This page describes Sudoku advanced solving methods, those that follow the completed assembly of candidates by the basic solving methods of Sysudoku Basic. The methods follow in a normally efficient order based on increasing effort and difficulty. The order also accounts for the use of visual solving aides that may be constructed in the process.

These Advanced pages identify spotting signals, your actions, the result, and why it works, with general schematics, grid pictures, and relevant post citations. Relevant posts can be called up by date from the monthly archive, by key words in a search box, and by tapping subject tags at the bottom of each post.

Remote Pairs

A remote pair is an XY chain of identical bi-value cells(bv). It is easy to spot among the line marked bv.

In this chain, a candidate of each value is true in every fourth cell, so any outside candidates seeing a candidate on both ends of a 4-cell segment is removed.

This pair of remote pairs were never posted, but can be reached following up Fick’s Trick in Alary’s More Extreme 200 of 5/5/15. Also see the 6 node remote pair in KrazyDad Super-Tough 555 posted 6/29/21. That gives you three 4 node segments to check for victims. Did I miss any?

Unique Rectangle (UR)

This is a practical technique to use immediately after line marking. It is four cells in a rectangle in two boxes, containing the same two values. Although analysis leading to eliminations follows many paths, UR comes early in the order of battle because spotting is easy, and a type table on the Solving Tools is available for quick reference.

In UR, candidates are removed, which if true, would allow a solution on two values on the corner cells to be interchanged for a second solution. For a puzzle composer, that would be a glaring mistake.

Either of these rogue 7’s would force the corner cells to 13. Besides these two, a 7 in c4 is also removed. The UR type table makes this a type 2, but you probably wouldn’t need to look it up.

Hidden UR, with slinks on the two UR corner values, offer additional eliminations, and you don’t need a type table. Here is an example, from ultrahardcore 135, posted 8/1/20. If 7r9c7 is true, 3 in that cell isn’t. The slinks make 3 in the adjacent corners true, making 7r9c2 false, and by slink, 7r8c2 true. The problem with that solution is,  3 and 7 can be interchanged, without effecting the solution outside of the corners, to make a second solution.

XY ANL Railways

XY chains are built with bi-value cell nodes. With a bv map, its easy to find them all, and to be ready when an elimination creates another one. With them you can build three types of loops. The three are pictured below, with the strong links, the slinks between bv partner candidates, represented by solid lines, and the weak links between bv, by dotted lines.

ANL is short for almost nice loop. In the nice loop, the alternation is perfect. In  the ANL, it fails at one point.

The eliminating ANL is a slink forming AIC with slink terminals joined to its victim by two weak links. One of the terminals is true, because, when one is false, the  inference is carried around to make the other terminal true. In the confirming ANL, the slink AIC terminals are slink partners with the confirmed candidate. The candidate between slinks must be true, because, were it false, the inference carried around would make it true.

In the confirming ANL, the slink AIC terminals are slink partners with the confirmed candidate. The candidate between slinks must be true, because, were it false, the inference carried around would make it true.

An XY chain, a slink connects bv partners, and a wink connects matching bv partners that see each other. A nice loop closes when matching bv candidates see each other. Then alternate candidates are true, and candidates between them are false, sometimes creating many eliminations.

On a 9 x 9  bv map, you can represent long XY chains with a single curve. Values change at every bv, but when a value is repeated along the curve, it marks another ANL. We connect the curves like railways where values match, and call the result an XY railway.

Here’s one for KrazyDad v.5 b.2 n.5 posted 6/8/21. It shows how complex the task of finding all ANL and nice loops can be, and how the railway can organize and simplify this task.

On the map, you can add winks  at matching values to form nice loops. On the grid, that may be a simple wink, or it may require an equivalent AIC.

In the KDST 525 grid, the map’s r4 wink was a simple wink, closing the nice loop around the blue section of the railway.

On the railway ANL and nice loops, you still have to find the victims. The red nice loop on the map had no victims, but the  long nice loop makes two removals and and a nice loop cluster.

