Techniques turn rules into moves. This page lists the first toolkit every Futoshiki player should own before chasing advanced propagation on 6×6 and 7×7. If definitions still blur, skim basic rules first; if habits need work, pair this with beginner tips.
Technique: extreme-digit placement
On an n×n board, digit n can never sit on the small side of an inequality, and digit 1 can never sit on the large side. Scan for cells that are “greater than” one or more neighbors—those are candidate homes for highs. Scan for cells that are “less than” neighbors—those favor lows.
On ProPuz Easy (4×4), placing 4 and 1 early often collapses the rest into {2, 3} puzzles. Practice naming the reason: “4 cannot go here because this edge requires a smaller digit.” Naming prevents cargo-cult clicking.
Technique: forced neighbor from a known digit
If a cell holds 3 on a 4×4 and shows > toward a blank neighbor, that neighbor is in {1, 2}. If the neighbor’s row already used 2, the neighbor is 1. One known digit plus one sign frequently yields a single legal value—or a tiny candidate set you can track.
Always combine with uniqueness: the forced neighbor must still be missing from its row and column. A “forced” 2 that duplicates a column is not forced at all.
Technique: leftover singles (Latin half)
When three cells in a 4×4 row are filled, the fourth is determined by uniqueness alone—even with no inequality on that cell. Beginners over-focus on signs and miss free leftovers. After every fill, ask: “Did any line drop to one empty cell?”
Hidden singles matter too: a digit that can fit in only one empty cell of a line belongs there, even if that cell still shows other pencil candidates. Clear the extras when you place it.
Technique: hotspot edges
An edge is a hotspot when both sides are blank but the comparison interacts with givens or near-complete lines. Prioritize those edges over quiet empty regions. Hotspots produce information; quiet regions wait.
A cell touched by two inequalities is a double hotspot. Its allowed range is the intersection of both constraints. Write the intersection before placing anything nearby.
Technique: reliable scan order
Adopt a fixed loop: (1) inequality sweep, (2) extreme candidates, (3) forced neighbors from givens, (4) uniqueness singles, (5) re-test edges, (6) repeat. Changing order every puzzle trains chaos. The loop is the technique that makes other techniques fire on time.
On Medium 5×5, the same loop works—only the candidate lists grow. Do not invent a new opener just because n increased.
Technique: mini contradiction test
When two candidates remain, ask “if this were 3, what breaks?” Mentally place it and watch one edge or one duplicate fail. That is not wild guessing; it is a short, reversible probe with a clear kill criterion. Limit probes to binary cells and undo immediately if both seem to survive—then you missed a constraint.
See also solve without guessing for the philosophy behind probes versus coin flips.
Technique: batch self-audit before Check
Place a small block, then audit rows, columns, and touching signs yourself. Only then use ProPuz Check grid. The audit is a technique: it builds the internal checker you need when printables or offline sheets have no button.
If Check fails, classify the error (duplicate vs broken inequality) before editing. Classification is deliberate practice; thrashing is not.
What to postpone
Long chain algebra, multi-cell fancy catalogs, and Expert-only bookkeeping can wait. Master the toolkit above on Easy and Medium until explanations feel boring. Boredom means readiness—then graduate to inequality chains and advanced methods.
Also postpone speed goals. Techniques exist to create correct fills; speed is a side effect of cleaner scans.
A one-week drill plan
Days 1–2: only extreme placement and forced neighbors on Easy. Days 3–4: add leftover singles and the fixed scan loop. Day 5: introduce binary probes. Day 6: one Medium board using the same toolkit. Day 7: review mistakes with common mistakes and replay a challenge from the daily calendar.
Journal one sentence per solved board: which technique unlocked the first digit? Patterns in that journal show which tool you underuse.
How the techniques work together
Think of extremes as your opening move, forced neighbors as midgame tactics, leftovers as free points, hotspots as attention management, and the scan loop as the coach that calls each play. Probes and audits are safety equipment. Using only one tool is like playing tennis with a single stroke—you will win easy points and lose real rallies.
On a typical ProPuz Easy grid you might place a 4 from extreme logic, force a 1 beside it from a sign, finish a row with a leftover single, then discover a {2,3} pair decided by one remaining inequality. That entire sequence is the toolkit in miniature. When Medium feels harder, it is usually because you skipped the loop—not because the techniques expired.
If you stall after learning the names, return to a live board and forbid yourself from placing anything you cannot label with a technique name. Labeling is training; unlabeled fills are how old guessing habits sneak back in.
Keep practicing
Open the Futoshiki hub, follow a concrete 4×4 walkthrough, deepen notes via pencil marks, and browse the full articles library. Techniques stick when they leave fingerprints on real grids—not when they stay as vocabulary.