The rules of futoshiki

The complete rules · prefer the illustrated guide with strategies?

Futoshiki (Japanese for "not equal") is a logic puzzle played on a square grid — here 4×4, 5×5 or 6×6. Every row and every column must contain the numbers 1 through N exactly once (N is the grid size), and every inequality sign between two cells must point toward the smaller number. Some numbers are given as clues; your job is to fill in the rest. Every puzzle has exactly one solution.

The six rules

1. Fill every blank cell with a number from 1 to N, where N is the grid size.
2. Every row must contain each number 1 to N exactly once.
3. Every column must contain each number 1 to N exactly once.
4. Every sign points toward the smaller number: < means the cell on its left is smaller, > means the cell on its left is larger.
5. Vertical signs work the same way: ∧ means the cell above is smaller, ∨ means the cell below is smaller.
6. There is exactly one solution, reachable by logic.

How to read the grid

Each sign sits between two neighbouring cells, like an alligator's mouth: it always opens toward the larger number and points at the smaller one. So a row that reads 3 < 5 is correct (3 is smaller than 5), and 7 > 2 is correct (7 is larger than 2). The signs between rows work the same way — a ∧ between two cells stacked vertically means the upper cell holds the smaller number, and a ∨ means the lower one does.

The signs point at the smaller number

This is the rule that gives futoshiki its name and its challenge. The signs are not decoration: every one of them must be true in the finished grid. A chain like 1 < ? < 4 tells you the middle cell can only be 2 or 3, and if the surrounding rows and columns rule out one of those, the other is forced. Long sign chains are often the fastest way to pin down a number.

One more thing: rows and columns

Just like sudoku, no row or column may repeat a number. That rule interacts with the signs: a number that fits every sign in its row may still be impossible because its column already contains it. The intersection of the two rules is what makes the puzzle solvable step by step.

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