Early in Sysudoku Advanced, we construct XY chain ANL and nice loops with a bv map as a guide, and X-chains with an X-panel as an aid. General AIC are built later, after scanning for local formations on the grid that can be spotted and exploited more readily. With so many possible ways to start and continue AIC, exploration is required. They are built, not spotted. What is spotted is promising chain starting situations, and as AIC are being built, productive endings. Systematic exploration in AIC building involves favoring likely continuations and updating maps, x-panels, coloring clusters and existing AIC with each removal or clue.
Strong links are central to every type of AIC. We start them on promising slinks. What makes a promising start is having promising AIC targets for ending the chain. Once started, we use every available means to keep the AIC going, and steering toward the targets when possible. We mentally backtrack and resume along an alternative path when progress fails.
This page is a series of examples illustrating AIC node types, and chain types. Citations with the examples will take you to posts with the full solution path, where you can see the starting grid and evaluate the starting decision. Headings will announce the new elements in each example, for scanning the page later.
Reverse bv, group node, coloring slink, path exploring.
From Rebecca Bean’s Extremely Hard Sudoku,# i-41, posted 3/27/18, the grouped AIC ANL could start with the blue 1. It takes wink out of the cluster along a crossing line to escape the cluster. We get to the internal slink of the bv. The next critical slink for continuing is the grouped 9 slink in the East box. The reverse bv 1 slink gets us to an ANL terminal. Two slinks are used as winks on this ANL.
The next two grids illustrate backing up to follow alternate paths, and how ALS provide them. It might be better to skip these on first reading, to get a more immediate view of the rewards of exploration, as opposed to the frustrations of it.
First, let’s back up one cell to 9r6c8 and try for alternate extensions. Along c8, no ALS 9 to wink to, but in r6, [3489]in 3 cells and a 4 group aligned with 4r6c9. Problem is the terminal 1r6c9 doesn’t work with 1r7c4 or 1r9c7 for a removal.
Going to another starting slink, the r9 8 value, the AIC ends in a dead (no victims) nice loop.
Taking another look, we can add a loop to the nice loop, but it’s still dead.
Backing up one more step on the original AIC, the aligned 9 value now slinks to value groups 5 and 3. No slink for 5, but 3 gets us into a crossing line ALS. We can use the 3 group as an ANL terminal, but the internal slink to other value groups of the ALS doesn’t function as a weak link. If a value group has a true candidate, it doesn’t mean that another value group doesn’t. To use an ALS as an inference chain node, you have to wink into it.
AIC Hinges, Nice Loop
With co many choices to be made, it may be useful to mark ahead the cells that many be AIC nodes. That’s any cell containing two or more slink partners. We call them AIC hinges. For cells having more than 2 partnering candidates, a single closed curve allows any slink partner to connect to any other, but the entering slink curves can select which connections are made.
To illustrate, let’s say a cell of 6 candidates, has three slink partners a, b and c. The hinge pictures the weak link connections between a, b and c.
The next two grids illustrate backing up to follow alternate paths, and how ALS provide them. It might be better to skip these on first reading, to get a more immediate view of the rewards of exploration, as opposed to the frustrations of it.
To illustrate, let’s say a cell of 6 candidates, has three slink partners a, b and c. The hinge pictures the weak link connections between a, b and c.
Hinges suggesting nice loop
Manuel Castillo’s Only Extreme 218 (11/14/17), the nice loop removals add coloring links to expand the red orange cluster. It soon leads to a bridge, then a merge, then a wrap.
The hinge in r2c1 points to the emerging slinks on 1 and 5 which identify three corners of the nice loop.
The Boomerang
You like the ANL? You’re gonna love the “boomer“. Start it with a slink leaving a cell of several candidates. A target is another candidate in the starting cell, as you watch for an ANL. You want to wink into one of those candidates, then add the wink to the starting candidate, closing the almost nice loop. So it’s another form of ANL, but you’re not looking for a victim. It’s the target candidate.
This early boomerang turned up in Royle 17-1020, posted 1/17/17 in a review of Berthier’s The Hidden Logic of Sudoku. It starts on 4r5c1.
Another way to boomerang is to return to 2r4c3, then wink into the target. In this case, you can turn that into a slink chain ANL with a slink from the target to 2r3c1, again removing the target.
Here are two more boomers with interesting origins.
This one was found by Sudoku expert Srmckr in “World’s Hardest Sudoku” 25 immediately after line marking. The start is 7r1c7. It’s posted 5/8/18 along with two alternative ways in which the removal could have been made.
This was before the boomer and its distinctive starting slink from a multiple candidate cell was recognized, so the spotting formula in the post is more complicated.
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 in the post of 7/23/13. The removal leaves a naked pair in c4.
This updated post actually was the introduction of the boomerang.
After the removal of 7r1c4 by the first boomer, you can back up to a new starting cell r1c6 and starting candidate 6r1c6 for another one.
One more boomerang. This one is from Andrew Stuart’s Challenge Puzzle 17 in The Logic of Sudoku. It shows that an aligned triple partner can be the returning value in a boomerang. If the chain slinks back to another triple partner, the boomer is back. The post of 9/25/18 reports the entire solution, including another boomerang following this one.
The Simple 1-Way
The 1-way is an AIC elimination technique as lethal as the boomerang, and more lethal from bv starting slinks. It is so named in Sysudoku because it uses the AIC inference transfer in one direction only, and for that reason, allows a branching form of AIC, the complex 1-way, to reach more targets.
The 1-way starts on a slink partner that sees other candidates of the same value. So if the starting candidate is true, these other candidates it sees are false. On the other hand, if the starting candidate is false, the AIC will carry that inference toward the target. If the AIC shows a target candidate to be false, it is, regardless of the starting candidate. You can end the AIC with a wink to the starting candidate, or a slink to a competing candidate.
