XY-Chains

XY-Chains is a way to connect two parts of the board that can't directly "see" each other. The "X" and the "Y" in the name represent these two values in each chain link. If we can connect the ends we can make deductions.

Originally these were found by extending Y-Wings into so-called Y-Wing Chains but it was quickly transformed into this more encompassing strategy. The same pincer-like attack on candidates that both ends can see by chaining bi-value cells. With Y-Chains the hinge relied on identical bi-value cells but in an XY-Chain the candidate can differ by one on each hop.

The example here is a very simple XY-Chain of length 4 which removed all 5's highlighted in yellow. The chain ends are 5 A7 and C2 - so all cells that can see both of these are under fire. It's possible to start at either end but lets follow the example from A7. We can reason as follows

• If A7 is 5 then A3/C7/C9 cannot be.
  
• if A7 is NOT 5 then it's 9, so A5 must be 2, which forces A1 to be 6. If A1 is 6 then C2 is 5.
  

Which ever choice in A7 the 5's in A3/C7/C9 cannot be 5. The same logic can be traced from C2 to A7 so the strategy is bi-directional, in the jargon.

The solver uses relatively simple chain notation with cells identified with [Row Letter+Column]. (There is an option to change to rYcX coordinates). In a chain we're alternating between strong and weak links but also turning candidates ON and OFF. To symbolise that the chain uses plus and minus. The number being turned on or off follows the symbol. The above example is
-5[A7]+9[A7]-9[A5]+2[A5]-2[A1]+6[A1]-6[C2]+5[C2]
5 taken off A3
5 taken off C7
5 taken off C9

In later documentation on Grouped X-Cycles you will see grouped cells denoted as +4[D4|E4] and when ALSs are used to make a link curly brackets are used: +7{H6|G6}. Rare exotic links like Unique Rectangles are named -9(UR[DF28]) as is an X-Wing -8(XW[-E3/-B3+B2-E2])

This next Sudoku puzzle contains an entertaining series of XY-Chains, starting with this rectangular one. It proves that 8 must be in either B3 or B8 and therefore we can remove the other three 8s in row B. Starting on B3 if that cell is either 8 or 6. If it is 6 then D3 must be 4 which pushes 2 into D8 which in turn makes B8 8. You can trace this from B8 back round for the same effect. A nice short XY-Chain, but as the next example shows, these four cells are a rich seam.

(Uncheck Rectangle Elimination)

Looking at exactly the same starting cell it appears we can make further eliminations, this time 6s in column 3. We go clock-wise, this time, round the rectangle. It proves 6 will either be on B3 or D3.

If you want to finish the puzzle by yourself, look out for a third elimination with those same four cells using 2s on column 8, or step through with the solver.

XY-Chains go way back to the earliest instance of the solver and had separate code for this strategy because the pattern is very simple and it pre-dates my work on AICs. But the selection of the “best” XY-Chain was very crude. I’d look for length 3 and return the first, then any length 4 and finally any length 5-12 and return the first. (Length here is the number of cells, not the chain links which are double the 'length').

I decided to see if I could use the AIC code to look for this pattern and re-use some code. The AIC Chain detection builds a list of the best 50 chains and allows the solver to “explore” other chains at the same step. This means XY-Chains share the same priorities: number of eliminations before length, length before same number of eliminations and so on. Same preference order as AIC and other chaining. This may change the score and solve order of puzzles that require XY-Chains.

Also the old XY-Chain code was not using Windoku and Sudoku X diagonal units which was a big miss.

There is also a side effect that URs and ALSs might be used in the links making a chain although it will be rare. In two minds about this since they are not part of the pattern but are logically fine. David Hollenberg found an example (choose second chain nect to 'explore').

XY-Chains Exemplars

These puzzles require the XY-Chains strategy at some point but are otherwise trivial.
New examples added here as of August 2025
They make good practice puzzles.

• Exemplar 1 (x5, score 81)
• Exemplar 2 (x2, score 88)
• Exemplar 3 (x2, score 89)
• Exemplar 4 (x2, score 90)
• Exemplar 5 (x2, score 93)
• Exemplar 6 (x2, score 98)