Links. Strong and Weak

A Link in the context of a Sudoku Puzzle is two nodes that directly affect one another.

Links between cells

When links are between two cells, the two cells must be in the same unit and they must contain a common candidate.The two blue cells contained in row 5 shown in Figure 1 both contain the candidate 7. These two cells are linked on candidate 7. Using the value 7 in either of the two blue cells has an immediate effect on the use of candidate 7 in the other blue cell.

If the two cells happen to have more than one common candidate then each common candidate is a link. The two red cells in row 6 shown in Figure 1 both have the candidates 2 and 8. The two red cells are linked on candidate 2. The two red cells are also linked on candidate 8. There are two links between the two red cells.

A link between two cells allows an inference to be made based on the candidates in the link. In row 8 in the example in Figure 1, if the candidate 5 was placed in the first yellow cell, the inference is "candidate 5 cannot be placed in the second yellow cell" and vice-versa.

This is called Weak Inference and is one of the basic rules of Sudoku. If a cell contains a value, then no other cells in the same unit (row, column or box) can contain that value.

When a link exists, and the two cells in the link are the only two cells in the unit that contain the candidate a second inference can be made.

In column 8 in the example in Figure 1, if the candidate 9 is not placed in the first green cell, the inference is "candidate 9 must be placed in the second green cell because no other cells in the column have the candidate 9" and vice versa.

This kind of link is called Strong Inference and occurs when the two cells in a link form a conjugate pair for a candidate.

A link between two cells with Weak Inference means "If the candidate is placed in one cell in the link then it cannot be placed in the other cell."

A link between two cells with Strong Inference means "If the candidate is not placed in one cell in the link then it has to be placed in the other cell."Links within Cells

Links and inferences can also exist within a cell. In this case the links or inferences are between different values but are contained in a single cell.If a cell has 2 or more candidates and one of the candidate is selected, that cell cannot contain any of the remaining candidates. All cells with multiple candidates have weak inference between all the candidates in that cell. "If candidate A is selected for this cell then the cell cannot contain candidate B."

The red cell in Figure 2 shows an example. If the value 5 is chosen for the red cell then the red cell cannot contain the value 6. If the value 5 is chosen for the red cell then the red cell cannot contain the value 9. There is a weak link between all the values in the red cell.

For a cell that has only two candidates there is also a strong inference between the candidates. "If the cell does not contain candidate A then it must contain candidate B."

The yellow cell in Figure 2 shows an example. If the value 9 is not the solution for the yellow cell then the solution must be the value 8. When a cell only has two candidates there is a strong link between those two candidates in that cell.

Some chaining techniques use links between cells as well as links within cells to build the chains.

About Links

If a link has Strong Inference it also has Weak Inference. If the inference "cell A does not contain the value 4 so cell B must contain the value 4" can be made on a link then the inference "cell A contains the value 4 so cell B cannot contain the value 4" can also be made on the link.

Both strong and weak inferences are bi-directional. The inference "cell A contains a 6 so cell B cannot contain a 6" also means "cell B contains a 6 so cell A cannot contain a 6".

The terms link and inference are often interchanged when talking about Sudoku. A Strong Link is a link that has Strong Inference. It also has Weak Inference by definition. A Weak Link is a link that only has Weak Inference.