Futoshiki: rules, tips and free puzzles online
Futoshiki is a Latin square with inequality signs. Fill the grid so every row and column holds 1 to N once, and every < or > between two cells is true. Play futoshiki online free here, on grids from 5×5 to 7×7.
The name is Japanese for “inequality”. The futoshiki rules take seconds to learn, but long chains of signs make for some very elegant logic.
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The rules
- The grid is N×N (5×5 to 7×7 here). Each row and column contains every digit from 1 to N exactly once.
- An inequality sign between two neighbouring cells must be obeyed. The narrow point faces the smaller digit and the open side faces the larger.
- Any digits already in the grid are given clues.
How to start
Ones and Ns
A cell that is larger than a neighbour can’t be 1, and a cell that is smaller can’t be N. Those two facts alone clear a surprising number of candidates.
Chains give bounds
In a chain of k cells each smaller than the next, the first is at most N − k + 1 and the last is at least k. Every cell in between is bounded from both sides.
Signs across lines
Chains can turn corners from rows into columns. A cell that is smaller than two different cells, each of which is smaller than something else, gets pushed down hard. Combine those bounds with ordinary row and column scanning, explained in our solving techniques guide.
Common mistakes
- Reading a sign backwards. The point always aims at the smaller digit.
- Stopping a chain at a corner. Signs between vertically adjacent cells continue the chain.
- Forgetting that every row and column needs all of 1 to N.
How the ranks scale difficulty
The same four ranks apply here as everywhere on the site. Lemma is a gentle warm-up, Proposition needs a real idea or two, Theorem is near-minimal with almost every clue load-bearing, and Conjecture is the sparsest of all, so expect to think hard.
Futoshiki grids run from 5×5 to 7×7. Larger grids give longer chains and more room for subtle interactions between rows and columns.
There are 99,999 puzzles per rank, each with its own number, so you can replay one or share it. Every type also has a 30-puzzle collection that climbs from Lemma to Conjecture. Every puzzle is checked to have exactly one solution that you can reach by logic alone, with no guessing.
Frequently asked questions
What are the futoshiki rules?
Fill an N×N grid so each row and column contains 1 to N once, and every inequality sign between neighbouring cells is true.
What does futoshiki mean?
It is Japanese for “inequality”.
How do I read a vertical futoshiki sign?
Exactly like a horizontal one: the point faces the smaller digit.
Professor Hoot’s hints teach the next logical step whenever you’re stuck.