Along the XY chain candidates of the same value are strongly linked by the AIC between them. At least one of them is true. Thus they  are terminals of an ANL. Multiple ANL can be defined along an XY chain

Here is the line marked XY chain map for ftpah 24, posted 10/15/19. It identifies three 6 ANL. Where two chains join, the joint angle signals the direction of travel. Three removals bring another bv, a different map, and four more 3 ANL removals.

If an XY nice loop fails to close or an ANL fails because of an single extra  candidate in one bv node  of the XY chain, you may be able to remove the extra candidate if another candidate of the same value is in the chain and sees the extra candidate.  In Denis Berthier’s The Hidden Logic of Sudoku, this is called an  xyt loop. Sysudoku defines a type of AIC chain, the complex 1-way,  by this phenomemon on the Guide/AIC Building page.

Here is an xyt chain/1-way from Royle 7269 from post 2/17/17. It was started as an XY 1-way from 9r9c7 because it has a target for removal.

The XY ANL that would remove 9r9c2 fails because of the extra 7r8c2. But in the chain direction a branch wink removes the extra 7.  The overall rationale for this complex 1-way is that 9r9c2 is removed if 9r9c7 is true, and if not, the repaired XY chain can be built to remove it.

In AIC building, complex 1-ways are started, and continued in every way possible, on little evidence of success. But applying them to XY chains much earlier is justified because the bv map or X-panel provides exact evidence of success.

Berthier has another “hidden logic” idea that is practical to watch out for on XY chains. He calls it the xyzt chain. Here’s one from Royle 4601, posted in the Hidden Logic review on 2/28/17.

The ANL with terminals 9r3c1 and r9c2 would eliminate 9r1c2, except for the extra 9 in r3c2.

But in the xyzt, the rationale is centered on the victim. If the victim is true, it erases the extra candidate and the repaired XY chain proves it false, going either way.

Make it automatic to look for a boxline on every elimination. The boxline is a bent region creature. If one remainder loses every candidate of a value, the other remainder loses all candidates of that value.

Here, a naked quad removes two 8 candidates in the East box, leaving an 8 slink to remove two in the West box. It’s a boxline because no 8’s remain in the Er5 box remainder, so 8’s are removed in the Er5 line remainder. The removals leave no 8’s in the NWc2 line remainder or the Wc3 box remainder. So 8’s in the line remainder and the NWc3 box remainder are removed. It’s the same 8.

This happens in Moito’s Very Hard III-34, posted 1/30/18.

Meet the Color Link and Sysudoku Coloring

A distinctive feature of Sysudoku on the advanced side is Medusa coloring. Yes, that is British colouring, but on this side of the Atlantic. Medusa coloring includes the internal strong link in bi-value cells and other ALS.

Coloring is a frequent solving tool and finisher in Sysudoku. In coloring, candidates are marked by two opposing colors to form a cluster, a connected network of candidates crossing boxes and lines,  One color of a cluster is true. Its candidates will be in the solution. The opposing color is false. This fact is used in solving before it is known which color is true. The early color marking of cluster candidates does not interfere with later solving operations, and clusters expand over more candidates as eliminations and confirmations occur.

An Alternate Inference Chain is built with alternating weak and strong links between candidates. Coloring requires a third type, the color link.  In a strong link (slink),  if one slink partner is false, the other is true. Thus at least one of them is true.

The color link is stronger. It’s a strong link in which only one slink partner is true. A color link exists when there are exactly two candidates of a value in a box, a line or a bi-value cell. Sysudoku uses Medusa coloring, the form that recognizes color links in bi-value cells (bv), as well as boxes and lines. A slink in a bi-value cell has partner candidates of different values. Medusa clusters generally have different candidate values because of bi-value color links.

Sysudoku friend Gordon Fick rates this Sudoku “not easy” despite the many clues. Seeing the many color links among the candidates, I marked a large blue/green cluster and filled in with a red/orange one.

Building clusters is easy. Start with a candidate that is a color link partner then continue that color with another color link partners of the partners, coming in later to add the opposing color.

My immediate reward here was a coloring trap. 6r3c2 sees a blue and a green 6, one of which is true.

