ALS-XZ Wings

Almost-locked-set constructions linked by restricted common candidates: XY-Wing through TUVWXYZ-Wing, plus the doubly linked ring forms.

XY-Wing

Three bivalue cells using three distinct digits, forming an ALS-XZ construction with an equivalent AIC proof.

XYZ-Wing

One trivalue and two bivalue cells forming an ALS-XZ construction, with the elimination digit present as a candidate in all three.

WXYZ-Wing

Two ALSs spanning four distinct cells and four distinct candidate digits, linked by one RCC.

A doubly linked WXYZ construction with two restricted common candidates

WXYZ-Ring

Two ALSs spanning four distinct cells and four distinct candidate digits, linked by two RCCs.

VWXYZ-Wing

Two ALSs spanning five distinct cells and five distinct candidate digits, linked by one RCC.

The doubly linked five-cell VWXYZ form with two restricted commons

VWXYZ-Ring

Two ALSs spanning five distinct cells and five distinct candidate digits, linked by two RCCs.

UVWXYZ-Wing

Two ALSs spanning six distinct cells and six distinct candidate digits, linked by one RCC.

The doubly linked six-cell UVWXYZ form with two restricted commons

UVWXYZ-Ring

Two ALSs spanning six distinct cells and six distinct candidate digits, linked by two RCCs.

TUVWXYZ-Wing

Two ALSs spanning seven distinct cells and seven distinct candidate digits, linked by one RCC.

The doubly linked seven-cell TUVWXYZ form with two restricted commons

TUVWXYZ-Ring

Two ALSs spanning seven distinct cells and seven distinct candidate digits, linked by two RCCs.

Almost locked sets joined by restricted commons

An almost locked set (ALS) is a house fragment of N cells holding N+1 candidates between them. When two such sets share a restricted common candidate (RCC) — a digit that cannot appear in both sets at once — each set is forced to lock onto its other digits wherever the RCC is absent, so a digit common to both can be removed from cells that see every possible position. The lettered wings are all instances of this one ALS-XZ rule at growing set sizes — each lettered wing fixing exactly N cells holding N distinct digits across its two ALSs.

In a doubly linked pair the two sets share two RCCs, which yields extra eliminations and locked-set consequences; the Ring in names like WXYZ-Ring means exactly that — two RCCs — and not a closed AIC. The visually central cell shown in the guides is a spotting aid only: the proof operands are the ALS sets, the RCC and the universal visibility condition for each target.

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