XY Chains and Remote Pairs

The AIC, or alternating inference chain, is a fundamental element of advanced Sudoku. The Advanced page illustrates how AIC are used directly for elimination and confirmation of candidates, as ANL and nice loops. AIC can also be constructed to serve as links supporting other methods. An outstanding example is the irregular XYZ wings

The XY chain is a natural form of AIC, a chain of bi-value cells connected by weak links. The two values of each bv are strongly linked. If one is false, the other is true. These internal slinks alternate naturally with the cell-connecting winks. The winks can be slinks or AIC functioning as winks between matching candidates. When the terminal candidates match in value, the chain is a slink.

If one terminal candidate is false, the other is true. Since at least one terminal must be true, the terminals form a toxic set. An XY chain and one of its toxic set victims is often referred to as an almost nice loop, or ANL.  “Almost nice” means the alternation is broken at only one point.

Going beyond that, if there is a wink between the terminals, alternation is perfect, and the XY chain becomes a nice loop, making all wink partners around the loop toxic to any matching candidate seeing them both.

The  XY-chain of three bv is generally called an XY-wing. There is no reason to consider it separately.

Looking over the Sysudoku collection review posts, you see many examples of XY chain ANL and nice loops, some having quite long XY-chains. Finding these results is easier than it looks, because of the XY Railway, a Sysudoku graphic tool easy to implement in office software, with which you draw a “railroad” network along which every XY chain must lie. Spurs are added to the railway to connect bv with the network, adding to the pool of XY chain terminals.

Here we walk through an example from the review of A.D Ardson’s Sudoku Diabolical, vol. 1, puzzle 30. You’ll need to reference this slink marked grid, as left by an 8-wing in the close of line marking.

In your 9 by 9 table, copy only the bv candidates.


I write the digits in increasing order, regardless of their placement in the cell. On the bv map, I’ll draw a single curve through each bv entry and from one bv to another, matching an entrance value to the last exit value. Starting in the Northwest corner, 3 is a possible terminal because there’s a second 3 on the map. The chain goes 35->59->94->45.  There I have to stop since the exit 5 sees no entrance 5.

In © PowerPoint, curves are drawn by clicking the screen at the starting point, and at points directing a curve drawing algorithm. Consult a hip 5-year old for details.

I like to go through the bv digits horizontally. I didn’t get to my second 3, but progress has been made. This rail is a path for XY chains, not an XY chain itself. The 59 bv is a starting  terminal to match my finishing exit terminal, and I have “found” an XY-wing. To see if there’s a victim, we must consult the full grid. 5r2c2 and 5r9c4 are toxic and 5r9c2 is in the crosshairs.

The record-keeping advantage of the bv map now becomes more apparent. I can look for branches off into other directions. The sequence 35->54 goes nowhere, as does 94->46. The dotted curves  represent looking, but not drawing. Moving down, I try 78->85->54. Then it’s 58->87->76, as I back up to make the chain rail as long as possible.

Get out your copy of the bv map,  start with 58, back up to 45 and see where your railway can go.  To be continued, after I talk about a special case of the XY chain, which is handled differently.

The Remote Pair

An XY chain in which all bv contain the same two values is known as a remote pair. Such a chain is, of course, very noticeable, and we need no railway to find it. A remote pair of two cells in a line or box is a naked pair, eliminating any other candidates of the two values from the containing unit. But longer remote pair chains can produce similar eliminations. The terminal cells of remote pair chains of 4, 6, or higher even numbers of cells also act as naked pairs. Their candidates of each value are toxic.  It’s because, if a candidate in the first cell is false, it’s bv partner is true, making matching candidates in the next cell and every other cell after, true. An outside candidate seeing both ends of any of these slinks is false.

How’s that again? Its made graphically clear in the following example.

From Rebecca Bean Extremely Hard, Part 7, Puzzle 6, here is a remote pair (black) on a chain of five matching bv. This chain allows a second 4-cell remote pair (red), which has no victim.  Following the argument above on the black remote pair, if 4r4c3 is false, 6r4c3 is true, making 4r5c2 and 4r9c8 true. The remote pair victim, 4r9c3 sees both ends of the slink between 4r4c3 and 4r9c8, one of which must be true.

Many remote pairs are found right out of line marking, but you can have a lurking ‘almost’ remote pair activated by an elimination.

From Funster Hard to Extreme by Charles Timmerman, here is remote pair with an interesting twist, in Extreme 102. One victim is regular, seeing both terminal 1’s in its lines. The second sees one terminal by a 1-chain wink.




X-chains are the type most frequently used for AIC victims and methods, but any AIC will do. Can you replace the 1-chain by an XY chain?  Answered later.

Back on Diabolical 30, . . .  the railway curve from 4r6c3 goes all the way to 6r2c7. We look along the rail for a second 4, then on the grid for victims. Did you go ahead and find two?

Now here’s a railway hobo trick: Reverse your direction and look for a spur over to that second 6. It’s another XY wing with two more victims.

Another productive idea is to seek a spur onto the railway on behalf of a bv being left out.  The spur sets the direction. Here we see that in the West box, 78->85 goe nowhere, as does 87->76, and if 37 gets on, its going in the wrong direction, away from the only 3.

If you notice part of the dotted chain and extended it as far possible in both directions you get the terminal 5’s, but no victims are watching.

This example illustrates well how the XY railway provides for systematic discovery of every available XY chain ANL.  For you, it isn’t at all necessary to draw the explicit XY chain slinks and winks I do for blog publication.  If you’d like to, some directions for that are available on the Tools page.

Did you find XY glasses for 1r5c3 to see 1r2c4 in funster Extreme 102? If not, you may be trying too hard.