Interactive Sudoku Technique
Alternating Inference Chains
An AIC is a logical conversation between candidate states. Strong Inferences say that two states cannot both be false. Weak Inferences say that two states cannot both be true. Alternate those promises correctly, and distant candidates begin to control one another.
From Specialized Chains to AIC
Keep the Logic; Remove the Training Wheels
The XY-Chain taught us to pass truth through Bivalue Cells. The Grouped X-Chain taught us to follow one digit through houses and grouped positions. An Alternating Inference Chain brings those ideas together. Its nodes are individual candidate states, and its path may move both within a cell and through a house.
Specialized Chain
Uses a recognizable structure: BVCs for an XY-Chain, or one repeated digit for an X-Chain.
Alternating Inference Chain
Follows any valid route through candidate states, provided Strong and Weak Inferences alternate without interruption.
An AIC is not defined by its shape. It is defined by the uninterrupted alternation of Strong and Weak Inferences.
Read the Candidate, Not Merely the Cell
The Language of Candidate States
A cell address alone is no longer precise enough. The notation
r3c4#9 means “candidate 9 in row 3, column 4.” The
number after the hash is the state being discussed—not a solved
value.
Candidate State
r3c4#9
Candidate 9 at r3c4 is either true or false.
Strong Inference
S:cell or S:row:5
The connected states cannot both be false. If one is false, the other must be true.
Weak Inference
W:cell or W:box:6
The connected states cannot both be true. If one is true, the other must be false.
r3c4#3candidate state
=
r3c4#9Strong inside r3c4
–
r3c6#9Weak across Row 3
Where Strong Inferences Come From
Two candidates in a Bivalue Cell, or the only two positions for one digit in a row, column, or box.
Where Weak Inferences Come From
Two different candidates in one cell, or two positions for the same digit that share a row, column, or box.
Important: “Strong” does not mean both candidates cannot be true. It means they cannot both be false. When the two states also see one another, the relationship may satisfy both conditions; the chain uses whichever inference is required at that point.
A Natural AIC in the Grid
Two Endpoints Corner Candidate 3
The highlighted route begins with candidate 3 at r3c4
and ends with candidate 7 at r9c4. The route changes
digits inside cells and carries repeated digits through conjugate
houses. Because the first and last links are Strong, at least one
endpoint must be true.
r9c4 is the elimination.
r3c4#3
Endpoint Br9c4#7
Thereforer9c4 ≠ 3
Candidate 3 at r9c4 conflicts with Endpoint A through
Column 4 and with Endpoint B inside its own cell. Whichever endpoint
is true, the victim is false. Remove 3 from r9c4.
Interactive Chain Trace
Let a False Endpoint Travel
A Strong Inference carries false → true. The next Weak Inference carries true → false. Continue that alternation and a false opening endpoint forces the far endpoint true.
Begin by testing Endpoint A as false.
Endpoint A false forces Endpoint B true. Therefore the two endpoints cannot both be false: they are strongly inferred.
Open the complete link ledger
| From | Link | To | Where |
|---|
The Endpoint Theorem
Test Both Roads to the Same Elimination
We do not need to know which endpoint is true. We need only prove that the victim loses candidate 3 under either possible state of Endpoint A.
Road One: Endpoint A Is True
If r3c4#3 is true, candidate 3 at
r9c4 is immediately false because both occupy
Column 4.
Road Two: Endpoint A Is False
The alternating path forces r9c4#7 true. The cell
cannot also contain 3, so r9c4#3 is false again.
Both roads agree: remove candidate 3 from r9c4. Any Single that appears afterward belongs to the Singles solver; the AIC has completed its elimination.
A More Demanding Ordinary AIC
A Cell May Return in a Different Candidate State
Longer-looking examples become readable when every colored cell is
treated as a collection of candidate states. In this chain,
r3c9#6 appears near the beginning and
r3c9#7 appears near the end. The amber cell is an
internal return junction, not an endpoint.
r1c7#4, returns through two states
of the amber cell r3c9, and ends at
r1c9#7. Candidate 4 at r1c9 is eliminated.
