XY-Wing

This is similar to a short forcing chain consisting of two links for each candidate, but instead of placing a number, it allows for candidate elimination. This a very common pattern in the harder puzzles.

In the partial Sudoku puzzle below, consider the cells that have only the candidates shown:

   
 XY 
   
   
 XZ 
   
   
 YZ 
   
   
 * 
   


It can be easily seen that whichever value is in XY, the cell marked with the asterisk cannot be Z.

if XY = X, then XZ = Z, so * cannot be Z
if XY = Y, then YZ = Z, so * cannot be Z

This allows Z to be eliminated from the candidates for the marked cell.

The cells don't need to form a perfect rectangle, but the cells containing XY and XZ need to be buddies, and the cells containing XY and YZ also need to be buddies. Once you've got this arrangement, you can eliminate Z from the candidates of all cells that occupy the intersection of the units containing XZ and YZ.

 

Other possible combinations:

 XY 
   
YZ  
 XZ 
   
***

 

*XY*
   
YZ  
 XZ 
   
   

 

The astute among you will notice both the above examples have XY, XZ and YZ in the same relative locations, and so can be combined to give:

*XY*
   
YZ  
 XZ 
   
***

All the cells marked with an asterisk can have Z removed from their candidates.

In the Sudoku puzzle below, the XY-wing in the green cells allows 7 to be eliminated from the blue ones.

Sudoku XY-Wing example

These Sudoku puzzles can be solved using XY-wings: (What are .sdk files?)