Sysudoku Advanced begins after line marking with easily spotted unique rectangles and remote pairs.
In the unique rectangle, two values appear in corner cells of a rectangle over two boxes. Candidates are removed that reduce the four cells to the same two candidates, or to a set of two candidates in diagonal corners that can be interchanged for a second solution.
The removals are justified by the belief that the puzzle composer would not leave a multiple solution that obvious. In this one, any of three 7 candidates being true leaves 13 in all corners. This is type 2 UR, but it takes a full page table to cover all types. One is in the Guide, and a copy starts the Solving Tools page.
A remote pair is a chain of 4, 6, 8, … bv cells of the same two values. Any outside candidate that sees its values in two of the chain cells four cells apart, (three between them) sees its true value, and is removed.
This 6 cell chain in A.D Ardson Very Hard 238 posted 5/23/17 houses two remote pairs. The four removals leave a naked pair c6s45 removing two 5-candidates. Less obvious are two bv eliminations.
In the Sue de Coq at left, the chute SEc7 must be 9(3+4)(5+8). Read that as 9 and (3 0r 4) and (5 or 8). This requires r9c9 to supply 3 or 4 to the box.
In the XYZ-wing on the right, weak links on values 3 and 4 means one of the three 7 (Z) candidates is true.
If Z=7 r4c5 is not true, 3 or 4 is, making one of the wing 7’s true. 7r4c6 sees all three 7’s, and has to go.
In this irregular 139-wing, an alternate inference chain (AIC) serves as a weak link, enabling 9r6c7 to “see” one of the wing 9=Z candidates.
With the strong links (slinks) in the hidden unique rectangle (UR) can you figure out why 9r6c2 cannot be true? This is from KrazyDad 575, posted 7/13/21.
The idea behind the XYZ wing goes further in the WXYZ wing, with four candidates in the center cell, and three bv wing cells.
More frequently encountered is the BARN, a naked set of n cells in a bent region (box and line) with a single value in both box and line. This one is from 12/7/21. The origin of the Bent Almost Restricted n-set, is posted 8/25/15.
Next in the order of battle are the two easiest-to-find alternating inference chains. What alternates is strong and weak links, slinks and winks, solid and dashed. They propagate an inference from one end to the other. The simplest AIC in concept is the X-chain.
X-chains connect candidates of the same value X. This 2-chain is an ANL, standing for almost nice loop. Join the chain ends and the victim with two winks, and the loop alternation is “almost nice”.
An AIC with a slink on each end is an ANL. It functions as a slink. At least one end candidate is true. If an outside candidate sees both, it must be false. It’s “at least” because both ends could be true, In that case, no inference travels on the chain.
By the way, for DIY fans, curve drawing in (C)PowerPoint gets some attention on the Solving Tools page.
Spotting long AIC can be aided by the X-panel. Here’s the 2-panel, with freeform zig-zag lines to represent dotted lines, for faster sketching. How close are other 2-chains?
A group is a set of candidates of the same value, which act as a true candidate if one of its members is true, and a false candidate if all of its members are false. Being simple, X-chains are easily extended by grouping.
In Nakimoto 63 of 2/18/2020, the 9-chain starting in 9r3c7 ends in the group r4c79. It is a group in r4, and is a slink partner with 9r4c1. It terminates a grouped ANL. 9r5c7 is a victim because it sees 9r3c7 or the true candidate in the group, regardless of which grouped candidate is true.
The x-panel remains useful for grouped X-chains.
XY chains connect bv cells with weak links. The bv provide the slinks. On a bv map, draw long curves connecting all the bv you can. These become XY railways. The XY ANL follow the railway on every repeated value. Here, two railways house 3 ANL, removing 5 candidates.
When you can, you always close the AIC you are building with a return to the starting point to close a nice loop, because then every outside candidate that sees both sides on any link is toast. That’s because it’s true of a slink anywhere, and if it’s a wink (weak link), the chain all the way around the nice loop is an ANL. The effect is often deadly.
The X-panel is the tool, not only for AIC , but also fish, both regular and finned. It’s a panel of 9 small grids, one for each value.
