Jeff wrote:Nice diagram, r.e.s.

Thanks ... I've edited the picture to illustrate how a forcing net can be drawn to have two uniquely identifiable nodes -- one node is the "starting digit" and the other is the "final digit". The starting digit is the only node with

no arrows pointing to it, and the final node is the only node with

both types of arrow pointing to it .

Jeff wrote:Can you shed more light on:

Why the starting cell r5c5 is chosen?

What trigger the '2' to be picked to start the forcing implications?

What would happen if '3' is chosen instead?

How do I know this particular contradiction and therefore concentrate on the 8s in box 7?

How do I apply this technique to another puzzle; is there a specific pattern to follow?

Thanks in anticipation.

I appreciate the smiley. I'll try to give concise answers ...

Since a forcing net is essentially just a graphic representation of a specific contradiction implied by the starting digit (forcing that digit's elimination), these nets allow the systematic elimination of incorrect candidates by examining

any or each candidate in the grid as possibly being the starting digit of a forcing net. (This seems to involve the same kind of "search-T&E" involved in examining digits as possibly forming one of an arsenal of basic "solving-patterns".) That,

in principle, if not in practice, is how a human

could discover the nice starting digit '2' in r5c5. But of course that would be infeasible for a computerless human, and indeed I used software to look for a cell with only two digits such that one of them leads to an easy solution by only simple moves, and the other digit (as determined by hand) leads to a forcing net. And of course there was no looking for a specific final digit, since any would do.

Actually, this suggests a kind of challenge; namely, to find an "optimum cell" such that

(1) one digit leads to the solution using naked- & hidden-singles only, and

(2) all other digits in the cell lead to forcing nets whose combined number of nodes is less than for any other cell satisfying (1).

In a sense the challenge amounts to seeking the "most elegant" set of contradictions inherent in a given sudoku puzzle. (I think the above example is an optimum cell, whose single forcing net has 27 nodes. The worst I've seen so far in other puzzles seemed to require about 50 nodes.)

One thing I'd like to ask those who consider "elimination by contradiction" (e.g. forcing nets) to be unacceptable ... Suppose you've just begun pencilling-in candidates, and the first five cells you complete are as follows:

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12 23 | 234

35 | 45

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Is there no candidate that you would eliminate (by logic alone) at this stage, given that the blank cells have as-yet-undetermined candidates?

tso wrote:r.e.s. wrote:Isn't exclusion by a naked pair an even simpler xy-chain? ...

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` [ab]`

/ |

[xa] |

\ |

[ab]

[...]I think it is implied that a forcing chain refers to patterns that are not also something more basic.

That usage would be a poor one, imo. If a pattern is a forcing chain, then surely it is correct to call it that, whatever else it might also be.