Futoshiki
Number placement with inequality clues
Futoshiki is a Japanese number placement puzzle that combines Latin square logic with inequality constraints. Fill an NxN grid so that each row and column contains every number from 1 to N exactly once — but with a twist: greater-than (>) and less-than (<) signs between certain cells must also be satisfied. These inequality clues add a layer of relational reasoning on top of the placement logic, creating satisfying deduction chains that are distinct from any other puzzle type.
History & Origins
Futoshiki (meaning "not equal" in Japanese) has been published by puzzle magazines since the early 2000s. It gained mainstream attention when The Guardian and The Times began featuring it as a daily puzzle alongside Sudoku. While its exact origins are debated, the puzzle builds on the centuries-old concept of Latin Squares (studied by Euler in the 1700s) by adding inequality constraints between cells — a deceptively simple addition that creates an entirely different solving experience.
How to Play Futoshiki
Fill the Grid
Place numbers 1 through N in each cell of an NxN grid (typically 5x5 or 6x6).
Row Uniqueness
Each row must contain every number from 1 to N exactly once. No repeats allowed.
Column Uniqueness
Each column must also contain every number from 1 to N exactly once.
Inequality Signs
The > and < signs between cells must be satisfied. If 3 > 1, the cell on the larger side must have a bigger number.
Given Numbers
Some cells start pre-filled. These anchor your deductions alongside the inequality clues.
Chain Reasoning
Inequalities can form chains (A > B > C), which restrict possible values for multiple cells simultaneously.
Strategy & Solving Tips
Expert Futoshiki solving means using inequalities not just as individual constraints but as chains that restrict ranges across multiple cells.
- Inequality chains restrict ranges: if A > B > C in a 5x5 grid, then A is at least 3, B is at least 2, and C can be 1-3
- Cells at the "small" end of long chains must have low numbers; cells at the "large" end must have high numbers
- A cell with > on both sides (larger than two neighbors) cannot be 1 or 2 in a 5x5 grid
- Combine inequality restrictions with row/column uniqueness to narrow candidates quickly
- If a number can only go in one place within a row or column (after inequality filtering), place it immediately
- Look for forced extremes: 1 must go on the small side of any inequality chain it participates in, and N on the large side
Common Mistakes to Avoid
- Only considering direct inequalities while missing chains — if A > B > C, then A must be at least 3 in a 5x5 grid
- Forgetting that the ABSENCE of an inequality sign also provides information in some variants
- Placing a number that satisfies all inequalities but violates row or column uniqueness
- Not tracking candidate ranges for cells involved in inequality chains — write them down!
Futoshiki FAQ
How is Futoshiki different from Sudoku?
Both use Latin square rules (unique digits in rows and columns), but Futoshiki replaces Sudoku's 3x3 box constraint with inequality clues between adjacent cells. The reasoning style is quite different — Futoshiki is about relational ordering rather than regional placement.
What does "Futoshiki" mean?
The name comes from Japanese and roughly translates to "not equal" or "inequality," directly referencing the puzzle's signature greater-than/less-than clues.
Are all the inequality signs shown?
Not necessarily. Only some cell pairs have inequality signs. The absence of a sign between two cells means their relationship must be deduced from other constraints.
Do I need to guess?
No. Every puzzle has a unique solution reachable through logical deduction alone, using the combination of uniqueness rules and inequality constraints.
Similar Puzzles You Might Enjoy
Sudoku
The most famous Latin square puzzle — similar row/column uniqueness but with 3x3 box constraints instead of inequalities.
Renzoku
Another Latin square variant where dots indicate consecutive numbers between cells.
Skyscrapers
Place unique heights per row/column with visibility clues from edges instead of inequality signs.
Ready to Play Futoshiki?
Explore the elegance of Futoshiki — where number placement meets inequality reasoning in a puzzle that's related to Sudoku yet entirely its own experience.