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Solving Techniques 11
X-Cycles

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

Strong link

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].

Weak link

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.

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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

A1A2A3A4A5A6A7A8A9
B1B2B3B4B5B6B7B8B9
C1C2C3C4C5C6C7C8C9
D1D2D3D4D5D6D7D8D9
E1E2E3E4E5E6E7E8E9
F1F2F3F4F5F6F7F8F9
G1G2G3G4G5G6G7G8G9
H1H2H3H4H5H6H7H8H9
I1I2I3I4I5I6I7I8I9

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