With that removal, there’s a second blue/green trap, but also, a conflict among the cluster colors. A red 3 candidate and a green 3 candidate compete to be the true 3 in c2. A logical expression of this conflict is

not(red and green) implies

orange or blue

That’s known as a coloring bridge. Now a candidate seeing any matching orange and blue candidates can be removed, because one or both of these colors is true. It’s a bridge trap.

There’s a different bridge. In c8 and in r2c8 we have not(orange and green) => red or blue.

Combining these two facts:

(orange or blue) and (red or or blue) => (orange or red) and blue = blue.

All blue candidates are true. Remove all green ones. Now it is easy to show that blue confirms  orange.

A wrap is any evidence that a cluster color is false. In the above bridges, we might say they wrap green, or that it is easy to show that blue wraps red. A typical wrap is candidates of the same value and wrapped color being forced into a box or line.

Clusters often merge to form a single cluster. This happens when colors of two clusters are shown to be color link partners. Then one of those colors becomes the opposing color of the link partner’s  color.

In UHC 309 Rounds the Bases, posted 9/13/21, we finally get two clusters to a decision point. There’s a bridge waiting in r6c1, but the r5c7 traps leave a naked single 5 and blue and orange 8’s alone in the East box. Either orange becomes green, or blue becomes red in a merge.

Let’s say, it’s not your day, and a big wrap refuses to happen. In AIC Building, a chain of clinks in a cluster that begins and ends on opposite colors functions as a single clink. You can represent it as a solid curve across lines and boxes. We call it a shortcut. It’s a wink, or a slink, as well. In shortcut examples in the posts, you can trace out the shortened AIC in the cluster.

After seeing this example, you might want to see other contributions made by Gordon Fick or some other mentioned contributor. Just enter the name in the main page search box. ©WordPress will return a list of posts containing the name. No telling what you might run across as you scan to find it.

There’s a bit more subtle trap that I identify in Sysudoku as the boyfriend trap.

Here’s an example from the KrazyDad Super Tough 585 post of 7/20/21.

9r1c3 is seen by blue girlfriend 9r1c8 in a cell with a green candidate. It’s in trouble, blue or green.

Off this page, Lite Coloring explains a Sysudoku technique to wring more traps from a cluster.

With or without a cluster in place, we move on to

Bent Region Methods

Several candidate elimination techniques concentrate on bent regions pictured above. The bent region methods are aided by a bv map, but also utilize almost locked sets (ALS).   An ALS is a set of n cells in a box or line holding (n+1) values. The bv is an n = 1 form of ALS. The candidate eliminations of bent region methods are often based on the fact that an ALS can lose only one of its values, and when it does, its other values are locked, i.e. cannot be removed, because one of their candidates is true.

Sue de Coq

In Sue de Coq we look for a box/line intersection that contains two bv or small ALS in its line remainders, each with two values matching a separate pair of values in the intersection.  Each ALS limits the intersection to only one of its values, so we call these alternates, and describe the contents of the intersection as

N(a+b)(c+d)

with a and b representing one alternative pair, and c and d representing the other. For a Sue de Coq, the intersection must be limited to a single value N left for the third cell after one of each alternative value is removed.  In the intersection N can be a slink, an aligned triple, or even a clue.

In this typical example from Frank Longo’s The Nastiest Sudoku Book Ever, puzzle 640, the Sc4 intersection is known to contain

9(3+4)(5+8).

Read that: 9 and 3 or 4, and 5 or 8. It’s not 3 and 4 because bv 34 at r1c4 must solve to 3 or 4. Likewise, bv58 in the S box doesn’t allow both 5 and 8. We don’t know if it contains 3 or 4 or if it’s 5 or 8, but each alternative value is in the intersection or the ALS, so any other alternate candidate in the remainder with  ALS sees a true value. 8r9c6 sees a true 8 in the intersection or in the bv r8c6.

It is possible to have a single alternate form  MN(a+b) with a single ALS, and bv (a,b) in one remainder.

A less restricted form of Sue de Coq, allowing more intersection values and ALS, is described in General Sue de Coq off this page.

A trial form of Sue de Coq, the Single Alternate SdC, is described in the Trials page of the Guide.

The XYZ-Wing

Along with SdC, also work the bent regions for  XYZ-wings. Two of three candidates in the hinge cell are seen by candidates in two wink connected bv, the wings. The wings also contain the third value Z in the hinge.

