An inequality chain is a sequence of cells linked by greater-than / less-than edges so you can order them even when some pairs do not share a printed sign. If A > B and B > C, then A > C must hold in the finished grid. That transitive step is ordinary logic, yet it unlocks many Medium and Hard ProPuz boards. This article shows how to spot chains, estimate ranges, and avoid overclaiming.
From one edge to a path
A single edge orders two cells. Two edges sharing a cell can order three. Continue until the path stops. Draw the path on paper as A → B → C (meaning decreasing, if each arrow is “greater than the next”) or use any notation you like—consistency matters more than symbols.
Chains can turn corners: horizontal then vertical. Futoshiki does not require chains to stay in one row. The only requirement is that each step uses a real printed inequality between neighbors.
Transitive payoff
Transitivity does not add a new printed clue between A and C; it adds a constraint on possible digits. On a 5×5, if A > B > C > D, then A is at least 4 if the four cells need four distinct values in a line—but be careful: chain cells are not always in the same row or column. When they are not, digits may repeat across the chain as long as Latin rules allow. The safe claim is ordinal: A is strictly greater than C, so if C is at least 2, A is at least 3, and so on.
When chain cells do share a row or column, uniqueness strengthens the story: a four-cell decreasing chain inside one row on 4×4 forces the permutation 4,3,2,1 along that path order.
Longest-path thinking for extremes
Ask which cell sits at the “top” of the longest greater-than path you can find. That cell is a prime candidate for digit n or other highs. The bottom of a long path favors 1. You do not always place immediately—you shortlist.
On Easy 4×4, longest paths are short and often decisive. On Expert 7×7, several competing paths exist; combine path tops with row/column leftover analysis before committing.
Range narrowing along a chain
Label each cell with a minimum and maximum possible digit given the chain and Latin bans. If B < A and A ≤ 3, then B ≤ 2. If B must also be greater than C, raise B’s minimum. Intersection of min/max with “digits still missing in the row” is where placements fall out.
Pencil marks shine here—see pencil marks in Futoshiki. Write ranges, not vibes.
Branching and merging paths
Sometimes two neighbors are both less than A. Then A exceeds both, but those neighbors are not ordered relative to each other unless another edge says so. Do not invent an order. Divergent “less than A” cells are siblings under A, not a sorted list.
Merging paths (two routes into one cell) tighten that cell’s range from multiple directions. Those merges are gold—treat them as hotspots in your scan loop from beginner techniques.
Common chain mistakes
Assuming A > C without a connecting path. Assuming chain cells must all hold distinct digits even when they do not share a Latin unit. Ignoring a given that breaks an assumed path top. Flipping a vertical glyph and reversing the whole chain.
If a chain “proof” contradicts a given, the proof has a false step. Walk the edges again with your mnemonic locked—see common mistakes.
Chains versus guessing
Following a chain is deduction. Placing a digit because “the chain vibes high” without range proof is guessing. Require either a unique legal digit or a unique cell for a digit before you fill. Chains supply constraints; uniqueness and extremes convert constraints into fills.
When two tops look equally plausible, compare their Latin conflicts. Often only one top survives column bans.
Practice drills
On the next three Easy boards, before any fill, outline the longest inequality path in each quadrant and circle the top and bottom cells. On Medium, spend one minute only on chain ranges, then resume normal solving. On Hard, when stuck, rebuild chains from unfinished edges instead of random probes.
Use daily challenges so you can compare notes with others on the same printed signs.
How ProPuz boards use chains
Generator density of inequalities rises with tier, so Hard and Expert boards tend to offer richer graphs of edges. That does not mean every edge joins one giant chain—many small chains still dominate. Your job is local graph reading, not memorizing a global template.
Check still validates the stored solution. A chain-based fill that satisfies signs and uniqueness should match; if Check disagrees, a misread glyph or duplicate is the usual culprit.
A short worked pattern
Suppose on a 5×5 you find A > B > C in one column, and A also sits greater than D to its right. A is a merge top: it exceeds B and D. If row constraints already ban 5 from every cell except A, place 5 in A. Then B and D are at most 4; continue trimming. You did not need a printed edge between B and D—the merge at A was enough to unlock the extreme.
Another pattern: a three-cell decreasing chain inside one row on 4×4 with the fourth cell given as 1 often forces 4-3-2 along the chain order. Confirm with uniqueness before celebrating. Patterns like these are reusable; they are not substitutes for reading the actual edges on today’s board.
Keep going
Combine chains with no-guessing discipline, speed habits in solve faster, and larger-board skill in advanced methods. Play from the hub or review the articles index for the full learning path.