Here is the 6-panel that reveals the five column (|) regular starfish occuring when the 6r3c4 fin is false. The fish removes X candidates in its rows (+) and outside of its columns. Of course, if the fin is true, and the starfish doesn’t exist, 6r1c5 is false.
So it’s all about seeing the fin. If a candidate sees the fin by AIC, it’s a finned fish victim. We call it a kraken fish, and a kraken victim. Are there any kraken victims for this fish? For the full grid picture of this ultrahardcore 221 finned starfish in post 11/02/2021, look in the monthly archive. The post date is all you need.
Before leaving the X-panel, Sysudoku order calls for examination of the panels for less populated edge regions, where patterns, the possible candidate distributions satisfying Sudoku rules, are limited
As an example, in the 4 panel of Nakamoto 103, posted 3/24/2020, the 3 freeforms leaving 4r1c1 and arriving at opposite side represent the 3 possible patterns that include that candidate. Since all 3 patterns also contain 4r3c7, in the cluster it is green as well. Slinks in c1, c7 and r3 bring the blue color to slink partners, the cluster traps 4r1c3, because it sees both colors, one being true.
After exploiting the relatively simple X and XY ASC, and before general AIC building, Sysudoku Advanced turns to Almost Locked Sets (ALS). A subset of the bypass is said to be locked because in its n cells holding n values none of those values can be removed by a candidate or group seeing a true outside candidate. An ALS is a set of n cells in a box or line holding (n+1) values. ALS candidates of the same value form a value group. An ALS is almost locked, because one and only one value can be so removed. This property forces eliminations in three Sysudoku ALS methods.
In the ALS_XZ, two ALS are found to have value groups of the same value X, that see each other. That is to say each member candidate sees all member candidates in the other ALS. The weak link signifying group seeing is called the restricted common. Because of it, one of the matching groups, and its ALS, has the true candidate of value X .The other ALS will eventurally lose its X value group, and have its other value groups locked. If the two ALS have a second group of matching Z value, any outside candidate seeing both Z groups is looking at a true Z value, and is removed.
Here are two ALS_XZ on the same grid, from the French Sudoku Maestro magazine.
In the ALZ_52, the restricted common X=5 group and candidate see each other in the NW box. Candidate groups are very productive in the ALS XZ.
One of the ALS in the ALS_35 is a bv. This is the case in most of them. Follow the above rationale in both ALS_XZ.
One or both of the ALS_XZ partners can be defined in boxes. This ALS_18 is from Heine’s ultrahardcore 91. Note the 1 grouping applies in the ALS, even though both box and line have more 8 candidates.
A second method based on the same “lose one group” ALS property is the Death Blossom.
In an example from Andrew Stuart’s sudokuwiki strategies section, the two ALS share values 1 and 7, but neither 1 nor 7 groups see each other, as required for a restricted common.
But each ALS has a group that sees a different candidate in a stem cell. One of the petal ALS will eventually give up that group, a 1 group or a 3 group, to the stem, and its 7 group will be locked. Whether this is Blue 3 or green 1, 7r3c3 will see a true 7, and will not be in the solution.
Bi-value cells make the best stem cells for death blossoms. If the stem has 3 values, 3 ALS value groups must see them, and a victim will have to see all 3 groups. It only gets less likely as the stem cell candidates increase.
One more way to use ALS here, is the Aligned Pair Exclusion (APE). It monitors how the combinations of values in two cells in the same unit are limited by the ALS that see both of them, and removes candidates missing from the permitted combinations.
In this deceptively simple example, from ultrahardcore 353, 11/30/21, combinations of 7r5c9 with 4, 5, and 8 of r6c4 are prohibited collectively by ALS 47 and 57 in C and 378 in r6.
What is aligned in the APE? Not necessarily the combination cells, but the ALS, including bv, with both the pair of combination cells.
By this time, we probably have underway the most distinctive feature of Sysudoku Advanced, its full version of Medusa coloring. Coloring marks slink networks called clusters, with candidate, groups and AIC strong links. Coloring clusters can be started early and built without interference with other methods.
Here we have three clusters. Blue or green candidates, red or orange, purple or grey, are true. Clusters can become related. In r6, we learn that grey and green cannot both be true. This is bridge means at least one of the opposing colors, blue or purple, is true. Since 1r9c9 sees blue and purple, it is removed. This trap brings a new bv, and a cluster expansion as the new r9 slink turns 1r5c5 grey.
