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User:83.204.155.225 wrote on the main page (which is moved here):

NO ! this is an incorrect conclusion. The only conclusion to be drawn here is that the starting grid is incorrect in the first place...and that there must be one or more additional candidates 1 in column 1 and one of those is true ! The reasoning is as follows : by simple coloring, it can be seen that both the candidates 1 in column 1 of the starting grid have the same parity. So either they are both 1 or neither is 1. By the sudoku column constraint neither can be 1, and there must be at least one further candidate 1 in column 1. The existence of at least one additional candidate 1 in column 1 means that the final strong inference above does not follow (that is : r6c1 <1> does not therefore imply that r1c1=1).

How do you determine that both candidates 1 in column 1 have the same parity? Note the diagram notation - there are other cells in row 6 and column 5 that may have 1 as a candidate. In particular, r6c2, r6c3, r6c7, r6c8, r6c9, r2c5, r3c5, r7c5, r8c5, r9c5 can have 1 as a candidate. --unkx80 20:05, 3 March 2007 (CET)

Apologies : didn't realise the significance of the marking system on the grid, and thinking that the candidates shown were all conjugates came up with an impossible situation. You are of course right in your conclusion. Best regards

I've fixed the incorrect deduction.

Should maybe add other examples ? XY chains, finned X-Wing... ?

In his original article on AICs, Myth Jellies made the point that the items in the chain are inferences--statements or assertions that can be either true or false--not candidate values for cells. I think this is key to a full understanding of AICs and how they differ from value chains (X-Chains, XY-chains) and Nice Loops (AICs are a superset of these, in that any X-Chain, XY-Chain or Nice Loop can be expressed as an AIC). This article, and the article on Inference, carry over terminology baggage from X-Chains, XY-Chains, and Nice Loops, and end up being too restrictive. A cell candidate can be expressed as an inference, for example, "r1c3 has the value 3", but inferences in AICs are not at all limited to that form. Similarly, the concepts of strong and weak links in AICs are not exactly the same as in X-chains, XY-chains, or Nice Loops. For an AIC, the strong and weak links are very simply and concisely defined:

Two inferences are strongly linked if both cannot be false simultaneously.
Two inferences are weakly linked if both cannot be true simultaneously.

That's it.

I have edited this article to make the above clear, and to provide examples of inferences and how they work. Also, I have added a proof of the key point of an AIC, which is that the endpoint inferences are always strongly linked.

Similar cleanup needs to be done on the Inference article.

Professor Prune 12:44, 25 March 2008 (EDT)

Missing Cases

Professor Prune, the mistakes on weak and strong inferences scattered throughout various entries here have been outstanding for a long time now, and I'm pleased you're now dealing with them. However your coverage is incomplete.

There are three theorems covering AIC deductions:

1. The end candidates of a chain starting and ending with strong inferences can't both be false (any sister seen be both is false, and if they are the same candidate it is true)
2. The end candidates of a chain starting and finishing with a weak link can't both be true (if they are the same candidate it is false)
3. All links in a perfect Alternating Inference circuit (or loop) are proved to be conjugate and either all the odd numbered nodes will be true or all the even numbered ones will be, leading to multiple eliminations of candidates seen by odd and even numbered sisters.

• No 1 is the most common and is the most useful.
• No 2 can almost always be reduced to No 1 by trimming off the end nodes.
• No 3 is rare but is really powerful and so is worth watching out for.

When an exclusion creates a single in a cell:

• (a)P = (a-b)Q -[..]- (c)R = (c)S => P <> (c)

I prefer the assignment form which usually makes notating the consequences briefer:

• (a)P = (a-b)Q -[..]- (c)R = (c)S - (c=a)P => P = (a)

Dpbobelisk 11:19, 29 March 2008 (CET)

Thanks for the encouragement, Dpobelisk.

You are correct that I ignored AICs starting and ending with weak links. Aside from the case where the chain links back to the starting premise (in which case we know that premise must be false), I don't see where such a chain results in eliminations.

I'll fix this up when I get a chance.

Professor Prune 15:52, 29 March 2008 (EDT)

ACK! Dpobelisk's recent edit of the description of AICs on the main solving techniques page prompted me to go back and re-read the original Myth Jellies treatise on AICs. I see now that I've made a major terminology mistake. What I have been calling "inferences" I should have been calling "candidate premises". The terms "link" and "inference" are synonymous when talking about AICs.

So when I said that "inferences can't be strong or weak--those are properties of links", what I meant to say is that "candidate premises can't be strong or weak--those are properties of the inferences (or links) between them".

I'm not very happy about the abbreviation of "candidate premise" to simply "candidate", as this causes confusion with the the use of "candidate" elsewhere in Sudoku to mean "candidate digit value". But this overloading of the term "candidate" occurs all over the Sudoku literature (including Myth's seminal article), so I'll learn to live with it.

I've edited the AIC article to:

1. Correct my misuse of the term "inference"
2. Remove the unnecessary restriction to starting and ending on a strong link

This article needs an example of an AIC resulting in eliminations that starts with a weak link.

Professor Prune 15:44, 30 March 2008 (EDT)

Excellent article! In "Non-Looping AICs" you write: "The power of an AIC is that the end points of such a chain are always strongly linked." IMHO "strongly" should be omitted.

hobiwan 10:10, 01 April 2008 (CET)

Good catch, hobiwan. I missed that one when I was editing out the restrictions to starting and ending on a strong link. Fixed now. Thanks!

Professor Prune 13:58, 1 April 2008 (EDT)