Sudoku HowTo

# X Chains

An X Chain is a looped chain built using alternating strong inferences and weak inferences all based on the same value. When the chain forms a loop there are possibilities for candidate reduction. An X Chain is a special kind of Nice Loop. The Standard Notation for Nice Loops can be used to describe X Chains.

An X Chain is considered Continuous when the loop is completed and the alternating strong and weak links are in tact. A continuous X Chain will always have an even number of cells and links.

An X Chain is considered Dis-Continuous when the loop is completed and there are two consecutive strong or weak links in the loop but all other links comply with the rule of alternating inference. The discontinuity is on the alternating inferences and there can only be one of them. The chain must still form a loop. A dis-continuous X Chain will always have an odd number of cells and links.

As long as the X chain complies with the alternating inference rule, the links used to create an X Chain can be applied in the reverse direction. The strong and weak inference logic will work in either direction around an X Chain.

Some more common solving strategies can be re-characterized as X Chains. An X Wing is a Continuous X Chain with 4 links. A SwordFish with a 2, 2 and 2 format is a Continuous X Chain with 6 links. Claiming often shows up as an Dis-Continuous X Chain with 3 links.Continuous X Chains

For a Continuous X Chain all instances of the candidate used to build the X Chain, that are not part of the chain, can be removed from the units that contain the weak inferences in the chain.
The example in Figure 1 shows a Continuous X Chain based on the value 2. Notice it could also be described as an X Wing. Notice also, that if you try and create the same X Chain going the opposite way around the loop the strong/weak inference logic still applies.

The Standard Notation for this X Chain is: = [R1C5] -2- [R3C5] =2= [R3C7] -2- [R1C7] =2= [R1C5] -

Now, why does this work?

The long winded description would be - "If the value 2 is placed in row one column five then the value 2 cannot be placed in row three column five. If the value 2 is not in row three column five then the value 2 has to be placed in row three column seven because there are no other cells in row three with the candidate 2. If the value 2 is placed in row three column seven than the value 2 cannot be placed in row one column seven and if the value 2 cannot be placed in row one column seven then the value 2 has to be placed in row one column five because there are no other cells in row one with the candidate 2."

This means that every cell at the end of a strong link in the chain will contain the value used to create that strong link. (The cells marked with an A as the chain is shown in this example would all contain the value 2.) If you apply the inferences in this example chain in the opposite direction, the chain will still contain a value 2 in all cells at the end of the strong links, but it would be all the cells not marked with an A because all the arrows would point in the opposite direction.

The value 2 will either be in all the cells marked with an A in this example chain if the inference logic is applied in an anti clockwise direction, or the value 2 will be in all the unmarked cells in the chain if the inference logic is applied in a clockwise direction. In either case, there will be a value 2 in every unit used to build the chain.

For the links in the chain that are based on weak inferences there could be other cells in the unit that contain the same candidate used to build the chain. We know that one of the two cells in each unit in the chain will eventually contain the candidate so it cannot be in any other cells in the same unit.

In the example the value 2 can be removed from all other cells in column five and column seven. (Shown as grey cells in this example)
The example in Figure 2 shows a much longer Continuous X Chain based on the value 3. All the same logic can be used on this X Chain. It just takes a lot longer to find

The Standard Notation for this X Chain is:

= [R1C6] -3- [R3C5] =3= [R3C8] -3- [R2C8] =3= [R2C1] -3- [R9C1] =3= [R9C4] -3- [R7C6] =3= [R1C6] -

Applying the inference logic in the direction shown in the example causes a 3 to be placed in all the cells marked with an A in the chain. Applying the logic in the opposite direction would cause a 3 to be placed in all the unmarked cells in the chain.

In either case there will be a value 3 in one of the two cells used for every link.

The value 3 can be removed from any other cells in the units containing the weak links. In this example, column one and box eight have candidates that can be removed.Dis-Continuous X Chains (Weak Links)

For a Dis-Continuous X Chain with two adjacent weak links, the candidate used to build the chain can be removed from the cell at the end of the two weak links.
The example in Figure 3 shows a dis-continuous X Chain with just three cells and two adjacent weak links. You might recognize this pattern as claiming.

The Standard Notation for this X Chain is:

[R3C7] -6- [R1C7] =6= [R1C9] -6- [R3C7]

Why does this work?

For this example, if the value 6 is placed in the start cell (row three column seven) then the value 6 cannot be placed in row one column seven. If the value 6 is not in row one column seven then it has to be placed in row one column nine (the end of a strong link). If the value 6 is in row one column nine then it cannot be placed in row 3 column seven (the last cell in the chain).

If you reverse the direction of the inferences in this X Chain the value 6 will be placed in row one column seven (the end of the strong link) and still cannot be placed in row three column seven (the last cell in the chain).

