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# Solving Techniques 11 X-Cycles

Before considering X-Cycles, it’s important to understand strong links and weak links.

As shown below, a link where only two numbers are possible in a row, column or box, is called a strong link. In the diagram, it is between [1] and [2].

As shown below, a weak link is a link where there are more than three numbers that are possible within a row, column or box. [1] and [2] are in three possible cells, which creates a weak link.

## X-Cycles

The most simple X-cycle is shown in the diagram below. The red line is the strong link, the dotted red line, the weak link.

This works the same way as an X-wing and it always works out to:
1
1
or
1
1
Hence, 1 can’t be entered in the areas marked X.

So the X-cycle, of a given number, refers to a
cell - strong - cell - weak - cell - strong - cell
or
cell - strong - cell - weak - cell - strong - cell - weak - cell - strong - cell
type of loop that begins and ends with a strong link. (It also works, if there are strong links, instead of weak within the pattern.)
The number in the weak links can be removed as candidates, but these cells change depending on the situation.

## X-Cycles 2

This is a slightly complex pattern. The red line is a strong link, the dotted red line is a weak link.

Where the 1 goes is in one of the two patterns below.

Pattern A

Pattern B

Hence, in the overlapping areas X, 1 can’t be entered, as shown below.

## X-Cycles 3

Below is another complex pattern. The red line is a strong link, the dotted red line is a weak link.

Where the 1 goes, is in one of two patterns below.

Pattern A

Pattern B

When the two patterns are combined, it becomes like the diagram below.

And in the overlapping areas marked X, a 1 can’t be entered. (The X’s of the strong link are not shown.)

## X-Cycles 4

Below is an easy pattern.

As you can see, 1 can’t be in the areas marked with X’s.
Examining the samples, in each case, the connection in the weak link, if it is a row or column, that row/column’s other cells are removed as candidates. If it is a box, the box’s other cells are removed as candidates.

If you can use this technique, you are quite accomplished at Sudoku.

## Names of cells in Sudoku

 R1C1 R1C2 R1C3 R1C4 R1C5 R1C6 R1C7 R1C8 R1C9 R2C1 R2C2 R2C3 R2C4 R2C5 R2C6 R2C7 R2C8 R2C9 R3C1 R3C2 R3C3 R3C4 R3C5 R3C6 R3C7 R3C8 R3C9 R4C1 R4C2 R4C3 R4C4 R4C5 R4C6 R4C7 R4C8 R4C9 R5C1 R5C2 R5C3 R5C4 R5C5 R5C6 R5C7 R5C8 R5C9 R6C1 R6C2 R6C3 R6C4 R6C5 R6C6 R6C7 R6C8 R6C9 R7C1 R7C2 R7C3 R7C4 R7C5 R7C6 R7C7 R7C8 R7C9 R8C1 R8C2 R8C3 R8C4 R8C5 R8C6 R8C7 R8C8 R8C9 R9C1 R9C2 R9C3 R9C4 R9C5 R9C6 R9C7 R9C8 R9C9