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## AIC with ALSsFrom sudokuwiki.org, the puzzle solver's site |

A Locked Set is a group of cells (that can all see each other) of size N where the number of candidates in those cells is equal to the size of the group. That is N cells contain N candidates. A solved cell or a clue is a Locked Set where N=1, but such a cell is not useful. The smallest useful Locked Set is a Naked Pair (where N=2) followed by a Naked Triple (N=3).

An Almost Locked Set (ALS) is N cells containing N+1 candidates. In the context of Alternating Inference Chains in this solver, an ALS is of size N=2 and the number of different candidates in those cells is 3, although bigger ALS groups are possible. So an ALS of size 2 will be a two Conjugate Pairs plus one other candidate.

An Almost Locked Set (ALS) is N cells containing N+1 candidates. In the context of Alternating Inference Chains in this solver, an ALS is of size N=2 and the number of different candidates in those cells is 3, although bigger ALS groups are possible. So an ALS of size 2 will be a two Conjugate Pairs plus one other candidate.

This might be easier to explain with a diagram. The highlighted yellow cells D7 and D8 contain [2,7] and [2,7,9] respectively. If D8 didn't contain a 9 the two cells would be a Naked Pair. The extra 9 makes these two cells an ALS.

While solving a puzzle I am hunting around for Inference Chains and perhaps I start with the bi-value cell J2 containing [2,9]. I pretend to turn ON the 2 which turns OFF the 9 in that cell. The only other 9 is in J8 which I turn ON. Looking up the column the 9 in D8 must be turned OFF. This creates an on-the-fly Naked Pair.

Now, a Naked Pair eliminates candidates in the row or column (or box) it is aligned on so we can use this elimination property as part of our chain. This is the trick! By removing the 9 in D8 we fix 2 and 7 into those two cells so we can look along the row at other 2s and 7 and turn them OFF. This I do in cell D3. From there I can continue the inference chain.

While solving a puzzle I am hunting around for Inference Chains and perhaps I start with the bi-value cell J2 containing [2,9]. I pretend to turn ON the 2 which turns OFF the 9 in that cell. The only other 9 is in J8 which I turn ON. Looking up the column the 9 in D8 must be turned OFF. This creates an on-the-fly Naked Pair.

Now, a Naked Pair eliminates candidates in the row or column (or box) it is aligned on so we can use this elimination property as part of our chain. This is the trick! By removing the 9 in D8 we fix 2 and 7 into those two cells so we can look along the row at other 2s and 7 and turn them OFF. This I do in cell D3. From there I can continue the inference chain.

A real life example now. This chain contains an ALS on the cells {G6,H6} (I used squiggly brackets to denote ALS as opposed to square brackets for Grouped Cells). 9s in row H are the entry point. We turn 9 ON in H2 which turns OFF the 9 in H6 - the extra candidate that makes the ALS an ALS. This gives us a Naked Pair of [5,7] that points up column 6 turning OFF the 7 in F6 and the chain continues.

Ultimately we use Nice Loop Rule 2 to place 4 in A4

AIC on 4 (Discontinuous Alternating Nice Loop, length 12):

-4[A4]+4[D4]-7[D4]+7[D2]

-7[H2]+9[H2]-9[H6]+7{H6|G6}

-7[F6]+4[F6]-4[A6]+4[A4]

- Contradiction: When 4 is removed from A4 the chain implies it must be 4 - other candidates 2/5 can be removed

Ultimately we use Nice Loop Rule 2 to place 4 in A4

AIC on 4 (Discontinuous Alternating Nice Loop, length 12):

-4[A4]+4[D4]-7[D4]+7[D2]

-7[H2]+9[H2]-9[H6]+7{H6|G6}

-7[F6]+4[F6]-4[A6]+4[A4]

- Contradiction: When 4 is removed from A4 the chain implies it must be 4 - other candidates 2/5 can be removed

This second example uses a chain to kill off-chain candidates, which is Nice Loop Rule 1. The ALS is in {F1,F4} and consists of [1/3/8] and [1/3] respectively. We turn off the extra candidate, 8 in F1 to enable the Naked Pair to be formed.

Alternating Inference Chain

AIC Rule 1: -3[B5]+6[B5]-6[B8]+6[D8]-8[D8]+8[D1]-8[F1]+3{F1|F4}-3[F3]+3[B3]-3[B5]

- Off-chain 6 taken off B9 - weak link: B5 to B8

- Off-chain candidates 1 taken off cell D8, link is between 6 and 8 in D8

- Off-chain 8 taken off F2 - weak link: D1 to F1

- Off-chain 8 taken off F3 - weak link: D1 to F1

- Off-chain 8 taken off J1 - weak link: D1 to F1

- Off-chain 3 taken off B4 - weak link: B3 to B5

AIC Rule 1: -3[B5]+6[B5]-6[B8]+6[D8]-8[D8]+8[D1]-8[F1]+3{F1|F4}-3[F3]+3[B3]-3[B5]

- Off-chain 6 taken off B9 - weak link: B5 to B8

- Off-chain candidates 1 taken off cell D8, link is between 6 and 8 in D8

- Off-chain 8 taken off F2 - weak link: D1 to F1

- Off-chain 8 taken off F3 - weak link: D1 to F1

- Off-chain 8 taken off J1 - weak link: D1 to F1

- Off-chain 3 taken off B4 - weak link: B3 to B5