The last cell in the chain is always in the same unit as a cell at the end of a strong link. Cells at the end of strong links will contain the value used to build chain. Therefore, the last cell in the chain is always in the same unit as a cell that will contain the value used to build the chain. The cell at the end of the chain that starts and ends with a weak link can never contain the value used to build the chain.

The candidate 6 can be removed from the yellow cell.

The same logic can be applied to much longer chains which contain two adjacent weak links.
The example in Figure 4 shows a longer dis-continuous X Chain build using the value 5 with two adjacent weak links.

The Standard Notation for this X Chain is:

[R1C6] -5- [R1C1] =5= [R7C1] -5- [R9C3] =5= [R9C6] -5- [R1C6]

We have shown above the cells at the end of strong links must contain the candidate used to build the X Chain, so either row one column one must contain the value 5 or row nine column six must contain the value 5. Either way, the value 5 cannot be placed in row one column 6.Dis-Continuous X Chains (Strong Links)

For a Dis-Continuous X Chain with two adjacent strong links, the candidate used to build the chain can be inserted as the solution in the cell at the end of the two strong links.
The example in Figure 5 shows a Dis-Continuous X Chain built on the value 8 with two adjacent strong links.

The Standard Notation for this X Chain is:

[R1C1] =8= [R2C3] -8- [R2C8] =8= [R4C8] -8- [R5C7] =8= [R5C3] -8- [R6C1] =8= [R1C1]

Why does this work?

The same inference logic applies here. The value used to build the X Chain has to be placed in the cells at the end of a strong links.

The X Chain can be constructed in either direction. In either direction the strong link forces the value used to build the X Chain to be placed in the ending cell.

In this example the value 8 has to be placed in the yellow cell which is at the end of a strong link regardless of the direction of the inferences in this X Chain.

30 March 2022
by John
RE: Comments and feedback for xChain

Thank you for your feedback. There are many ways to solve each puzzle and different people find different solving methods easier to look for. I believe W Wings are implemented and called WXYZ Wings. I guess it just depends which order I setup the solver to look for things.
29 March 2022
by Searching for W-wings would simplify at least one of your puzzles
RE: Comments and feedback for xChain

Hi:

I suggest you implement W-wings in your solver. They make many puzzles simpler. For example, in your page https://sudoku.ironmonger.com/howto/xChain/docs.tpl?setreferrertoself=true Example 5, there's a W-wing for r3c9=r5c7=48 (strong link for 8 in row 7) while allows for the elimination of 4 in r1c7 and r4c9.

Best regards,
Andy Potvin
afpotvin@hotmail.com
23 January 2022
by John
RE: Comments and feedback for xChain

The link between R2C8 and R4C8 is strong because the two cells contain a naked pair containing 5 and 8. That means that the 8 has to be in one of the two cells and the 5 has to be in the other one of these two cells. You arrived at the same conclusion by finding the hidden pair in R5C8 and R6C8 removing the extra 8s in the column. Either way, the link is strong.
23 January 2022
by Nelson Heinzen
RE: Comments and feedback for xChain

In Figure 5, the link between R2C8 and R4C8 is indicated as a strong link (double lines).
But I think it would be a weak link as there are other cells in column eight (C8) with the digit 8, R5C8 and R6C8.
The X-Chain in Figure 5 would not be correct if the link in column eight (C8) is indeed a weak link.
If a weak link cannot be used as a strong link, how do you explain that?
However, in the Figure 5 puzzle, there is a possibility of turning the (supposed) weak link in column eight (C8) into a strong link.
We can see that there is a Hidden Pair of 1 and 6 in column eight (C8), in cells R5C8 (168) and R6C8 (1568).
With R5C8 (16) and R6C8 (16), there would be only two cells left with the digit 8 in column eight (C8). Thus, we would have a strong link between R2C8 (58) and R4C8 (58).
I am learning advanced Sudoku puzzle techniques and I would like to clarify the doubt about this X-Chain link in Figure 5.
Sorry for some glitch in English, because I used an online translator to translate my text from Portuguese (pt-BR).
I appreciate if you can reply to my email (nelsonheinzen@gmail.com).
30 September 2014
by John
RE: Comments and feedback for xChain

Thanks for your feedback Shahapurkar.

As you can see towards the bottom of the links definition page any Strong Link is also a Weak Link. In these kinds of chains a Strong link can be used as a Strong link or a Weak Link. A Weak Link cannot be used as a strong link.
29 September 2014
by Shahapurkar
RE: Comments and feedback for xChain

In the figure 5 under x chain, you have indicated link between r2c3 and r2c8 as weak line. I believe strong links are indicated by double lines and weak links are indicated by single lines. Also, link between r28 and r4c8 is indicated as strong link.

How can this be, as per the definition of strong link and weak link?

Since I refer to my mail everyday and not this web page on x link, I shall be grateful if you could send me an explanation to my email address i.e. dshaps01@gmail.com. the character before 1 is zero.

Regards

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This page last updated on March 6, 2015