An Alternating Inference Chain, or AIC, is a connected chain of alternating slinks and winks. X-chains, including grouped chains, and XY-chains, including remote pairs, are made of links of a single type. If we refer to a chain as an AIC, we generally mean it is a mix of link types. Any mix of link types is fine. The elimination and confirmation effects of chains depends on the alternating sequence of slinks and winks, and not their types.
In the progression of advanced techniques from easier to harder, chains of a single link type come first. With aids like the bv map and the X-panel, we can readily spot ANL, nice loops, and inference chain links. AIC’s can extend these regular chains occasionally, but when we’re past those, and need the puzzle to give us a clue to the next clue, we’re ready for the AIC building stage. In this stage, it’s more a matter of piecing chains together, and when an objective is recognized, finding some combination links to fulfill it.
Here is an AIC building example from Rebecca Bean’s Extremely Hard Sudoku, # I-41. The chain begins with the XY node r9c7 going through the cluster to stop at the blue 1. Then continue from r9c7 with the idea of ending on any 1 in the NE or E boxes. The grouped 9-chain does that with a reversed bv node r6c8 to terminate at 1r7c9.
An ANL to brag about, but constructed, not found.
Now about that reversed bv node r6c8 in the i-41 AIC ANL. In the AIC building stage, bv take on that additional role. But such an AIC node does not depend on having only two candidates in the cell. Any cell with a slink partner, and either a second slink partner or an aligned triple partner, is potentially an AIC node. Sysudokies prepare for AIC building by marking such cells as AIC hinges, by drawing a weak link curve between the slink partnering candidates. There are two unused hinges in the diagram above. For cells having more than 2 partnering candidates, a single closed curve marks all of the AIC hinges.
For illustration, let’s say a cell of 6 candidates, has three slink partners a, b and c. Two slink chains, a to b and b to c, are represented by the attached slinks. Note how a to c is excluded.
In the Sysudoku Order of Battle, the “keep it going” construction crew of AIC is held in reserve until the lower hanging fruit of the single form chains is gathered. For one reason, they may get the weekend off. But also, to have the tools available for the job. Coloring clusters are a resource for mixed AIC, so AIC building generally follows coloring. It does come before the Limit Pattern Overlay methods described a later in this Guide on the pattern analysis pages, but after the more easily spotted X-Pattern Analysis methods that are signaled on the X-panels.
And so, at the last minute, when it is time for “Hail Mary” AIC building, the AIC hinges are added to the grid.
AIC Blue Print Examples
AIC hinges make some ANL’s much easier to spot. Take this ANL from Royle 17-7295(1/3/17), from a vast collection of 17-clue puzzles, and cited in Berthier’s The Hidden Logic of Sudoku.
You just don’t get to this simple ANL without the hinges.
AIC hinges can extend already productive chains. In the MegaStar Maestro puzzle 47 (6/12/12) introducing AIC, a 49 to 4 chain combination ANL is extended through a reversed bv and three hinges for one ANL (green), and an extension from that chain grabs another, more decisive one
Earlier in Maestro 47, the AIC hinge field produced two confirmation ANL’s in a row.
First, an XY to X combination was extended into a loop to confirm 3r5c8.
Then, the XY is switched earlier to a 2-chain for a 2 confirmation ANL.
It’s AIC flexibility supplementing uniform chains, via AIC hinges.
In a more recent review, Manuel Castillo’s Only Extreme 218 (11/14/17) brings a similar example. The AIC hinge field rescues a pair of stalled clusters with a nice loop.
The hinge corner includes a trap, and the nice loop removal also enables a skyscraper, but there is something much more decisive afoot. The cluster expansions produce the bridge:
Not(green and red) =>
blue or orange,
but more decisively, both of the South band hinges make a slink between blue and orange. This brings a merge. It can’t be right, unless red is blue and orange is green. The merged cluster quickly wraps green.
To summarize, the flexibility of AIC’s makes the AIC scan a challenging task for human solvers. It is well to undertake it after more structured methods are completed. AIC hinges, placed on the grid beforehand, provide visual guidance and additional logical resources for the AIC scan.
The Boomerang ANL
Compared to the usual ANL, in which we do not even draw the victim winks, this ANL is something entirely different. The victim “sees” both slink terminals of an AIC, but here, one of them is in its cell, and has a different value.
To find this ANL, start with any cell containing a slink partner beginning a chain, and try to extend the chain to reach any candidate seeing any of the other candidates in the starting cell. Boomerang ANL is a good name for it.
This one turned up in Royle 17-1020, a huge collection of puzzles with 17 givens. Here are two more boomerangs with interesting origins.
This one was found by Sudoku expert Srmckr in “World’s Hardest Sudoku” 25 immediately after line marking, hence without benefit of AIC hinges. It starts with a column slink and depends on what would become two AIC hinges and a reversed bv. This is reported in the post of May 8, 2018, along with two alternative ways in which the removal, which brought an immediate collapse, could have been made.
The next example was found in an update of KrazyDad’s Insane collection. The puzzle is Insane volume 4, book 2, puzzle 5. This grid begins a series of three boomerangs reported in the post of July 23, 2013. The removal leaves a naked pair in c4.
A second boomerang uses most of the inference chain of the first to remove a second 7. Can you put it together before peeking?
One more boomerang ANL. This one is from Andrew Stuart’s Challenge Puzzle 17 in The Logic of Sudoku. It shows why an aligned triple partner is included in the definition of the boomerang. The post of 9/25/18 reports the entire solution, including another boomerang following this one.
The ALS Node
A rare resource for the AIC scan is the ALS Node. It uses an Almost Locked Set, an ALS, as if it were a bv, providing a connecting slink and changing the value. It requires the entering and exit candidates or groups to “see” the entire group of like value candidates in the ALS. The internal slink is between value groups. Because the removal of an ALS value locks the remaining groups, there is a group slink between every pair of ALS value groups.
The ALS node is rarely needed, but ALS are so numerous that chains of nodes are described in Sudoku literature and built into digital solvers. The problem for human solvers is the low return on investment in time, for individual ALS nodes or chains of them.
Here is an excellent example showing how ALS nodes are found, Figure 30.1, from Andrew Stuart’s The Logic of Sudoku.
Only after you put together an XY from 7 to 2 with a 2-chain would you realize that the ordinary ALS in NW links up an ANL confirming NW7.
The ALS hinges are no help in this case. Each one reaches a point of failed alternation on both ends.
However, after the modest follow up of NW7, starting with
(NW1, NW5m, r3s56),
three hinges come alive to involve the same ALS again, this time as a grouped terminal of a hinged ANL.
And with this much done, it’s no stretch to uncover the ALS as a node again, this time connecting values 2 and 4 in a nice loop
The full disclosure of the 30.1 “before and after” this series, is reported in the post of 9/18/2018.
After this ALS node assisted nice loop, the puzzle of Logic Figure 30.1 is solved with a finishing move you are now familiar with
And finally, a last ALS node example is also courtesy of Andrew Stuart. This time, from his solver’s solution of World’s Hardest (just kidding) Sudoku, puzzle 200.
This confirming ANL looks appropriately unlikely as a human solver finding. Yet you can imagine finding exit chains from the ends of the ALS and seeing them come together at SW9.