I was going to post this on Eureka, but the boards are down this week-end.

Stephen's deduction is anything but a conventional AIC. In fact, without extending conventional AIC wisdom, it's not an AIC at all. More like an alternating set chain. The problem with interpreting it as an AIC is that it doesn't rely on the same kinds of inferences. Conventional AICs alternate between "not both false" and "not both true" inferences, but these notions don't apply here. In this new case, the AIC does not transport one truth : it transports 2 truths at the same time. Basically, it extends the chain patterns to accounting for inference sets that transport two truths at once. And it could be further extended to an arbitrary number of truths.

There are generic formulas for shaking up sets, but I don't want to elaborate on those. Just for example, here's the formula to split a set :

AB{m:n} - A{x:y} = B{m-y:n-x}, with any required adjustments to keep all variables within a reasonable range of values (there's the obvious lower bound of 0, the obvious upper bound of the number of possible truths in the set, etc.).

Meaning : if you extract from set 'AB', that has at least 'm' truths and at most 'n' truths in it, a subset 'A' that has at least 'x' truths and at most 'y' truths, then the remaining subset 'B' must have at least 'm-y' truths and at most 'n-x' truths (plus adjustments).

To get a general idea of the principle, here are a couple examples :

In a very basic AIC case (where all the strong inferences are conjugates, and all the weak inferences are same-cell or same-unit-number) :

A strong inference is 'A = B' where the AB set is {1:1} (exactly one is true).

A weak inference is 'A - B' where the AB set is {0:1} (at most one is true).

So the sets push around the truth like so :

Either A is {0:0}, and we get : A{0:0} = B{1:1} - C{0:0} = D{1:1} - E{0:0} ... = Z{1:1}

Or A is {1:1}, and we get : A{1:1} = B{0:0} - C{0:1} = D{0:1} - E{0:1} ... = Z{0:1}

In Stephen's deduction (where the case is pertty basic for N=2, all strong inferences correspond to pairwise conjugates) :

A strong inference is 'A = B' where the AB set is {2:2} (each holds exactly two true candidates).

A weak inference is 'A - B' where the AB set is {0:2} (each holds at most two true candidates).

The A set has three possibilities :

Either A is {0:0}, and we get : A{0:0} = B{2:2} - C{0:0} = D{2:2} - E{0:0} ... = Z{2:2}

Or A is {1:1}, and we get : A{1:1} = B{1:1} - C{0:1} = D{1:2} - E{0:1} ... = Z{1:2}

Or A is {2:2}, and we get : A{2:2} = B{0:0} - C{0:2} = D{0:2} - E{0:2} ... = Z{0:2}

The applicable AIC deductions in the "N=2" case are :

Strong loop : If A and Z are actually the same set, then A must be a {2:2} set (since AZ is either {2:2}, {2:3} or {2:4}). It's the great attractor : any truths taken out of A transport back to it.

Regular loop : If there is a weak inference between A and Z, then AZ must be a {2:2} set, and recursively all inference sets in the chain must be {2:2}. Blind justice, all inferences are assimilated.

Weak loop : If there is a set X such that A - X - Z (weak inferences to A and Z), then ... well this one is funny : we conclude that X is {0:1} (the gray hole : A and Z siphon truths away from X but they do so independently; any truths taken out of A transport to Z, but in the worse case both hold one truth and these target the same candidate in X, so X is still left with one truth.)

In the general case, for sets 'N' : we have strong inferences of at least N truths, and weak inferences of at most N truths.

The strong loop forces A into a {N:N} set.

The regular loop forces all inference sets into {N:N} sets.

The weak loop makes any target set into a {0:N/2 rounded down} set. This conveniently works out to {0:0} when N=1.

Well known patterns can usually be interpreted as cases of set transport :

Basic chains have already been looked at, when N=1. I'll only add that ALS and other original means of transport fall under the set-shaking formulas as well.

Interactions and singles are single inference set patterns with N=1 : AB{1:1} - (A{1:1} = B{0:0}.

Bigger set patterns are single inference set patterns with N>1. For instance, a swordfish relies on AB{3:3} - A{3:3} = B{0:0}.

Set overlap wreaks havoc in all cases, but can be tractable : like the X in the weak loop, some information can still be used, although I usually can't get my head around it.

Stephen's deduction is a simple case of chaining sets where N=2. Its simplicity is what makes it amazing : there are only two truths transported, the weak(cover) sets consistently alternate between cell pairs and boxes, and the strong(base) sets consistently alternate between number pairs in rows and columns, and there is no nasty overlap.

Here's a matrix representating the deduction as both a chain and a set problem, for those who are familiar with these constructs :

- Code: Select all
` | (27)b1 r56c2 (16)b7 r8c45 (27)b9 r45c8 (16)b3 r2c56`

-------+-------------------------------------------------------------------------------

(27)c2 |(27)r13c2 (27)r56c2

(16)c2 | (16)r56c2 (16)r79c2

(16)r8 | (16)r8c13 (16)r8c45

(27)r8 | (27)r8c45 (27)r8c79

(27)c8 | (27)r79c8 (27)r45c8

(16)c8 | (16)r45c8 (16)r13c8

(16)r2 | (16)r2c79 (16)r2c56

(27)r2 |(27)r2c13 (27)r2c56

-------+-------------------------------------------------------------------------------

|(Put the column remainders down here, they are the eliminated.)

Where rows are "at least two" sets (well, actually, "exactly two" in this case), and columns are "at most two" sets.

As a side note, Stephen's alternations between cell-box and row-column suggest this pattern has affinities with braiding. Perhaps an interesting pattern can emerge from studying this.