Closed Loops

A closed loop is formed by an even number of cells, which contains the same candidate (=possible solution). All the cells which are a part of this loop contain 2 candidates. I will explain this to you more clearly by using some examples:

A Simple Example :

example 1 solving sudoku with closed loops

To find a closed loop you start with selecting a candidate. In this example I have chosen the number (3).

Continue by looking for two cells which contain two numbers and have the number(3) in common. Those two cells should also have a strong link. This means that it is absolutely certain that one of the two cells will have the candidate as their final solution. Cell B3 and cell B8 both contain the number(3) and because there are no other cells in column B containing this number, one of the two cells will have the number(3) as the solution. Thus the link between those two cells is a strong link.

A second strong link is formed by the cells F3 and F8.

Finally if the cells of the strong loops can "see" each other, then they can be connected. Cell B3 and cell F3 can "see" each other and form a weak link, because another number(3) is present on the same line as those two cells. B8-F8 also form a weak link.

The closed loop is formed by the cells B3-F3-F8-B8. Notice also that a closed loop consists of strong-weak-strong-weak links.

In the final solution of the puzzle all the red cells or all the green cells will be number(3). It is not possible that number(3) will be the solution for the red cells and the green cells at the same time. In the yellow area no other cells can contain the number(3). If there are any cells containing the number(3), then this number can be eliminated from that cell.

Second Example :

example 2 solving sudoku with closed loops

In this example the three strong links can be found : B3-B5, D3-D8 and G6-G8.

The cells in the strong links are connected to the cells in the other strong links as shown by the light coloured lines. The link B3-D3 is a weak link, because there is also a (5) in cell H5. The cells B6 and G6 are also connected with each other and form a weak link.

Cell B6 can not be connected with D8, because they do not "see" each other (They are not in the same row, column or box). De link between cell D( and cell G8 is also a weak link.

When you walk round the loop, you will find a weak leak after each strong link. This means all conditions of the closed link are fulfilled. Thus, all numbers(5) can be removed from all cells which are located in the yellow area.

A Real Puzzle :

example 3 : finding closed loops

In this puzzle a strong link has been found between the cells B3 and B7. I continued looking for other strong links with the common candidate (8). I have found two other strong links formed by the cells G3-G9 and C8-I8. The other links which are light coloured, represents the weak links.

Walking around all the links you will find the pattern strong-weak-strong-weak. Thus this is a closed link according to our definition.

Column B is coloured yellow, because the cell B3-B7 are located in column B (if one of the two cells becomes 8 then all the other cells can not become 8). The same can be said for column G, row 3 and row 8.

The cell B7 and C8 are in the same box, thus this box is coloured yellow. Cells G8 and I8 also appear in one box, thus this box is also yellow. As I said before the six cell form a closed loop and therefore all the numbers(8) can be removed from all the cells which are located in the yellow area. As shown above this technique is a very powerful method to eliminate many candidates.