What is a link in a Sudoku puzzle?
A Sudoku link is a logical relationship. Links are building blocks used in advanced puzzle solving techniques. Links are most used in chaining and looping techniques such as Remote Pair, X-Chain, Turbot Fish, X-Chain Variants (2-String Kite, Skyscraper, Empty Rectangle), XY-Chain, Nice Loop/AIC, Group Nice Loop/ AIC, Almost Locked Sets (ALS), and Forcing Chain.
A “link” is sometimes called an “inference”. There are four types of links:
- Weak Links
- Strong Links
- Either-Or Links
- Group Links
1.0 Weak Links
Consider the following Sudoku in progress. All the cells having a possible 9 candidate have been highlighted.
Weak Link: If one candidate is true then the other candidate must be false. Both can be false.
With this puzzle there is a weak link relationship between cell R2C3 and cell R4C3 shown here with the possible 9 candidate highlighted in dark green:
The relationship is between the two candidates within the two cells. We indicate a weak link relationship between the two cells by having a dotted line with an arrow between the cells. The arrow indicates the direction of the week link.
So, what this means is if cell R2C3 has a value of 9, then the cell R4C3 cannot have a value of 9 as shown here:
Here is the case when both cells in the relationship are false. Both R2C3 and R4C3 are false because cell R7C3 has been set true with a value of 9 as shown:
A Weak Link can also occur between to candidates within the same cell shown here:
For a Weak Link to occur with candidates within a cell it must be a “bivalue” cell. A bivalue cell is just a fancy term for a cell having exactly two possible candidates.
If we look closely at the possible 9 candidate in cell R2C3 we see it is participating in nine different Weak Link relationships shown here:
There are five cells having a relationship with the possible 9 candidate in cell R2C3 and one more relationship between the 9 and the 5 candidates within the same cell.
2.0 Strong Links
Consider the following Sudoku in progress. We have highlighted a Strong Link relationship between cell R2C3 and cell R2C8 as indicated by the black outline and solid black line between the two cells:
An arrow is used to indicate the direction of the Strong Link. With a Strong Link if one candidate is false then the other candidate must be true. Here is what this means:
We have a value of 5 in cell R2C3 as indicated by the red circle. And it is false there is a 9 in cell R2C3 as indicated by the red X, then because there must be at least one value of 9 in the house making up row 2, then cell R2C8 must be 9. There is no other cell in the house making up row 2 where a value of 9 can be placed.
A Strong Link can also occur between to candidates within the same cell shown here:
Consider the following Sudoku where a Skyscraper is currently being highlighted. For the cell R2C5 we are showing three Strong Links:
At most we have ever seen is three Strong Links for a particular candidate in a cell. Be able to see all the Strong Links occurring for a particular candidate is important in building out advanced puzzle solving techniques involving chaining sequences.
Just as with the both-can-be-false rule with a Weak Link, with a Strong Link both candidates can both be true at the same time. However, this violates the most basic rule of Sudoku of only having one value be set in each of the 27 houses. On the Hodoku Wiki page for chaining it indicated that it is possible with advanced Strong Links both elements of the relationship could both be true. We have never seen it and it is hard to imagine it could ever possibly happen.
3.0 Either-Or Links
Consider the following Sudoku with all the cells having a possible 9 candidate highlighted in green.
The cell R2C3 and the cell R2C8 have a Strong Link relationship in both directions. When we have a Strong Link in both directions, we call this a “bi-directional” link. A bidirectional Strong Link is indicated by a single solid line with an arrow at both ends as shown here:
With the two cells above there is also a bidirectional Weak Link relationship between the two cells as well:
Whenever we have both bidirectional Strong and Weak Links between two cells, we can label this as an Either-Or type link indicated by a single line with no arrows:
When you have two sets of Either-Or type links lining up in the same columns you get an X-Wing pattern as shown:
4.0 Group Links
We will demonstrate what is a Group Link by example. Consider the following Sudoku:
The possible 2 candidate in cell R1C6 has a Weak Link relationship with the two 2’s in cells R1C8 and R1C9. We are going to treat the two cells R1C8 and R1C9 as a single “node”. As a single node, the cells R1C89 form a Strong Link relationship with cell R3C9 as shown by the solid line with an arrowhead. Using Group Links are common in more advanced puzzle solving techniques involving chaining and looping. A Group Link also occurs with the Empty Rectangle puzzle solving technique.
Let’s review the Empty Rectangle diagrammed above. There is an Either-Or type link between the possible 2 candidates in cells R3C2 and R6C2. There is a Strong Link relationship between the possible 2 candidate in cell R6C2 and the possible 2 candidate in cell R6C4. There is a Group Strong Link relationship between the possible 2 candidate in cell R3C2 and the node of possible 2 candidates in cells R4C4-R4C6 which are highlighted in purple. There is a Group Weak Link relationship between the possible 2 candidates in cells R4C4-R4C6 highlighted in purple forming a node and the possible 2 candidate in cell R2C4 as indicated by the dotted line. And then there is a Strong Link relationship between the possible 2 candidate in cell R2C4 and the possible 2 candidate in cell R6C4 as indicated by a solid line.
The way the logic of the Empty Rectangle works is because there is an Either-Or link between the cells R3C2 and R6C2, there must be a value of 2 in either cell R6C2 or the cell R2C4. In both cases, this results in the possible 2 candidate in cell R6C4 as being a non-possible candidate as indicated by being highlighted in dark red.
5.0 Related dxSudoku Videos
