THE TWO RULES, AND WHY THE INEQUALITY IS THE REAL CONSTRAINT
Futoshiki has two rules: (1) every row and column contains each digit from 1 to N exactly once (the Latin-square rule). (2) The inequality signs between adjacent cells must be satisfied — if a sign points from cell A to cell B, the value in A is greater than the value in B.
Most beginners focus on the Latin-square rule (familiar from Sudoku) and treat the inequalities as hints. The better mental model is the reverse: inequalities are the primary constraint, and the Latin-square rule is the elimination mechanism that turns inequality chains into forced placements. On hard grids, almost all progress comes from reading inequality chains rather than from simple row/column elimination.
TECHNIQUE 1: CHAIN BOUNDS — DIRECT UPPER AND LOWER LIMITS
An inequality chain is a sequence of cells connected by consecutive inequality signs in the same direction. For example, A > B > C in a 5×5 grid (digits 1–5).
A chain of length L forces the first cell to be at least L and the last cell to be at most N − L + 1. In the A > B > C example (chain length 3 in a 5×5), A must be at least 3 (since it must be greater than B which must be greater than C, so A ≥ 3, B ≥ 2, C ≥ 1) and C must be at most 3 (since A ≤ 5, B ≤ 4, C ≤ 3).
More precisely:
- The largest cell in a chain of length L must contain a digit ≥ L.
- The smallest cell in a chain of length L must contain a digit ≤ N − L + 1.
- Every cell at position k from the bottom of the chain (where the smallest is position 1) must contain a digit ≥ k.
Scan for chains before anything else. A chain of length N − 1 in an N×N grid forces the largest cell to contain N and the smallest cell to contain 1. A chain of length N − 2 forces the largest cell to contain N or N − 1. These bounds cascade into eliminations across the row and column.
TECHNIQUE 2: GIVEN DIGITS AS BOUNDARY CONDITIONS
When a given digit is at one end of an inequality chain, the chain bounds become exact. If a cell has value 4 and a sign reads 4 > X, then X must be 1, 2, or 3. If the grid is 5×5 and X is at the bottom of a chain of two (X > Y), then X ≥ 2 — combined with X ≤ 3, X must be 2 or 3. If Y also appears in a row where 2 is already placed, X must be 3, and Y must be 1 or 2.
Given digits anchor chains. Start with every inequality sign that has a given digit on one side, derive the range of the other side, and propagate that range through the rest of the chain.
TECHNIQUE 3: SINGLE-SIGN CONSTRAINTS AND CROSSINGS
Even a single inequality between two cells (A > B, no chain) gives information. In a 5×5 grid:
- If A > B, then A cannot be 1 (there's nothing smaller in the valid range 1–5 below it). B cannot be 5 (there's nothing larger for A to take).
- If A and B are in the same row, eliminating 1 from A and 5 from B immediately reduces the candidates for every other cell in that row.
When a cell is involved in two inequality signs — one on each side — it is "crossed". For example, A > B < C means B must be smaller than both A and C. In a 5×5 grid, if B < A and B < C, B's maximum value is min(A, C) − 1. If the row already contains several digits, B's range may collapse to a single value.
Crossings are particularly powerful because they constrain the crossed cell from both directions simultaneously. Scan for crossings as a second pass after chains.
TECHNIQUE 4: COMBINING INEQUALITIES WITH ROW/COLUMN ELIMINATION
The Latin-square rule acts as a multiplier on inequality reasoning. Once you derive that a cell must contain a value in range [2, 4], check the row and column: if 2 and 3 are already placed in that row, the cell must be 4. If 4 is already placed in that column, the cell must be 2 or 3 — but from the inequality, it must also be ≥ 2, so nothing is ruled out yet, but the range is exactly {2, 3}.
The propagation works in both directions: inequality bounds narrow the Latin-square candidates, and Latin-square elimination narrows the possible satisfying values for inequality signs. The solve alternates between these two modes.
In practice: after every cell is filled (by either inequality forcing or Latin-square placement), check every inequality it touches. Does the placed value change the bounds on the adjacent cell? If yes, propagate immediately.
TECHNIQUE 5: WHAT THE LARGEST AND SMALLEST POSSIBLE VALUES REVEAL
For each empty cell, determine its current valid range from the inequalities and Latin-square eliminations. If any cell's range collapses to a single value, place it immediately.
More subtly: if a cell's upper bound equals its lower bound after accounting for inequalities on both sides, it is forced. This happens more often than it appears — a cell with a chain-derived upper bound of 3 and a chain-derived lower bound of 3 must contain 3, even if the row hasn't directly eliminated any values.
On hard grids, maintain a running upper and lower bound for each empty cell as you solve. Update both whenever a neighbour is placed or a chain constraint tightens. This is the key discipline that separates systematic Futoshiki from guess-and-check.
A PRACTICAL SOLVING SEQUENCE
- Pass 1: Identify all chains. For each chain of length L, record the bounds: largest cell ≥ L, smallest cell ≤ N − L + 1. If any chain is long enough to force a specific value, place it.
- Pass 2: Apply given digits as boundary conditions on adjacent inequality signs. Propagate through chains from any given digit.
- Pass 3: Identify crossed cells (constrained from both sides). Derive their combined upper/lower bounds.
- Pass 4: Alternate inequality reasoning with Latin-square elimination — each mode feeds the other. After each placed cell, update bounds for all neighbours.
- If stalled: Find the cell with the narrowest current range and enumerate its candidates. Trace which assignment produces a contradiction and eliminate it.
Most beginner and intermediate Futoshiki grids resolve entirely in passes 1–3. The combining of inequality and Latin-square reasoning (pass 4) handles intermediate-hard grids. Contradiction analysis is only needed for the hardest grids — well-constructed Futoshiki puzzles have a unique solution reachable by pure logic.