Sudoku Solving Techniques

XYZ-Wing

The XYZ-Wing technique involves three cells. One cell (called the pivot) has three candidate digits. The other two cells each have two candidate digits. The candidate digits in these three cells must fulfill certain relationships. We will demonstrate such relationships below in different patterns.

1. XYZ-Wing (Block-Row)

Consider the following partial Sudoku puzzle:

In Figure 1a, the pink cell (the pivot) and the two yellow cells form an XYZ-Wing (Block-Row) pattern. The pink cell has three candidate digits X, Y, and Z. One of the yellow cells shares the same 3x3 block with the pink cell and has two candidate digits X and Z. The other yellow cell shares the same row with the pink cell and has two candidate digits Y and Z. Notice that all these three cells have a common candidate digit Z. The pink cell has two other candidate digits X and Y, and each of those is also a candidate digit for one of the two yellow cells.

Let's see what happens to the orange cells under this XYZ-Wing (Block-Row) pattern.

The orange cells in Figure 1b are those cells falling in the common related area of the pink cell and both yellow cells. That means the orange cells are in the same row, in the same column, or in the same 3x3 block as the pink cell and each of two yellow cells.

There are three possibilities for the pink cell:

• If the pink cell is X, then the XZ yellow cell in the same 3x3 block cannot be X and therefore must be Z. As a result, the orange cells, which are in the same block as the XZ yellow cell, cannot be Z.
• If the pink cell is Y, then the YZ yellow cell in the same row cannot be Y and therefore must be Z. Again, the orange cells, which are in the same row as the YZ yellow cell, cannot be Z.
• If the pink cell itself is Z, then once again, the orange cells, which are in the same row and in the same block as the pink cell, cannot be Z.

So, in any of these three cases, the orange cells cannot be Z.

Example for the XYZ-Wing (Block-Row) technique in a real Sudoku puzzle

2. XYZ-Wing (Block-Column)

Consider the following partial Sudoku puzzle:

In Figure 2a, the pink cell (the pivot) and the two yellow cells form an XYZ-Wing (Block-Column) pattern. The pink cell has three candidate digits X, Y, and Z. One of the yellow cells shares the same 3x3 block with the pink cell and has two candidate digits X and Z. The other yellow cell shares the same row with the pink cell and has two candidate digits Y and Z. Notice that all these three cells have a common candidate digit Z. The pink cell has two other candidate digits X and Y, and each of those is also a candidate digit for one of the two yellow cells.

Let's see what happens to the orange cells under this XYZ-Wing (Block-Column) pattern.

The orange cells in Figure 2b are those cells falling in the common related area of the pink cell and both yellow cells. That means the orange cells are in the same row, in the same column, or in the same 3x3 block as the pink cell and each of two yellow cells.

There are three possibilities for the pink cell:

• If the pink cell is X, then the XZ yellow cell in the same 3x3 block cannot be X and therefore must be Z. As a result, the orange cells, which are in the same block as the XZ yellow cell, cannot be Z.
• If the pink cell is Y, then the YZ yellow cell in the same column cannot be Y and therefore must be Z. Again, the orange cells, which are in the same column as the YZ yellow cell, cannot be Z.
• If the pink cell itself is Z, then once again, the orange cells, which are in the same column and in the same block as the pink cell, cannot be Z.

So, in any of these three cases, the orange cells cannot be Z.

Example for the XYZ-Wing (Block-Column) technique in a real Sudoku puzzle