The thing is, one of the three Z candidates is true.  If both Z wing candidates are false, both bv (or ALS) X  and Y wing values will be true, and the Z value will be placed in the hinge cell.

In this regular 347-wing 7r4c9 sees the hinge and wings 7’s. This  comes from Antoine Alary’s More Extreme 4, posted 2/17/15.

We call this a regular XYZ because all its winks are between candidates, and are limited to a single bent region.

When you build a bv map, it is worthwhile to make a copy and add possible XYZ hinges, including all of those having wings anywhere on the grid. Then you can look for forcing chains, AIC with terminal winks, which function logically as winks. Irregular XYZ are much more numerous than regular ones.

In this i675-wing in Moito Very Hard III-15,  posted 1/23/18, the r4c4 bv57 isn’t a workable wing, because no outside candidate can see its 5. But there’s another 57 bv on the XYZ map at r6c8, and a forcing chain winks it to 7r5c6. A victim is found.

Alary’s More Extreme 4 has a similar wing cell attachment. There are many 47 wings for the 347 hinges, but we only look at the one 37 wing. The victim 7’s  line up for 347r1c1 only, and whadoyouknow,

there is a forcing chain attachment to that hinge. But that’s not all. There’s another 7 that sees two out of three of the toxic 7’s. So can it see the attached wing’s 7? In the grouped forcing chain the critical slink is between the 7 candidate and 7 group in the E box.

Also consider the classic WXYZ-wing, built along the same plan, but with four hinge candidates and a third WZ wing. Even the irregular version is rarely seen, with a victim needing to see three candidates.

Here’s one we see just before the green/orange bridge trap it causes wipes it out. The classic WXYZ demonstrates how Z’s in 3 wings and a hinge can all be seen. Two are in the hinge box, with the victim.

I didn’t make it up. It’s Nakamoto Extreme 143, posted 4/07/20.

Bent n-Sets

Bent regions are also sites for BNS0 and BARN eliminations. The BARN, or Bent Almost Restricted n-Set, with n = 4, has been the more productive in our reviews. It is a set of 4 cells containing 4 values, with 3 restricted to a box or line, and one value appearing in both remainders of a bent region.

Here is a BARN parade from Nakamoto Extreme 83 posts of 3/3/20&27/20 and after extensive AIC building.

BARN 1

BARN 2

BARN 3

BARN 4

In The Logic of Sudoku, Andrew Stuart interprets the  BARN has a WXYZ-wing, so we call the  WXYZ-wing with single cell hinge, and three bv wings a classic WXYZ. Sysudoku protested the labeling of BARN 4-sets as WXYZ in the post of 8/25/15.

Introduced by Bob Hanson’s Student Assistant solver, a BNS0 is an n-set, a set of n cells containing exactly n values, with all values confined to the box or the line of the bent region, and locked within their units, that is to say, with no value in both remainders.

BNS0 are very rare, and not necessarily easy to spot, a double whammy. But here is Bob Hanson’s BSN0 5-set from his Student Assistant Report. The report labels the BARN as a BNS1.

It turns out that a regular XYZ is a 3-set BNS1or BARN, as well as a form of ALS_XZ. Here is Hanson’s regular 145-wing at left, and its marking as a BNS0 3-set in Wc3 at right. 5-candidates are restricted to the W box, and 1 to the c3 column, while 4 is not restricted to either.

The X-Panel Methods

Leaving the bent region,  we build our X-panels and use them to see the patterns of remaining candidates of the same value.

An X-Panel is a single digit 9×9 table, giving the position of each remaining candidate of a value X. A ©Word template is available with a sheet of 12 X-panels, allowing for three spares for analysis operations. X-panels concentrate the visual attention on remaining candidates of a single value X, and help with X-chain AIC, fish, and pattern analysis.

There’s good reason for doing X-chains plus fish, for each value as each X-panel is completed.

X-chains

X-chains are the simplest form of AIC, but not the easiest to find, because there is no structure like bv to mark their path. X-panels makes it easier, limiting the path of candidates of a single value. Slinks are critical to constructing them, and as that last irregular 347-wing demonstrated, X-value grouping creates new slinks for more X-chains.

