Xwing

X-wing, Swordfish and Jellyfish are an interesting family of useful Sudoku patterns which are more difficult to understand than the others described on this site. Don't worry if you don't understand them straight away from the defintions or at the first reading. It is best to make sure X-wing is clear to you before going on to Swordfish because the possible Swordfish variations are very much more extensive. I've laboured the Swordfish explanation to try to cover its full complexity. In consequence I've given a very short outline of Jellyfish because if you get Swordfish, Jellyfish is easy to understand (though difficult to recognise).

An X-wing is formed when 2 columns only have 2 occurences of a symbol S, and for both of these columns these S's occur on the same 2 rows. Please see Figure 1.

Explanation.

Suppose we ignore all cells except those involved in the X-wing pattern (so, we keep only the intersections of columns 4,9 and rows 1 and 2 in Figure 1). Also we ignore all symbols except for the one forming the X-wing pattern (6 in the figure). Then for a symbol S we get the following arrangement of cells:

SS
SS

Every column must have an S. For the 2 columns here these are the only occurences of S, so their S's must come from these rows. But, as is explained below, fixing the S for one column automatically sets it for the other.

Suppose we put S in the top row for the first column. Then S cannot occur elsewhere in that column or on that row. But the other column has to have an S, so it must be on the second row. And vice versa. Diagramatically:

SS    S-    -S
SS    -S    S-

      (1)   (2)

These are the only 2 possibilities.

Notice that the argument is completely independent of the existence of any other occurrences of S on these 2 rows: we are forced into one of these two choices in order to provide an S for each of the two columns. Both choices leave an S on each of the two rows. Hence we can remove any other S's on both rows.

Obviously the rows and columns do not need to be adjacent (for example see Figure 1.)

Finally, the same argument can be made if we reverse rows and columns: an X-wing is also formed in a grid containing 2 rows which only have 2 occurences of symbol S, and for both of these rows these S's occur in the same 2 columns.