gsf wrote:rep'nA wrote:why are these "the" seven tri-cycles...? What doesn't work, for instance, with

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`(0)`

/ \

1 3

/ \

*---2---*

this is a mirror image of one of the 7 tri-cycles, so it does work

the ones that don't work (provide eliminations or assignments) are 111, 222, 333,

and tri-cycles with any 0-edges -- I fixed the note to emphasize this, thanks

I don't see that this is a mirror image. The tri-cycle in your list provides another information on another vertex:

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`(1)`

/ \

2 3

/ \

*---1---*

123

Actually this combination is the only tri-cycle that provides information for two of its vertices.

So I think you either have to make it 8 tri-cycles (plus mirror images) that provide coloring information or you may put it this way:

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`*`

/ \

1 2

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(0)--3--(1)

123

Or maybe you think of mirroring like in Physics not only as space inversion (Parity P) but also as possible charge conjugation (C), exchanging 1-edges <=> 2-edges and also coloring information (0) <=> (1)?

In this case your tri-cycles may be CP invariant.

But then you need only 4 information-yielding tri-cycles and can derive the others by "charge conjugation".

Thinking further about it, you need only the leftmost tri-cycles in your list and can derive the others be replacing any of the 1- or 2-edges by a 3-edge.

BTW, it is not true that the 333 tri-cycle doesn't provide any coloring information, it provides every coloring information. So it would prove for any of its vertices that it must and must not have the color in question, which is a contradiction.

Actually, the 133, 233 and 333 tri-cycles are impossible for a valid puzzle or in other words would prove a puzzle to be unsolvable.