Working on the X-panels, its easier to copy the candidates to a larger, single value table in ©PowerPoint . Even then,  it’s seldom necessary to draw separate dotted and solid lines. Just use a freeform to map out the route of a chain, and if it helps, put a little bar across a segment to signify a wink. That marks the intent of grouping well enough.

Here’s the analysis panels on two ungrouped ANL.

Grouped X-chains

More X-chains are available when we find groups of X candidates that create grouped slinks. When two X-groups in a box or line cover all X candidates, a group slink is formed. If the true X vlue is not in one of the groups, it is in the other.

In World’s Hardest Sudoku 200, 4r2c5 and group 4r2c13 form a grouped slink, then that group and 4r3c2 are a grouped weak link, because they don’t cover the 4’s in NW. Then 4r3 and 4r46c2 group slink, and two candidates wink in W. Finally 4r5c3 and 4r5c56 slink, and the both victims see the 4r4c56 group.

The grouped ANL can be easier found on the 4-panel. Interpreting the marked freeform is a bit harder.

On this page you’ve seen an X-chain used as a simple weak link in an irregular XYZ-wing. The X-panel isn’t useful for that kind of X-chain application, because there, we were looking for any form of AIC wink, not necessarily an X-chain between two given candidates. Grouped AIC nodes are best found on the grid in AIC building,  where they are the only means of getting the next slink.

Here’s a telling example  from another World’s Hardest Sudoku 134 at 7/3/18 right after line marking. The grouped X-chain lets the West box 8-group see a single 8 in the BARN. The group already sees all the other 8’s, in the BARN, so the 8-group becomes the victim, That leaves boxline Er6 eliminations.

Regular Fish

For human Sudoku solvers, it is very helpful to be aware of the contests among lines, boxes and cells for true candidates, the ones that become clues in the solution. In the basic process of populating cells with possible candidates, we mark subsets like the naked quad in c7 in the line marking of Alary’s More Extreme 144. As we complete line marking, we will lock 1, 4 5 and 7 out of r89c7.

We could fill c7 now. Cells r89c7 can only contain complementary values 6, 8 or 9, so r8c7 must contain 8, and r9c7, 6 or 9.

Regular fish define a similar contest, in which a set of n rows or n columns lock candidates of a single value out of the line positions they need to hold their n candidates of that value. Candidates of that value in other lines in those n positions are removed. Each such contest is a regular fish. The “regular” means it’s between lines only, without boxes or fins.

Here is a very productive swordfish, from. Alary’s More Extreme 144, posted 3/10/15.

The swordfish is about rows r157 needing to reserve positions c369 for their 3 9-candidates, thus requiring other rows to give up their 9-candidates in those columns.

Here are the three names associated with the number of lines. Names for n = 5, 6 and 7 seldom appear because a fish of n lines and m lines free of candidates comes with a complement fish of, at most, (9 – m)/2 lines.  In this example, m =2, and we can have a complement swordfish. For n = 5, the complement is a smaller fish.

On Sysudoku grids, fish are represented by icons pointing toward the removals. But imagine looking for a fish on a grid as full as the above.  For an X-wing it’s easy enough, because the 2 lines are marked as slinks in the cell bottom corners. As you mark the line two candidates of a line slink, you check for a parallel line for slink candidates in the same two positions.

For higher order regular fish, we look instead at the X-panel, as we finish looking for X-chains. Without the distracting candidates of other values, we look for n lines of n candidates in each direction, smaller n’s first.

A very helpful aid to finding and recording regular fish is the blank line tally. In a blank line crossing the fish lines, we place n dashes marking them. In a blank line in the fish line direction, we place plus signs marking candidates. Then every crossing line candidate in a + line and not in a dashed line is removed.

Here is the 9-panel display of the Alary More Extreme swordfish on rows r157 and the complementary swordfish on columns c157. One may be more obvious than the other, but in either one,the removals are the same.

The simplest and most frequently encountered regular fish is the X-wing, in which two lines reserve two crossing line positions for themselves. These are easily spotted during line marking, as you mark the second line.  Pull up KrazyDad 535 posted 6/15/21 for a decisive example.

