, its elimination from r6c3. (e.g. 1r6c3 is eliminated by (1)r7c3 = ... = (1)r6c456, and so on for 5, 8)
...assuming you have proven that:
(1r7c3 false => 1r6c3 false) & (5r7c3 false => 5r6c3 false) & (8r7c3 false => 8r6c3 false)
I don't understand how you deduced that: 1r6c3 & 5r6c3 & 8r6c3 are false.
My chains above can be written in current language:
Whether 1r7c3 is True OR 1r7c3 is False, 1r6c3 is False. No need to have 1r7c3 False & 5r7c3 False & 8 r7c3 False, to have 158r6c3 False.
Three chains => three eliminations. I can't catch your concern with this logic.
The main point in these Remote Triple is to catch their derived strong links. The subsequent inferences are just "business as usual"
Regards.
