Wings

The Core Principle Behind Every Wing Technique

All wing techniques share a single foundational idea: if a candidate digit must appear in at least one of several cells, and every instance of that digit is visible from a particular target cell, then the target cell cannot contain that digit. The pivot is typically a bivalue cell that acts as a fork in the logic. The wings are cells forced to take on a particular digit under one branch of the pivot's decision. The elimination digit (Z) must land in at least one wing regardless of which branch the pivot takes. The Restricted Common Candidate (RCC) formalizes the connection between pivot and wings. The RCC is the digit shared between the pivot and the wing group where all instances are visible to the pivot, ensuring mutual exclusion.

XY-Wing: The Classic Three-Cell Wing

The XY-Wing uses three bivalue cells. Pivot {A, B}, Wing 1 {A, C}, Wing 2 {B, C}. The wings see the pivot but need not see each other. If Pivot = A, Wing 1 must be C. If Pivot = B, Wing 2 must be C. Either way, C appears in at least one wing. Eliminate C from cells seeing both wings. Rated Level 6 (Hard). The power comes from the two wings not needing to share a house, enabling cross-grid eliminations.

W-Wing: The Strong-Link Connection

Two identical bivalue cells {X, Y} connected through a conjugate pair on digit X. They must not share a house. At least one must be Y. Eliminate Y from cells seeing both. Rated Level 6 (Hard). Easier to spot than XY-Wing because you scan for repeated bivalue cells.

XYZ-Wing: Three Candidates in the Pivot

Pivot {X, Y, Z}, Wing 1 {X, Z}, Wing 2 {Y, Z}. All three cells contain Z. In every case, at least one of the three cells must be Z. Eliminate Z from cells seeing all three cells. This is more constrained than XY-Wing because the target must also see the pivot. Rated Level 7 (Very Hard).

The Extended Wing Family: WXYZ-Wing Through STUVWXYZ-Wing

The same logic works when the wing side is expanded into an Almost Locked Set (ALS). The pivot remains a bivalue cell; the wing is an ALS. XY-Wing: 3 cells, Level 6 WXYZ-Wing: 4 cells, Level 8 VWXYZ-Wing: 5 cells, Level 10 UVWXYZ-Wing: 6 cells, Level 10 TUVWXYZ-Wing: 7 cells, Level 11 STUVWXYZ-Wing: 8 cells, Level 11 The naming adds letters in reverse alphabetical order. Each letter represents one additional cell in the wing ALS.

How to Find Wing Patterns

Step 1: Identify bivalue cells as potential pivots. Step 2: Check for XY-Wing patterns (two bivalue wings sharing one candidate each with the pivot). Step 3: Check for W-Wing patterns (identical bivalue cells connected through a conjugate pair). Step 4: Check for XYZ-Wing patterns (three-candidate pivot with two bivalue wings). Step 5: Check for extended wings (bivalue pivot with ALS wings). Tips: Start with bivalue cells. Look for repeated digits. Verify the RCC restriction carefully.

Difficulty Progression

XY-Wing: Level 6, Hard W-Wing: Level 6, Hard XYZ-Wing: Level 7, Very Hard WXYZ-Wing: Level 8, Expert VWXYZ-Wing: Level 10, Master UVWXYZ-Wing: Level 10, Master TUVWXYZ-Wing: Level 11, Extreme STUVWXYZ-Wing: Level 11, Extreme

Wings and Almost Locked Sets: The Algebraic Connection

The extended wing family is mathematically equivalent to ALS-XZ where one ALS is a single bivalue cell. An engine implementing ALS-XZ automatically finds all extended wing patterns. The XY-Wing is the degenerate case where both ALSs are bivalue cells. The XYZ-Wing has a three-candidate pivot and is better understood as its own pattern.

Summary

The wing family provides elimination tools scaling from XY-Wing to STUVWXYZ-Wing. Every member relies on the same core logic: a pivot creates a two-way fork, and regardless of which branch it takes, a specific candidate must end up in at least one wing cell. Cells seeing all possible locations of that digit can eliminate it. Start with XY-Wing, internalize its logic, and the larger wings follow naturally.