Off this post, Finned, Kraken and Sashimi Fish continues the fishing expedition. These are combinations of fish and extra candidates (fins) that eliminate candidates when fish victims see them. It shows that panel 6 above has a kraken fish.

Freeform Pattern Analysis

After you scan each X-panel for X-chains and fish, there something else to scan it for, and that is, signs of limited numbers of patterns.

In Sysudoku, a pattern is a set of remaining candidates of a value, which provides one for every box and line not containing a clue of that value.  Each pattern candidate provides its value to a box, and two crossing lines. An inherent property of patterns is that no candidate of a pattern can see another. Solving starts with large numbers of patterns for most values, which decrease until the one true pattern is known for each value. The true patterns for all values fit together in the solution.

A plausible sounding idea now recognized as humanly impractical was the Pattern Overlay Method, of finding all patterns and eliminating patterns in unresolvable conflict. A general tool for finding patterns in the POM represented patterns by letters. Sysudoku adopts only a few, more accessible, POM techniques, some based on the coloring network, with freeforms as the graphic representation of patterns.

A freeform is a graphical entity, a series of connected lines. A single pattern of value X can be represented on an X-panel or the grid by a freeform starting on one side of the grid, crossing one X candidate in each line and each box where one is available.

Freeforms can reveal orphans, the candidates not crossed by an  available pattern. In this example from Andrew Stuart’s The Logic of Sudoku, only two patterns exist for value 7. When this is discovered, 11 orphans can be removed.

It is possible to find all patterns on an X-panel, by drawing all free forms from one of the four edge lines to the other side. It’s not humanly practical to do that on all X-panels, but with practice, you can detect where freeform crossings are severely limited.

This rather spectacular example makes the process simple enough to explain.

On the 7-panel here, pick an edge from which freeforms are limited. In Logic, Andrew picked North to South. I can say the almost swordfish of rows 3, 5 and 7 don’t let many across, but columns 4, 5 and 8 look almost as tight. I like row 9 and getting across the South box.

From r9c3, one of two gets through, and from r9c9, one of five. Cells r1c1, r3c4 and r8c5 are new clues. And you can build a cluster from the fact that either the blue crossed 7’s or the green crossed 7’s above are true.

On World’s Hardest 36 of 5/29/18, top down freeforms, once the freeform starting at 5r1c9 touches 5r3c4, it can’t get through r7. 5r1c9 is an orphan, and can be removed.

The 9r1c9 ff is forced to touch r4c8, and loses its chance to cross r8.

Off this page,two pages show how freeforms support limited pattern overlay and combine with coloring in pattern analysis. Adjacent Line Freeforms assembles likely candidates of one value for trials, based on pattern restrictions on another value. Patterns and Coloring demonstrates pattern analysis techniques based on coloring.

Fireworks

X-panels are useful in elimination methods known as fireworks. A series of posts from 8/28/21 to 1/25/22 show how matching bv, slinks, and elbows across X-panels create n-sets and resulting fireworks eliminations. An elbow is a crossing line with a single candidate in each line remainder. The preliminary conclusion of this review, is that fireworks eliminations are too infrequent to justify searching for matching elbows on the X-panels.

The series continues with post 2/2/22, that reviews the firework’s sixth forum example, the only one that goes beyond matching elbows. It is found to be a trial. Two concluding posts demonstrate that results of the other forum examples introducing fireworks are duplicated by Sysudoku Advanced  methods.

Leaving Exhaustive Methods

The Sysudoku Order of Battle has come this far with the lower hanging fruit, the methods with constructed aids that limit spotting search to practical limits, and suggestions to “find them all”. One example is the irregular XYZ, where we filter the search down to the few hinges with necessary wings on the grid, then try to construct the AIC wink that makes them work.

In many cases, this order has  progressed far enough that we have accumulated bv and slinks for coloring clusters. Now we move to methods not as practical to do exhaustively, where personal preference decides how far to pursue them, before AIC building.

Aligned Pair Exclusion (APE)

Pick two target cells in a bent region with value combinations matched by ALS in the region..  No combination of values, one from each target cell, can be seen by two value groups of any ALS.