As eliminations and confirmations occur, clusters expand, with color bridges and mergers bringing new traps, past promising toward overwhelming, with each elimination and clue placement. Clusters cross lines and boxes, confirming half, and removing half (a wrap) when a true color is discovered. Expect many difficult puzzles to collapse by coloring wrap, rather than the next candidate elimination.
Sysudoku extends the ability to trap to candidates outside the cluster by lite coloring. Coloring applies to aligned groups that are seen by single candidates or other groups. AIC slinks also apply in coloring, and can extend clusters and cause wraps.
Having checked for more immediate eliminations, and built resulting clusters, we are ready to build AIC beyond X and XY chains. Not by finding them, but by starting at slink partners and finding a next wink then and slink of any kind, while watching for a direction to a possible ending. What makes a good starting candidate? A slink partner seen to have several wink/slink alternations in another direction, and several possible ending targets. Of course, an ANL ending is very popular. For that you want ending candidates of the same value, and corresponding victims. But don’t agonize too long. There are several alternative endings.
Sysudoku labels this ending, this type of AIC, a boomerang. As you build and follow up branches, you’re looking to return to the starting cell, and wink at a rival of the starting candidate. This one starts on 2r1c4 and could have had 6r1c4 as a target. Another boomerang starts at 6r1c6 and follows the same AIC to rival 4.
The post of 7/23/13 also has the color wrap of a previous puzzle, and some wildly irregular XYZ snaps.
AIC starting from bv are usually targeted as a 1-way. You start with several like value candidates in view and hope to wink into one of them. If the starting candidate is true, then all of the targets are false. The AIC starts with the assumption the starter is false and carries this inference to the target, which is false either way. So the AIC is being used in only one direction.
This time, a branch off eliminates 8r4c5 as well, and the resulting boxline wipes out two more. The puzzle is ultrahardcore 45. Post 8/31/21 features cluster building to a double cluster wrap, as well.
As you try to keep an AIC going, you can include the value groups of an ALS as a resource. You wink into the ALS to a value group, and from there, have slink access to its other value groups. That’s because if the entry group is false, all the other groups are locked, meaning they contain a true candidate of the group’s value. You can wink back out from the second value group, to continue the ASC, or to close a 1-way, or to use it as the terminal group of an ANL.
In this tricky example from ultrahardcore 221, posted 11/2/21, the s-chain from 6r9c9 goies through the candidate 6r7c2 to the bv slink, then winks to orange 7r7c7, then the coloring slink to red 1r6c1, winks into the single 1 group of the ALS, and slinks to the 6 group, for the ANL removing 6r9c2.
These AIC examples, and many more in the later posts of the blog, were found by two computer solvers coded to duplicate well known human methods, and available online. These are Andrew Stuart’s Sudokuwiki solver, and Philip Beebe’s PhilsFolly, nicknamed “Beeby” here. Neither of these solvers use coloring. I know you’re going to point out politely that the slink from orange 7 to red 1 above is coloring. Yes, my version uses a coloring shortcut. Beeby did a bv slink to 1r7c6, a c6 slink to 1r6c6, then an r6 slink to 1r6c1. Having a cluster in place helps AIC building.
It probably takes more investment in ALS than you’d care to make, but you can build an AIC entirely of ALS nodes. PhilsFolly calls it an ALS-wing.
In this ALS-wing ANL, both terminals are value groups. The circles are marking value singles, an asset in building ALS chains.
One more Beeby trick borders on AIC criminality. It is PhilFolly’s Complex Discontinuous Loop, which is a 1-way. To illustrate, here is a misdemeaner.
The prospective 1-way from 3r8c3 runs aground when the final 7 slink fails. But in the 1-way, a branch from 7 actually cancels the preventing 7 and restores the slink confirming the 3’s rival 7r4c3, and making 3r4c7 false regardless of 3r8c3 being true or false. The post of 11/5/21 and several others have much more elaborate 1-way branches hacking the jungle to let 1-way AIC through.
These examples have outlined a whirlwind of advanced solving ideas for DIY Sudoku solving. A more careful account of each, with more citations to posts, is available in the Guide.