Read the amber cell twice
Early in the chain, r3c9#6 participates in the
candidate-6 route. Later, r3c9#7 participates in the
candidate-7 route. One address may therefore appear more than
once without repeating the same logical node.
r1c7#4 = r1c7#6 – r3c9#6 = r3c5#6 … r3c1#7 – r3c9#7 = r1c9#7
This remains an ordinary AIC. Visual complexity by itself does not make a chain an Advanced AIC.
Beyond the Visible Chain
What Makes Sudoku Paradise’s A-AIC Advanced?
An ordinary AIC follows one readable path through individual candidate states. Sudoku Paradise’s Advanced Alternating Inference Chain begins one level deeper. It first builds a proof network containing individual candidates, grouped premises, Strong Inference Sets, and Weak relationships. It then searches that network for valid alternating paths and compiles only the eliminations those paths prove.
Select a stage to inspect the separation between proof and action.
The proof system cannot alter the puzzle while it is reasoning. Evidence is built first; action occurs afterward through the same recorded elimination process used by the rest of Sudoku Paradise. This prevents an unfinished proof from changing the candidate grid upon which that proof depends.
A-AIC is not simply a longer AIC. It searches for alternating proofs inside a richer logical environment.
Sudoku Paradise reserves that additional machinery for late escalation—after recognizable techniques and ordinary candidate-level AIC have had their opportunity.
A Disciplined Search
How to Look for an AIC
1. Start with Strong Inferences
Mark BVC pairs and conjugate pairs. They are the engines that can turn a false state into a true one.
2. Alternate, Never Improvise
After Strong comes Weak; after Weak comes Strong. If two Weak or two Strong steps appear in succession, recheck the route.
3. Inspect the Endpoints
When an open chain begins and ends Strong, ask which candidate is weakly linked to both endpoint states.
Before accepting the elimination
- Every token names a precise candidate state.
- Every Strong Inference is supported by a BVC or conjugate house.
- Every Weak Inference joins states that cannot both be true.
- The link types alternate from the first edge to the last.
- The victim conflicts with both endpoint conclusions.
Check the Inference
Four Quick AIC Decisions
Choose the statement that follows from each relationship. The goal is to read the logic, not to memorize a drawing.
A Strong Inference
Candidate A is false. What follows?
A Weak Inference
Candidate A is true. What follows?
A Bivalue Cell
r3c4 contains only candidates 3 and 9. How are they related?
The Endpoint Victim
A candidate is weakly linked to both strongly inferred endpoints. What follows?
0 of 4 decisions confirmed
Questions Worth Asking
AIC Frequently Asked Questions
Must an AIC use only Bivalue Cells?
No. BVCs provide useful Strong Inferences inside cells, but an AIC may also use conjugate pairs in rows, columns, and boxes.
Must every candidate in the chain be the same digit?
No. That restriction belongs to X-Chains. An AIC may change digits through links inside cells and continue through houses.
Does the AIC decide which endpoint is true?
Not necessarily. For this elimination, it is enough to prove that the endpoints cannot both be false and that the victim conflicts with either endpoint being true.
Why does the AIC eliminate rather than place a digit?
The chain proves the victim false. If that removal leaves a cell with one candidate, a Naked Single makes the placement. Keeping those responsibilities separate preserves a clear proof record.
Is every complicated-looking AIC an A-AIC?
No. An ordinary AIC can cross many cells or revisit a cell through another candidate state. A-AIC differs by the richer proof network it searches, not by appearance alone.
From Candidate States to Candidate Sets
Next, Let an Entire Set Carry the Proof
An AIC moves through individual candidate states. ALS-XZ gathers several cells into an Almost Locked Set, connects two such sets through a Restricted Common Candidate, and uses their shared candidate to prove an outside elimination.