An ideal situation for APE.  Two target cells (rounded outline), two large ALS with value groups 2568 and 2367. Combinations of 2 and 6 with 2567 are present in one of the two ALS. Since 4 is not present in the ALS, combinations of 4 with 2567 are safe.

This untypical example comes from the ultrahardcore 89 post of 9/7/21. Go to this post, or the interesting Coloring Sue de Coq and APE of 6/5/12, and tap APE on the “Tagged “ list  at the bottom, for a host of APE examples.

Grouped AIC

A set of candidates of the same value in a box or line is a group. A group is considered true when a candidate of the group is known to be true. This makes grouped weak and strong links possible. Every member of a group A sees every member of another group B in the same box or line, If group A contains a true candidate, group B cannot. That’s a grouped weak link. We sometimes say that group A sees group B. If group A is false, i.e. has no true candidate, and there are no candidates of the group value in the row or column outside of the two groups, then group B must have the true value. So those are the conditions for a grouped strong link.

Value Groups in ALS

An ALS value group is the set of all candidates of that value in the ALS. Every ALS is contained in a a box or line.  Value groups of ALS in the box or line are group slink partners , but value groups  in ALS in different units may see each other by sharing value and a line or box. So when we say that an ALS can only give up a single value, and mean that it can only give up one of its value groups, we are talking about ALS in different units, with matching value groups sharing a third unit, with only one value group in one ALS getting the true candidate.

ALS_XZ Partnering

The predominating ALS elimination method is the ALS_XZ, where two ALS have value groups of the same value X that see each other, so that only one of them can have the true X candidate. This grouped weak link is generally known as the restricted common. In Sysudoku grids the RC is a thick, short-dashed black curve.  The two ALS must also share a second pair of groups of value Z, not necessarily seeing each other.

Since one ALS will get the solution candidate, the other ALS will eventually lose the X value. An ALS can give up only one value, so its other groups, including its Z group has a true candidate.  We don’t know which ALS has the true Z candidate, but any outside candidate seeing both Z groups will see it, and can be removed.

To find an ALS_XZ, look for a restricted common value X, then for value groups of a second value Z.

Here’s an ALS_14 from Heine’s ultrahardcore 311 posted 3/30/21. The RC is a simple wink. One ALS is a bv cell. X=1, Z=4.

In this case, the 4r6c4 removal leaves an Er6 boxline.

This example illustrates the hardest part of ALS_XZ spotting, the identification of ALS among the scattering of values among candidates.  Once you recognize the “right” 3 cells of 4 values, the RC and matching Z group leap out at you. Is there a way to break it down into one thing at a time, and easiest first?

First of all, instead of looking for pairs of ALS of all types, scan for possible ALS partners for ALS of the simplest type, and advance through first ALS from easiest to hardest. Most would agree that would put bi-value cells in the lead as a first ALS. Then as we look for ALS partners, filter the search by having to have a single in a crossing line, or a single or aligned group in a box. With that satisfied, if there is a matching value in the line or box, build the smallest ALS around those matching candidates. That prescription covers the ALS_14 above, and in fact, most reported ALS_XZ.

That prescription covers the ALS_14 above, and in fact, most reported ALS_XZ.

Here is another bv to box example,  found by reader Gordon Fick in Antoine Alary’s More Extreme Sudoku 200, posted 5/15/15. The hard part is identifying the SE box ALS and Z group.

Stuart’s The Logic of Sudoku Figure 29.1, RC matching value is a single in a crossing line. We find the 5 matching the bv, then add in the 28 for a 3-cell ALS_15 partner. 5r4c5 can’t be included in a larger ALS.

Another possible first ALS in the XZ scan is a long line ALS with a single of a box confined slink.

In this ALS­_15 from Stuart’s The Logic of Sudoku the single 1 sees group 1 in the same box and bv 25 is added to form ALS 1257 in r2c279. It works even when the second line is far down the grid.

Then we notice another ALS partner  with a 5 group another r1 5 can see. Sometimes the large population of ALS is an advantage.

