How to play Futoshiki

What is Futoshiki?

Futoshiki is a number-placement puzzle on an N×N grid. You fill each cell with a digit from 1 to N so that every row and column contains each digit exactly once. That part works like sudoku rows. What makes Futoshiki different is the inequality symbols between cells: they tell you which of two adjacent cells must hold the larger number.

The name comes from Japanese (不等式) and means "inequality." In English you might see it called greater-than sudoku, the inequality puzzle, or the unequal puzzle. Grid sizes range from 4×4 for beginners up to 9×9 for people who want a serious challenge.

The two rules

1. Latin square: Every row and every column must contain each digit from 1 to N exactly once. In a 5×5 grid, each row and column has exactly one 1, one 2, one 3, one 4, and one 5.
2. Inequality constraints: Where an inequality symbol appears between two cells, the relationship must hold. If cell A has a < pointing toward cell B, then the digit in A must be less than the digit in B.

That is the complete rule set. No 3×3 boxes, no cage sums, no arithmetic. Just row/column uniqueness and a handful of directed inequalities.

Reading the inequality symbols

Symbols sit in the gaps between adjacent cells. On this site, we use four symbols depending on the direction:

The pointed end of each symbol faces the smaller value. Think of < as an open mouth eating the bigger number — the wider side points at the larger cell.

Not every pair of adjacent cells has a symbol. Where no symbol appears, the two cells have no special relationship beyond normal row/column uniqueness.

Worked example: 4×4 grid

Suppose you have a 4×4 grid with digits 1 through 4. Row 1 already has a 3 in column 1 and a 1 in column 4. Between column 2 and column 3, there is a < symbol.

Step 1 — Use row uniqueness. Row 1 already has 3 and 1. The remaining two cells (columns 2 and 3) must hold 2 and 4 in some order.

Step 2 — Apply the inequality. The < between columns 2 and 3 means column 2 is less than column 3. With only 2 and 4 available, column 2 must be 2 and column 3 must be 4.

Step 3 — Propagate. Now that row 1 is filled (3, 2, 4, 1), each column has one fewer candidate for the remaining rows. Continue this process across the whole grid.

This is the core loop in every Futoshiki puzzle: use row/column elimination to narrow candidates, then use inequalities to decide between the remaining options. On harder puzzles you chain multiple inequalities together before anything resolves.

Basic strategies

Naked singles

When a cell has only one possible digit after accounting for its row, column, and any inequalities, place it. This is the bread-and-butter technique. After each placement, re-check neighboring cells because new constraints emerge.

Inequality forcing

If A < B, then A cannot hold the grid's maximum digit and B cannot hold 1. In a 5×5 grid, A ≤ 4 and B ≥ 2. When a cell's candidates already sit near the top or bottom of the range, a single inequality can pin it to one value.

Inequality chains

Follow sequences of inequality symbols. In a 5×5 grid where A < B < C, the chain length is 3. A can be at most 3 (needs room for B and C above it), and C is at least 3 (needs room for A and B below it). Chains of length equal to the grid size force every cell in the chain to one value.

Hidden singles

If a particular digit can only fit in one cell of a row or column (even if that cell has other candidates), place it. Hidden singles often appear after inequality constraints have eliminated candidates from most cells in a row or column.

Futoshiki vs Sudoku

The Latin square constraint is identical: each digit once per row, once per column. But Sudoku adds 3×3 box uniqueness as its extra constraint, while Futoshiki uses inequality symbols instead. This changes the solving logic significantly.

In Sudoku, you reason about which cells can hold a digit. In Futoshiki, you also reason about relative size: "this cell must be biggerthan that one." Inequality chains — where you follow a sequence of symbols across the grid — have no equivalent in standard Sudoku. If you enjoy Sudoku and want something that feels related but uses a different part of your brain, Futoshiki is a natural next step.

Other puzzles in the same family include KenKen (arithmetic cages on a Latin square) and Killer Sudoku (cage sums on a Sudoku grid). Each variant layers a different constraint type on top of the basic number-placement framework.

Common mistakes

Ignoring the direction of the symbol. It is easy to read < as "less than" and accidentally apply it backwards when the cells are stacked vertically. The pointed end always faces the smaller cell.

Forgetting row/column uniqueness. When you focus on inequalities, it is tempting to place a digit that satisfies the symbol but duplicates another digit in the same row or column. Check both constraints.

Guessing instead of deducing. Well-constructed Futoshiki puzzles are fully solvable by logic. If you feel stuck, there is a deduction available that you have not spotted yet. Look for inequality chains or hidden singles.

Frequently asked questions