Inequality (Futoshiki): rules, strategy, and free play
Inequality — known internationally as Futoshiki ('not equal' in Japanese) — is a Latin-square logic puzzle with comparison constraints. Fill an N×N grid with digits 1–N so each row and column contains every digit exactly once. The twist: between some neighbouring cells, a < or > sign tells you which cell must hold the smaller digit. Every inequality must hold true in the final grid.
THE RULES
- Fill every cell with a digit 1–N. On an N×N grid (typically 4×4, 5×5, or 6×6), every cell holds a digit from 1 to N.
- Each row and column contains every digit exactly once. Latin-square constraint — same as Sudoku rows + columns, without the 3×3 box rule.
- Every inequality sign must hold true. A '<' between two cells means the left/upper cell holds a smaller digit than the right/lower cell. '>' is the reverse. The signs appear between adjacent cells only (horizontal or vertical neighbours).
- A few pre-placed digits seed the deduction. Most puzzles start with 0-3 digits already filled. The rest emerges from the row/column rule + inequality signs.
BEGINNER STRATEGY
- Find forced extremes. A cell with a chain of '>' signs pointing away (>, >, >) must hold a large digit — it's bigger than at least 3 cells in that line. On a 5×5 grid, that cell must be 4 or 5. Conversely, a cell with chains of '<' pointing in is small.
- Use the inequality to eliminate. If A < B and you can place A = 4, then B must be 5+. On a 5×5, B is forced to 5. Cascade by eliminating impossible candidates per cell.
- Treat inequality chains as ordered runs. A run like a < b < c forces three strictly-increasing digits. On a 5×5, that's at minimum {1,2,3} — meaning the first cell of the chain can be AT MOST 3 (since two cells must follow with larger digits).
- Cross-reference rows and columns. Pencil-mark candidate digits per cell using the row/column Latin-square rule, then prune via inequalities. The intersection of 'allowed by row + column' and 'allowed by adjacent inequality signs' usually leaves 1-2 valid digits per cell.
- Never guess. Inequality / Futoshiki has a unique solution by deduction. If you're guessing, look for an unevaluated inequality-chain extreme or a Latin-square contradiction in your candidates.
COMMON MISTAKES
- Reading the inequality sign direction backwards. A '<' means left/upper is SMALLER. Beginners frequently flip this — especially when the sign is rotated in a vertical pair. Always re-read: the arrow opens toward the LARGER value.
- Applying each inequality only once. After placing a digit, all inequalities that touch that cell update: neighbours' candidate ranges narrow. An inequality chain is not a one-step clue — cascade every placement through every adjacent inequality sign.
- Forgetting the Latin-square constraint. The row/column uniqueness rule is as important as the inequality clues — but it's invisible on the grid. After every placement, eliminate that digit from the rest of the row AND column before evaluating inequality signs.
- Evaluating inequality chains only at the end. A run of three inequalities a < b < c forces strictly increasing values — evaluate the whole chain at once. The minimum value at the start of the chain and the maximum value at the end are constrained by the chain length, not just the adjacent pair.
HOW TO THINK ABOUT IT
Inequality is a bounds puzzle, not a value puzzle. Every cell holds not a digit but a range [min, max] — and the inequalities are range constraints. Treat each inequality as 'the smaller side's max = larger side's min - 1'. Chain-apply those bounds from the extremes of each inequality chain inward. The Latin-square rule then narrows the surviving candidates to a single digit per cell. Think in ranges first, digits second.
WHY THIS PUZZLE REWARDS YOU
Inequality / Futoshiki is GridJoy's puzzle for fans of comparison logic. Where Sudoku reasoning is purely 'this digit goes in this cell', Inequality reasoning is 'this digit must be SMALLER/LARGER than this neighbour' — relative deductions on top of absolute placements. The puzzle was popularised in Japan in 2001 and saw a brief Western boom via newspaper puzzle pages in the late 2000s. GridJoy's variants run 4×4 (warm-up) through 7×7 (genuinely hard); the deduction style closely mirrors Sudoku scanning, just with an extra constraint layer on top of the Latin-square base.
VARIANTS
- Skyscraper. Latin-square grid with edge-clue visibility constraints instead of cell-pair inequalities. Same base, entirely different constraint style — Skyscraper tells you HOW MANY cells are visible, Inequality tells you which adjacent cells are BIGGER or SMALLER.
- Calcudoku. Latin-square grid with cage arithmetic (+ − × ÷) instead of comparison clues. Calcudoku rewards numerical intuition; Inequality rewards ordered reasoning.
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The classic Latin-square base — play it free in the browser.
Skyscraper →
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Calcudoku →
Latin square with arithmetic cages — a maths-flavoured cousin.
Inequality / Futoshiki strategy for beginners →
Count arrows per cell to bound min/max — forced cells appear first.
Inequality / Futoshiki: how to think →
Trace full chains — the extremes lock first and cascade inward.
Sudoku strategy for beginners →
Latin-square scanning — the foundation Inequality builds on.
Inequality vs Skyscraper →
Both add one rule to Sudoku — comparison signs vs. edge view counts.
Beginner mistakes in logic puzzles →
Five habits that make logic puzzles feel harder than they are.
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