Same-Digit AIC Ring
A Same-Digit AIC Ring — traditionally listed as Bidirectional X-Cycle — follows one digit through alternating strong and weak inferences and closes through a final weak inference verified independently of the path. The candidate propositions and links define the deduction; a count of highlighted cells or a familiar drawing does not.
What is a Same-Digit AIC Ring?
A representative ring is (A=B)-(C=D)-...-(Y=Z) closed by a weak inference from Z back to A. Each equals sign is a strong link (XOR) between two exhaustive positions of the digit; each dash is a weak inference (NAND) — not both. The closing weak inference must hold on its own; a drawing that closes geometrically proves nothing by itself.
Walkthrough example
A walkthrough of one Same-Digit AIC Ring: one digit whose strong links and weak inferences close through a weak inference verified independently of the path.
Step 1
Consider these cells.
Step 2
The highlighted candidates form the closed Same-Digit AIC Ring (B1=B9)-(E9=E6)-(I6=G5)-(G2=C2). Each equals sign is a strong link (XOR); each dash is a weak inference (NAND). The last link, C2=5 back to B1=5, is a weak inference that holds independently of the path — it closes the loop, and the endpoint theorem supplies its complementary strong inference, so every weak edge of the ring becomes an exact-one pair. The numbered implication trace walks the ring in one direction to display the alternating links; it illustrates the closed structure and is not an extra forcing assumption. 1. If C2 contains the value 5, then B1 cannot contain the value 5 (the value can occur only once in the block) 2. If B1 does not contain the value 5, then B9 must contain the value 5 (only remaining possible position in the column) 3. If B9 contains the value 5, then E9 cannot contain the value 5 (the value can occur only once in the row) 4. If E9 does not contain the value 5, then E6 must contain the value 5 (only remaining possible position in the column) 5. If E6 contains the value 5, then I6 cannot contain the value 5 (the value can occur only once in the row) 6. If I6 does not contain the value 5, then G5 must contain the value 5 (only remaining possible position in the block) 7. If G5 contains the value 5, then G2 cannot contain the value 5 (the value can occur only once in the column) 8. If G2 does not contain the value 5, then C2 must contain the value 5 (only remaining possible position in the row)
Step 3
So value 5 can be removed: closure turns every weak edge of the ring into an exact-one pair, so exactly one of its two alternating assignments holds — and each red candidate conflicts with a true ring proposition in both.
A same-digit ring alternates strong links (XOR) with weak inferences (NAND) and closes through a weak inference verified independently of the path. The endpoint theorem saturates the closing pair, so the digit can be removed from any cell that sees both sides of a weak edge; the ring itself never selects which alternating assignment is the solution. You can play sudoku online to practise with the current candidate state visible.
How to spot it
Choose one digit, list its strong nodes — pairs of positions where the digit must occupy at least one — and connect consecutive nodes through weak inferences between occurrences that see each other. Then verify the closing weak inference from the last node back to the first.
The logic
With the closing weak inference verified, the endpoint theorem supplies the missing strong inference between the closing pair, so every weak edge around the ring gains a complementary strong inference. Remove the digit from any cell that sees both sides of one of those saturated pairs; the ring itself never selects which alternating assignment is the solution.