X-Wing Sudoku: the rectangle that removes candidates

An X-Wing uses one digit that is restricted to the same two columns in two different rows, or the transposed arrangement. The digit must occupy opposite corners, so eliminate it from other cells in those columns without choosing which diagonal is true.

Start with a single digit

Pick seven and look across unfinished rows. In row A it can occur only in columns P and Q. In a different row B it can also occur only in P and Q. Those four candidates form the corners of a rectangle.

If A places seven in P, B must place it in Q because column P cannot repeat seven. If A places it in Q, B must place it in P. Either way, columns P and Q receive their sevens from rows A and B. Every other seven candidate in those columns is impossible.

Separate base units from cover units

Role	What must be true
Base rows A and B	Each has exactly the two candidate columns P and Q
Cover columns P and Q	They may have extra candidates elsewhere
Elimination targets	Other rows in P or Q
Corners	Remain candidates until further deductions

An extra seven in a base row outside P and Q breaks this row-based X-Wing. Extra sevens in the cover columns do not break it; those are precisely the candidates you want to remove.

A practical scan

1. Choose a digit with several remaining placements.
2. Find rows with exactly two positions for it.
3. Compare their column locations.
4. When two rows match, check both complete rows again.
5. Remove the digit from other cells in the matched columns.

If row scanning finds nothing, transpose the method: two columns confined to the same two rows permit eliminations in those rows. Keep the direction of the proof consistent.

Why a rectangle alone is insufficient

Four visible marks at rectangle corners are not enough. A row might contain a fifth candidate elsewhere, allowing its seven to escape the rectangle. Then the argument forcing opposite corners fails.

Nor can you use different digits at the corners. The restriction comes from the no-repeat rule for one digit. A visually pleasing arrangement of mixed candidates has no equivalent force.

What happens after the elimination

A removed seven can expose a naked single in a target cell or a hidden single elsewhere in its row. Update all affected notes and return to ordinary scanning. You do not have to decide the X-Wing diagonal before benefiting from the pattern.

When practicing, write the two possible diagonal cases in words. Both cases should rule out the same external candidates. If one case leaves a target possible, recheck the base rows: you may have missed a candidate or mixed row-based and column-based reasoning.

Trace both diagonals on a candidate grid

Use rows A and D as the base rows, and columns B and H as the cover columns. We are tracking seven alone. A dot means seven is not a candidate in that square; an X marks a corner, and a T marks an external target.

Row	Column B	Column H	Any other column
A	X	X	No seven
D	X	X	No seven
F	T	.	Not relevant
I	.	T	Not relevant

If A places seven in B, D must place seven in H. If A places seven in H, D must place seven in B. F's candidate in B is ruled out in either case, and so is I's candidate in H. Remove seven from both targets while keeping all four corner candidates.

Suppose F's target cell had candidates two and seven. It now becomes two. I's target cell might have three, seven and nine; it shrinks to three and nine and stays unresolved. The same X-Wing can produce both an immediate placement and a useful reduction.

Rotate the argument without rotating the rules

For a column-based example, imagine columns C and G can place four only in rows B and F. Each base column must contain four. Their fours therefore occupy opposite corners across those rows. Remove four from other columns of rows B and F, rather than from unrelated rows of C and G.

The strict restriction always belongs to the base units. The targets belong to the cover units. Stating those roles before erasing notes prevents the common mistake of removing candidates along the wrong direction.

Test one extra candidate

Now allow seven in another column of row A. The first rectangle is no longer an ordinary row-based X-Wing: A could place its seven outside B and H, freeing D to occupy either corner without filling both cover columns. You cannot remove F's or I's seven on that argument. By contrast, adding another target in column B outside A and D leaves the valid pattern intact and gives you another elimination.

Where to play

Try a harder track in Ninefold to practice scanning candidate positions across rows and columns. Its iOS technique guides are available to revisit while learning.