Outcome
You can recognize an {X,Y,Z} pivot with {X,Z} / {Y,Z} pincers, prove all three pivot cases, and apply the stricter common-peer rule.
Intro copy
XYZ-Wing changes one thing from XY-Wing:
The pivot can now be Z.
That one extra candidate changes the elimination geometry.
Step 1 — Transform XY into XYZ
Start with:
Pivot {X,Y}
Pincers {X,Z} {Y,Z}Ask:
In XY-Wing, where can Z be true?
Correct:
One of the pincers.
Now add Z to the pivot:
Pivot {X,Y,Z}Ask:
Where can Z now be true?
Correct:
Pivot or either pincer.
Step 2 — Three-case proof
Use concrete values:
- pivot
{2,7,5}; - pincer A
{2,5}; - pincer B
{7,5}.
Learner taps:
- pivot = 2 → pincer A = 5;
- pivot = 7 → pincer B = 5;
- pivot = 5 → pivot itself is 5.
Merge all branches:
One of the three pattern cells is 5.
Step 3 — Visibility comparison
Show two candidate-5 targets.
Target A
Sees both pincers, not pivot.
Target B
Sees pivot and both pincers.
Prompt:
Which target is valid in XYZ-Wing?
Only B.
Feedback:
Target A would work for XY-Wing, but not when the pivot itself can be Z.
Step 4 — Guided board
Learner:
- finds a trivalue pivot;
- identifies XZ and YZ pincers;
- labels Z;
- checks common peers of all three;
- eliminates valid Zs.
Step 5 — False pincer
Provide one pincer that does not see the pivot.
Prompt:
Can this cell participate?
No.
Feedback:
Each pincer must be a peer of the pivot.
Practice
Include:
- row/box geometries;
- several trivalue decoys;
- valid and invalid target visibility;
- XY-vs-XYZ comparison cards.
Hint ladder
- “Look for a trivalue cell.”
- “Treat it as
{X,Y,Z}.” - “Find
{X,Z}and{Y,Z}peers.” - “Which candidate appears in all three pattern cells?”
- “The target must see all three.”
Completion
Three XYZ-Wings, two visibility comparisons, and one false-pincer rejection.
Next
Skyscraper & Two-String Kite
Now move from multi-digit Wing roles to one-digit Strong-Link patterns.
Reference
Read XYZ-Wing for the three-case proof and direct comparison with XY-Wing.