The 1-way starts on a slink partner that sees other candidates of the same value. If the starting candidate is true, these other candidates it sees are false. On the other hand, if the starting candidate is false, the AIC will carry that inference toward the target. If the AIC shows a target candidate to be false, then it is, regardless of the starting candidate being true or false . You can end the AIC with a wink to the starting candidate, or a slink to a competing candidate.
In this 1-way example from Stefan Heine’s ultrahardcore 1, posted 8/10/21, the starting candidate is 1r9c8. If it is false, then every candidate ending a slink on the AIC, including 7r9c9, is true. So 1r9c9 is false, whether 1r9c8 is true or not. We all agree its one or the other.
With this 1-way from uhc 133, the removal at r5c6 triggers a Wr5 boxline.
In a second simple 1-way in uhc 133, the start and scaffolding is available for a boomer.
AIC starting points and structure can sometimes be re- applied later. So prior AIC on previous ©PowerPoint slides are assets akin to maintained bv maps, X-panels, and coloring clusters.
The ALS Node
Another resource for AIC building 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” two singles or value groups of the ALS.
The internal slink is between value groups. The removal of an ALS value locks the remaining value groups, so at least one of every pair of ALS value groups is true. This puts a group slink between every pair of ALS value groups, regardless of alignments.
Because ALS are numerous everywhere in the candidate field, they are often the means of continuing the chain. For the same reason, listing all ALS offers the human solver little return on the time spent. A better DIY approach is to identify the ALS value groups that continue the chain, and then the ALS that provide them.
Here is an example of how ALS nodes are found, as needed. It’s the Sysudoku version of Figure 30.1 from Andrew Stuart’s chapter Almost Locked Sets in Chains, in his The Logic of Sudoku.
You mentally leave r1c2 on the 2 slink fo a possible boomer and come back to the 7 slink. Then you see the single 2 in the ALS 425, the triple partner winking into r1c2! and the wink to the ALS 4 group. Hey, the slinks into 7 make a confirming ANL.
After the modest follow up, the same ALS provides a grouped terminal of an eliminating ANL.
More ALS value group ANL terminals to follow.
It’s not a stretch to uncover the ALS as a node again, this time connecting values 2 and 4 in a nice loop with much the same AIC structure. After the ALS node assisted nice loop, the puzzle of Logic Figure 30.1 is solved with a boomer, a finishing move you are now familiar with.
The full disclosure of the 30.1 “before and after” this series, is posted 9/18/2018.
Another ALS node example is courtesy of Andrew Stuart. This time, from his solver’s solution of World’s Hardest (just kidding) Sudoku, puzzle 200.
This confirming ANL looks unlikely as a human solver finding.
Imagine starting from the r8 9-slink, 8r7c8, slinking to 8r2c8, failing and backing up to 8r7c8 and needing another slink out. It seems a stretch only because it’s written out. To your searching eye, it’s a few seconds. 8r2c8 is an ALS value group.
The ALS node is central to a rather spectacular series of backup boomerangs in KD Insane 4X5 – standing for KrazyDad volume 4, book 10, number 5. It comes near the beginning of 4X5 Holds Out for Pattern Trials of October 15, 2013.
A more likely recruitment of an ALS node occurs in this confirming ANL from KD Insane 485, posted 10/1/18, where a slink from bv, start toward a confirming ANL , is continued to a 7-chain.
Also in Insane 485 is an ALS value group acting as an ANL terminal. Starting 9r2c9 with the r9 slink, the AIC enters ALS 379 on single 7. An internal slink leads to the 9 value group, which incidentally, includes 9r2c9. That doesn’t interfere with ANL logic. The value group, or 9r2c9 or both, and we really mean both this time, are true, and the other 9’s in r2 are toast.
The ALS ANL terminal is no fluke. The Insane review yields two more. Here is the one from KD Insane 495, the next week.
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The Complex 1-Way
Now to see why the simple 1-way is “simple”. Because the AIC 1-way carries the “starter is false” inference in only one direction, X-chain branches in that forward direction can be used to create slinks for more AIC paths forward. A striking example is PhilsFolly Beeby’s version of the KrazyDad Insane 485 ALS ANL above.
Starting the 1-way on 9r2c9, the AIC gets to r2c2. Looking back along the path, a branching wink from 3r2c9, which is true by the AIC assumption, erases 3r2c2. Providing the slink for 3 1-ways.
Of course, this ability can lead to some very complex 1-ways indeed. There’s a post on it 5/11/21, and more examples by search on “complex 1-way”. For a real challenge, upload a complex 1-way puzzle to PhilsFolly and read Beeby’s text solution notes. It’s a new experience.
The ALS-Wing
Phillip Beeby includes in his solver an option to find ALS-wings, AIC with an ALS at every node. This is less specialized than it sounds, because in almost all cases, these are three node chains, and the ALS_XZ is a two node chain.
Here’s a typical ALS-wing example in Heine’s ultrahardcore 47, posted 4/13/21. The ALS-wing extends the restricted common connection between ALS to more ALS to form an AIC, in this case an ANL. The connecting winks are group winks and the slinks are ALS value groups.
Unlike the Death Blossom, the wings can be large ALS. Here, the difficulty of recognizing large ALS is evident. The Sysudoku posts on ALS-wings is mostly about a systematic way to identify all ALS, so as to duplicate solver code in finding ALS_XZ and ALS-wings. Not really a DIY project, but you could spot one.
The page titled “ALS-Wings”, off AIC Building, displays a variety of forms of ALS-wings that appear in the blog reviews, and shows how many of them that are found by solvers simplify to AIC with one or two ALS nodes.
Systematic Sudoku 