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. Or you might have sent a wink to the East box 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. After it’s done, you might notice the shorter ANL for the same elimination. Happy accidents by focused chain building is the theme of AIC Building.
Why this reliance on the less than systematic activity of building chains any way you can? Because you’ve done the systematically exhaustive exploitation of logical networks and single type chains, and the puzzle is unsolved.
The size of the space of anything goes AIC is far too large for exhaustive searching. For publishable examples of what you can find, sudoku researchers call on computer coded solvers. It’s rather pointless to do the same, unless you’re interested in the means of human discovery of what the computer code finds.
When you solve a very hard puzzle by AIC building, enjoy and share your victory. The mystery of what you might also have found remains.
Before you plunge into AIC building, there is a routine enhancement of the grid that would increase your success.
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. The curve allows any connection between a, b and c, but the slink connecting curves suggest a to b and b to a and c.
The outside a, b and c here are usually candidates, but you could also pre-mark winks connecting to pre-marked groups.
AIC ANL Examples
AIC hinges make some Almost Nice Loops 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 the Sysudoku review of 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.
The boomerang above 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? For the answer, look after the next boomerang.
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 can be the returning value in a boomerang. If the chain slinks back to the triple, the boomer is back. The post of 9/25/18 reports the entire solution, including another boomerang following this one.
Now for your confirming peek at the second boomer in Insane 425, eliminating 7r1c6. After the removal of 7r1c4 by the first boomer, you can back up to a new starting cell r1c6 for another 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.
This is to suggest you copy a scratch grid to doodle possible AIC on.
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.
A more likely recruitment of an ALS node occurs in this confirming ANL from KD Insane 485, where a 1-chain and a 7-chain are joined by an XY node, but the XY node to 9 goes nowhere until the ALS node carries a confirming 9 slink bac k to the 7-chain.
The ALS Boomerang
Another combination of Almost Locked Sets and the boomerang idea makes the ALS a target in AIC building. Here is the first encounter in the KD Insane series, again from KD Insane 485.
What happens is that the starting 9r2c9’s AIC doesn’t come back to 3r2c9, but it does get back to the 9 value group of the ALS 379 of which the starting 9 is a member. Instead of an ANL removing 3, we have an ANL with the starting starting candidate and the ALS group it belongs to, being the terminals.
Of course the 9’s seeing both terminals are toast, but can we remove one of the remaining 9’s? The logic of the ANL says that one or both of the terminals is true. The group can be true without the member 9r2c9 being true, but if the member is true, the group is true. The only certainty is that the ALS group is true. At least one terminal is true, and the group is true, regardless of whether it’s both, or which one. Just don’t attempt to explain it to anybody you know.
The ALS boomerang is no fluke. The Insane review yields two more. Here is the one from KD Insane 495.
The Subset Node
What do you make of this AIC node? Its from KD Insane 475. It’s an ALS boomerang with its closing node an ALS group, but for the purpose of spotting, you can think of the last node as a subset node. If 3r5c7 is true, and 3 is taken from the ALS 123 in c1, It becomes a naked pair removing 2r8c1. We get the same result by explicitly drawing the ALS group boundaries and the strong link between 3 and 2 groups, but there may be cases in which the subset node interpretation is easier to see.
The subset node is an ALS node. It can appear in any AIC.