Introduction
Nice loops — also called Alternating Inference Chains (AIC) — are a flexible way to place or eliminate candidates by reading an alternating chain of strong and weak links. Unlike pattern-based techniques, AICs work directly with candidates and the logic that ties them together. This article explains how to read a nice loop, when to look for one, and walks through a clear candidate-grid example you can practice on.
What a nice loop / AIC is
AICs are chains of candidate nodes where each link between nodes is either a strong link or a weak link. A strong link means that, in the relevant unit (row, column, or box), if one node is false then the other must be true — typically because the digit can appear only in those two positions in that unit. A weak link means that both nodes could be false at the same time — they are merely alternatives, not mutual exclusives.
A nice loop is a closed AIC: it starts and ends on the same candidate (or on candidates that see the same cell). Reading the loop with alternating strong and weak links gives a powerful conclusion. Two common outcomes:
- If the loop alternates and closes on the same candidate with opposite parity, you can place that candidate in some cells (a forcing chain).
- If it closes so that two nodes imply the same digit must be true, any candidate that sees both of those true nodes can be eliminated.
Strong vs weak links — how to spot them quickly
- Strong link: Only two candidates for a digit in a unit. For example, if digit 5 can only be in cells A and B of a box, the link between A5 and B5 is strong. If A5 is false, B5 must be true (and vice versa).
- Weak link: Two candidates in different cells for different digits or when more than two candidates remain in the unit for that digit. If A3 and B7 are alternatives but neither forces the other, the link is weak.
Remember: an AIC alternates strong, weak, strong, weak. When you trace the loop, follow this alternation mentally.
When to look for a nice loop / AIC
- You’re stuck after using singles, pairs, triples, and fish techniques. AICs come next as a candidate-level step.
- When you see pairs of cells in a unit that create potential strong links (two-options for a digit in a row/column/box). Those pairs are the most useful building blocks.
- When a suspicious candidate appears that is seen by several cells connected by candidate relationships. If two chain endpoints both force that candidate to be false or true, an AIC can confirm it.
Example: step-by-step on a candidate grid
Below is a small candidate-grid excerpt. Focus on the highlighted nodes: they form a closed alternating chain connecting the same two digits (3 and 7). The conclusion is an elimination of 7 in a target cell that sees two different nodes of the loop.
Step 1 — Identify nodes
Each node is a single candidate (digit in a specific cell). In the visual, the loop nodes are six cell-digit candidates that alternate between digit 3 and digit 7.
Step 2 — Mark strong links
In practice you label a link strong when the digit has only two possible positions in a unit. For example, if digit 3 can appear only in two cells of a row, the link between those two 3-candidates is strong.
Step 3 — Trace the loop alternating strong and weak
Follow the chain: strong link, weak link, strong link, weak link, and so on, returning to the start. Because the chain is closed and alternates correctly, the truth value (true/false) forced around the loop produces a contradiction unless the target candidate is removed. That contradiction is exactly what allows you to eliminate a candidate seen by both sides of the loop.
Step 4 — Apply the elimination
Any candidate that sees both of the last two linked nodes (the ones that would both be true under opposite assumptions) can be removed. In the example here, the candidate 7 in the labeled target cell can be eliminated.
Why this works (brief intuition)
If you assume one node in the loop is true and follow the alternating chain of implications, you end up forcing the opposite state of that same node — a contradiction. That means the initial assumption must be false, which implies its linked alternative must be true. When both possible assumptions force a particular remote candidate to be false, that remote candidate can be safely eliminated.
Practice tips
- Build AICs from small strong links first (pairs inside a row/column/box), then connect them with weak links. Strong links are the anchors.
- Keep your chains short at first — 4–8 nodes. Short loops are easier and often sufficient for everyday puzzles.
- Use pencil marks neatly. It’s hard to trace AICs without clear candidate lists.
- If uncertain, try both truth-assumption readings: assume one node true and follow implications; then assume it false. If both lead to the remote candidate being impossible, you have a valid elimination.
Conclusion
The nice loop / AIC sudoku technique is a compact way to use local candidate forcing to reach global conclusions. It rewards careful candidate management and gives you a systematic method to place or remove candidates when standard methods stall. Practice spotting strong links (pairs) and stitching them with weak links — once you see a few loops, building and reading them becomes natural.
Caption reference: follow the highlighted nodes in the example grid to see one clear loop and the single candidate it allows you to eliminate.