Finally, what happens when two value groups in ALS XZ partners see each other. Back in Alary’s More Extreme Sudoku 200 again, Gordon finds  crossing line ALS partners sharing two values 7 and 8. In the solution, one ALS gives up 7 and the other gives up 8, leaving the other values in each ALS locked. 8r7c7 sees all 8’s in both.

A home page search on ALS_XZ brings many examples, and posts devoted to ALS mapping, the attempt find and record all ALS, and in the process, all ALS_XZ. The examples show that the key ot most ALS_XZ are singles, the candidate value groups, and aligned value groups enabled to see value groups in other ALS and be seen by outside candidates.

The Death Blossom

There is a way around the restricted common requirement of ALS groups seeing each other. Less usefully, its mechanism also allows more than two ALS Z groups for a victim to be required to see.

On the first point, here’s a DB from Andrew Stuart’s Strategy page on Sudokuwiki.org. Neither value group 1 or 7 see each other, but both value groups see a candidate in a bv and a common value groups seen by an outside candidate.

In DB talk, the bv is labeled a stem cell, and the ALS become petals. Now if 7r3c3 is true, it takes a value group from each petal, value groups 1 and 3 in the two petals are locked and the stem loses both of its values.

This strategy can be expanded to include stem cells of three or more values, but then three or more ALS have to have value groups that see the stem cell candidates, and  three or more Z groups must be seen by a single victim candidate.

Unless one ALS matches 2 of three stem values, as in ultrahardcore 47.

Hodoku comes up with another type of victim alignment, where the Z=6 groups are themselves aligned, enabling 6r5c5 to see both.

When it runs out of prepared examples, Hodoku can construct an example on your order, and build a puzzle around it. A coding triumph, but not a common occurrence.

Another way to stretch the death blossom is to find superimposed ALS, a decidedly inhuman task. Here is another Hodoku wonder. Cell r8c4 belongs to three ALS, and two 4 cell ALS share three cells.

The AIC Building page features chains with ALS nodes, and ALS-Wings off AIC Building describes inference chains of ALS nodes alone, with examples. The first Death Blossom above is an ALS-wing.

AIC Building

Now that you’ve had a look at all the local area methods in the order of battle, and ALS pairs, as well as exhaustively generated XY and X AIC, and gotten your coloring clusters in place, it’s time to tackle AIC building. We say “building”, not  “looking for”, because you don’t spot the typical AIC. You start one with potential outcomes in mind, then find ways to extend it and branch off of it, until terminates, with the result you had in mind or some other.

In computer solvers, AIC building comes early, because the solver has its own priorities and menus. And because AIC building is straightforward to code, and explain to users. In human DIY solving, we want to recognize easy eliminations, and build visual aids for more,  before watchfully building AIC, node by node.

There are actually two distinct AIC building  situations. One occurs when you have an “almost” repertoire method you need a weak or strong link to complete, and can start an AIC that may satisfy that need. In that type, you have the right AIC, or you don’t.  It’s a short search. There’s a specific target terminal. The solver is not going to help, because it is not coded to recognize  “almost “ methods.

Then there’s AIC building,  the late Sysudoku Advanced phase, where the elements are identifying starting candidates, with potential targets, recognizing nodes that continue chains, choosing branches that move toward targets, and returning from branches that fail, to continue the AIC. Now you have aids and experience to follow up on AIC building results.

The nature of AIC Building is illustrated by this AIC ANL from Royle 17-7295 posted in the review of Berthier’s The Hidden Logic of Sudoku on 1/3/17. It comes after two hidden UR’s, a finned swordfish and a X and Z reversable ALS_XZ.

This build started at 4r1c8.  Among the targets are 4r8c8 by ANL, and 5r8c8 by 1-way. Only 3r1c7 leads to a slink.  From 3r1c3 there’s 9 to r1c1 or r4c3, and 3 to 5r3c3.

You could choose r1c1 and see the return to start, or the hop to 4r7c1, but let’s say you head for the ANL, avoid the blue return to start again, and arrive at 4r8c3.

We might lean on this success by checking if there is another slink into 4r1c7. The reward is a second ANL.

This is how it goes in AIC building. Coloring takes over at this point, and the AIC wraps green.

Your reference for AIC types beyond the ANL and node types is AIC building, off the Guide page.