XY-Wing

Hard

A pivot cell with two candidates {X,Y} sees two wings {X,Z} and {Y,Z}; digit Z is eliminated from cells seeing both wings.

How It Works

A "pivot" cell has candidates {X,Y}. It sees two "wing" cells: one with {X,Z} and one with {Y,Z}. If pivot takes X, the first wing must take Z. If pivot takes Y, the second wing takes Z. Either way, Z must be in one of the wings — so any cell seeing both wings can have Z eliminated.

This technique uses "pincer" logic and is very powerful.

Example

79	2	1	4	57	8	3	9	8
4	6	5	124	2	3	1	4	6
2	9	4	579	8	6	5	138	7
6	158	279	3	4	1	459	2	8
37	469	168	249	35	268	149	57	6
9	145	8	157	267	9	358	6	7
1	5	357	6	9	5	679	8	2
5	389	6	279	1	458	239	5	3
3	7	5	348	7	2	4	3	1

Key cells of the techniqueCells where elimination is appliedCandidate to be placedEliminated candidate (crossed out)

Pivot R5C5={3,7}. Wing1 R5C1={3,9}, Wing2 R2C5={7,9}. → R2C1 sees both wings → Eliminate 9 from R2C1.

Practice with a Real Puzzle

This 9×9 puzzle is solver-verified to require this technique on its solution path.

Medium26 givensStrict

79	2	1	4	57	8	3	9	8
4	6	5	124	2	3	1	4	6
2	9	4	579	8	6	5	138	7
6	158	279	3	4	1	459	2	8
37	469	168	249	35	268	149	57	6
9	145	8	157	267	9	358	6	7
1	5	357	6	9	5	679	8	2
5	389	6	279	1	458	239	5	3
3	7	5	348	7	2	4	3	1

79	2	1	4	57	8	3	9	8
4	6	5	124	2	3	1	4	6
2	9	4	579	8	6	5	138	7
6	158	279	3	4	1	459	2	8
37	469	168	249	35	268	149	57	6
9	145	8	157	267	9	358	6	7
1	5	357	6	9	5	679	8	2
5	389	6	279	1	458	239	5	3
3	7	5	348	7	2	4	3	1

79	2	1	4	57	8	3	9	8
4	6	5	124	2	3	1	4	6
2	9	4	579	8	6	5	138	7
6	158	279	3	4	1	459	2	8
37	469	168	249	35	268	149	57	6
9	145	8	157	267	9	358	6	7
1	5	357	6	9	5	679	8	2
5	389	6	279	1	458	239	5	3
3	7	5	348	7	2	4	3	1