X Chains and Loops

X-chains are alternating inference chains (AIC) of candidates of a single value X. As other types of AIC, X-chains build almost nice loops(ANL) and nice loops. They are also the most prevalent form of AIC as a link, weak and strong. The reason is their simplicity. When you are trying to link two matching candidates X with an AIC,  an X-chain is the most direct, the most likely and the easiest to verify.

From the reviews, Krazy Dad’s SuperTough vol.5, book 6, no. 5 provides several examples.

Along with an XY ANL (green), we have a regular 583-wing (orange) with an irregular victim 3r9c1. After finding  the XYZ-wing, a sysudokie would be looking for a Z=3 seeing all three of its toxic set members. 3r9c1 sees the two c1 members, and you have the two points you want to wink. Here’s where slink marking helps, as the position of  3r9c6 signals a vertical slink to follow. We interpret the C box slink as a wink and remove 3r9c1. There is an XY chain wink, but the 3-chain is more direct.

The very next grid in KDST 565 gives up two skyscrapers, the most frequent form of X-chain ANL.

“Skyscraper” is one of many names applied to different forms of X-chain. The 2-chain shows why the name is apt, but sysudokies don’t look for forms of X-chain by name. Instead, they look for X-chains of all lengths and forms.

In support of that search, sysudokies transcribe candidates of each value to a 9 x 9 table, an X-panel, leaving clues and other candidates behind.

Here are the X-panels on which the skyscrapers appear, not taking either removal into account. Slinks and fish lines stand out  clearly.


In XY chains the slinks and winks are separated for you. In X-chains, a slink is exactly two candidates in the box or line. The X-panel makes it easier to connect slinks by winks into inference chains.

Either skyscraper finishes KDST535, but you can find more 3-chain ANL on the 3-panel.  I’ll show mine at the end of this page.

Grouped X Chains

Candidates of the same value X within a box or line can be nodes in X chains. A group is considered true if it contains the true candidate, and false if it doesn’t. This convention allows slinks and winks to be defined between groups and individual candidates, and between groups. If two X-groups in a box or line include all X candidates of the unit, there is a slink between the groups. That’s because, if the true candidate is not in one group (false) it must be in the other (true). If all X candidates are not included in the groups, the link between the groups is weak. It’s a wink, because if one group contains the true candidate, the other group cannot.

An example from World’s Hardest Sudoku (WHS) 200 illustrates candidate/group inference transfer abilities of grouping. Groups are denoted by closed boundary curves. Note which unit defines each link of the chain. Starting at 4r2c5, we have a candidate/group slink in r2, a group/candidate wink in NW, a candidate/group slink in c2, a group/candidate wink in W, and a candidate/group slink in r5. Now the victims in C see the entire group in C and the terminal candidate in N.  Be sure to account for every link by the grouping principles above.

Group to group links are also useful. Here is a far out example from World’s Hardest Sudoku (WHS, just a title) 134 in which a group to group slink makes an turn around that enables the West group to see all of the unrestricted 8’s of a Bent Almost Restricted n-set, a BARN. The East eliminations are a boxline on Er5.

While X-chain almost nice loops (ANL) are well represented in Sysudoku collection reviews, X-chain nice loops are almost entirely absent. A likely reason is that a long X-chain nice loop is a railway for ANL. When the solvers encounter the first of these with a victim, they are unlikely to continue the effort to close the nice loop, following up on the removal instead. The nice loop often loses its opportunity to be counted, but solving progress not necessarily slowed.

WHS 134 has one of those rare X-chain nice loops. It is actually shown here, but in the review post, two skyscrapers are adjoined for the 5 link  ANL eliminating 5r2c2.  Here, the missing wink is added between the terminals in NW to create the nice loop, gaining another elimination, 5r7c2.



More often, grouping is a key to closing the nice loop.

 X-panels are not transcribed for help with X-chains alone. As each X-panel is completed – and I  do it by increasing X – the panel is examined for fish and for easy to exploit pattern restrictions, as described on separate pages.

Returning to further revelations for KrazyDad SuperTough 5 6 5, there are two routes for an ANL (green) to eliminate 3r4c1. In combination with the previous ANL, these combine to form an ANL confirming 3r4c6 as a clue. This happens when the departure from alternation is a pair of slinks converging on the new clue.