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) as in the [2,8] set in the diagram. The next smallest Locked Set is a Naked Triple (N=3) and so on.

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. In the diagram above the [2,8] paisr are joined by a stray 6 which stops it being a useful Naked Pair.

Lets continue with this ALS.

While solving a puzzle I am hunting around for Inference Chains and perhaps I find my chain turns ON the 6 in cell C4. That will remove all other 6s in the box including the 6 in our ALS. If that 6 is OFF then we create 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 6 in B5 we fix 2 and 8 into those two cells so we can look along the row at other 2s and 8s and turn them OFF. This I do in cell B9. From there I can continue the inference chain. You get two cracks of the whip: check both branches - the 2s and the 8s in the pseudo Naked Pair.

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

## Comments

Comments Talk## Tuesday 4-Jun-2013

## ... by: Nono

Question like Str8tsFan.the last solver version 1.95 does not eliminate the 8 in J1.

No problem for good sudoku players !

## Wednesday 3-Oct-2012

## ... by: Str8tsFan

I hope Andrew will ever read (and answer) these comments... About the second example:(1) Using the link to the solver leads to a sudoku with a tiny little difference: B9 has an additional candidate 6, which is missing at the example. At the solver this candidate will be eliminated with the very same example:

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

(2) What about the candidate 8 at J1? As far as I can understand the theory, the weak link "D1 to F1" should not only eliminate the 8s at F2 and F3 (weak link in same box) but also the 8 at J1 (weak link in same column). More interesting: the solver doesn't eliminate that 8 at J1 either. Why? Did I make any mistake, or is it a flaw of the solver? As far as I understood, a weak link can be part of two entities (box and row or box and column) and thus should be able to eliminate candidates at both entities, maybe the solver fails to check the second entity?

## Tuesday 10-Jul-2012

## ... by: Mr Turner

The last paragraph seems flawed since there is an 8 at F8. The 8s at F2 and F3 could be colored off as an inference from D1 being on. This implies F8 is on. F8 is also inferred to be on from D8 being off.## Wednesday 22-Apr-2009

## ... by: Mr Archibald

I call this the 6 pack.the first three in a line on rows 1,2 is the some as the last six on row 3. this can work upsidedown and backtofront and sideways.

a b c d e f g h i

i h g f e d c b a

d e f a b c i h g