An XY-Chain is a chain built entirely from bivalue cells.
Each chain cell has exactly two candidates.
The chain moves by alternating:
- a Strong Link inside each bivalue cell;
- a Weak Link between peer cells that share one candidate.
The first and last cells contain the same endpoint candidate Z.
That proves:
At least one endpoint must be Z.
Candidate Z can therefore be removed from every cell that sees both endpoint cells.
A simple candidate sequence
Consider:
A {3,9}
B {9,8}
C {8,2}
D {2,3}Requirements:
- A sees B;
- B sees C;
- C sees D;
- each peer connection is on the shared candidate.
Start with A not 3.
A not 3
A is bivalue → A = 9.
A = 9
B sees A → B cannot be 9.
B is bivalue → B = 8.
B = 8
C cannot be 8 → C = 2.
C = 2
D cannot be 2 → D = 3.
So:
A not 3 → D = 3.
Read the chain in reverse:
D not 3 → A = 3.
Therefore at least one endpoint is 3.
Any candidate 3 seeing both endpoints is impossible.
Why every cell is bivalue
The chain depends on a Strong Link inside each cell.
Cell {3,9}:
not 3 → 9.
If the cell were {3,9,5}:
not 3
would leave:
- 9;
- or 5.
9 would not be forced.
So standard XY-Chain requires every chain cell to have exactly two candidates.
Step-by-step board example
XY-Wing is the short introduction
XY-Wing uses three bivalue cells:
{X,Y}
{X,Z}
{Y,Z}It can be interpreted as a short XY-Chain.
That makes the learning progression natural:
XY-Wing
↓
understand bivalue propagation
↓
extend through more cells
↓
XY-ChainHow cells connect
Suppose:
- A
{2,7}; - B
{7,9}.
If A sees B and A becomes 7:
- B cannot be 7;
- B must be 9.
The inter-cell 7 relationship is Weak.
The within-cell {7,9} relationship is Strong.
This repeated pattern lets the candidate value “walk” through the chain.
How to find XY-Chains
1. Highlight bivalue cells
2. Pick a start cell
Choose one candidate as the desired endpoint digit Z.
3. Follow the other candidate
Find a peer bivalue cell containing it.
4. Switch candidates
Inside the new cell, follow its other value.
5. Continue through peer bivalue cells
6. Stop when the endpoint digit Z returns
7. Check common peers of the endpoint cells
A productive XY-Chain needs a Z elimination or another valid endpoint deduction.
Avoid looping aimlessly
The candidate graph can branch quickly.
A manual solver should:
- prefer short chains;
- avoid revisiting the same state;
- stop when a useful endpoint appears;
- return to simpler deductions after an elimination.
The objective is not to enumerate every bivalue path.
XY-Chain vs X-Chain
X-Chain
One digit stays fixed.
Strong Links normally live in units.
XY-Chain
Candidate digits change.
Strong Links live inside bivalue cells.
Both can end by proving that one of two endpoints contains the same candidate.
Common mistakes
Using a trivalue cell
Breaks the standard strong within-cell implication.
Connecting cells that do not see each other
The shared candidate must create a valid Weak Link.
Repeating candidates without checking peer visibility
Candidate sequence alone is not enough.
Ending on a different endpoint digit
The common-peer elimination taught here requires the same candidate at both endpoints.
Eliminating from a target that sees only one endpoint
Invalid.
Recognition drill
Start from one bivalue cell:
{3,9}
Choose 3 as endpoint.
Then verbally follow:
9 peer → other value → peer → other value...
until you reach another {...,3} cell.
If you cannot state every shared-candidate peer relationship, the chain is not proven.
FAQ
Do XY-Chain cells all need two candidates?
Yes, for the standard form taught here.
Can XY-Chain use many different digits?
Yes.
Is XY-Wing an XY-Chain?
It can be represented as a short one.
What candidate gets eliminated?
The candidate shared by both chain endpoints, from cells seeing both endpoint cells.
What to learn next
Full AICs generalize chain inference beyond the restricted X/XY forms, but they belong to a later Expert expansion.
For the current curriculum, mixed Advanced Practice is the better next step.