How to Solve Futoshiki: Rules, Tips and Strategies for Beginners
Futoshiki ("not equal") is the Japanese logic puzzle of inequalities: a square grid where every row and every column must hold the numbers 1 to N exactly once, and every < > sign between neighbouring cells points toward the smaller number. The signs do half the work for you — they tell you which way the numbers lean — so the puzzle is more approachable than it looks. Here's how it works and how to solve one cleanly, step by step.
The rules in one minute
Fill every blank cell with a number from 1 to N, where N is the grid size (4, 5 or 6 on our generator). Every row must contain each number 1 to N exactly once, and every column must too — no repeats, just like sudoku. Every sign points toward the smaller number: < means the cell on its left is smaller, > means the cell on its left is larger. The vertical signs work the same way — ∧ means the cell above is smaller and ∨ means the cell below is smaller. Every futoshiki puzzle has exactly one solution.
How to read the grid
The signs sit between neighbouring cells, and each one is a mini-clue: the smaller number is always on the side the sign points at, like an alligator's mouth opening toward the bigger meal. A row that reads 3 < ? can only end in 4, 5 or 6; a row that reads ? < 3 can only start with 1 or 2. On our futoshiki generator the signs are drawn between the squares exactly this way, and you can type numbers directly into the blanks with instant checking. If you just want the plain rules without the strategies, our rules page has them all in one place.
1. Start with the extremes: 1 and N
The smallest and largest numbers are your most powerful clues. A > sign always points at a smaller number, so any cell that must be smaller than a neighbour can never hold the largest number, N. Work the other way too: a < sign pointing away from a cell means that cell is larger than its neighbour, so it can never hold the 1. Sweep the grid marking which cells can't be 1 and which can't be N — you'll often find a cell that is forced to one of them outright.
2. Follow the sign chains
When several signs line up in a row, chain them together: 1 < ? < ? < 5 means the two blanks must be two numbers strictly between 1 and 5 — so they're 2, 3 and 4 in some order, with the chain forcing the middle ones to be 3 and 4 (2 can't sit between two larger numbers in a three-cell chain of increasing signs... actually it can — what the chain guarantees is that the blanks are from {2, 3, 4} and each sign picks the smaller one). The practical payoff: a chain of signs rules out whole ranges, and a chain that starts or ends at an extreme often fills itself in.
3. Use the rows and columns like sudoku
Every row and every column is a complete set of numbers 1 to N. Once you know four of the five numbers in a row (in a 5×5 grid), the fifth is whatever is missing — no sign needed. This missing-number trick combines beautifully with the signs: when a sign tells you a cell is smaller than its neighbour, and the row tells you the cell can only be 2 or 3, the sign decides between them.
4. Let signs eliminate candidates
Think of each sign as a filter. If a cell is < 4, its candidates are 1, 2, 3 — cross out everything else. If it's also in a column that already contains 2 and 3, it's forced to 1. The signs give you the range, the rows and columns give you the exact digit. Work both together and nearly every blank resolves through elimination rather than guessing.
5. Pencil in the candidates
In the corner of each blank, jot the numbers its row and column still allow, then apply each attached sign to narrow the list. A cell with > on both sides can't be N; a cell with < on both sides can't be 1. Keep the lists short and cross them off as you solve — futoshiki grids are small, so tidy pencil marks turn the whole puzzle into a simple elimination chart.
6. Avoid the classic mistakes
Three errors trip up beginners. First, reading a sign backwards — always remember it points at the smaller number. Second, forgetting the vertical signs: ∧ and ∨ are easy to miss in the middle of a busy grid. Third, filling a number that satisfies every sign but repeats in its row or column. Check each finished line in both directions, and if you're stuck, the answer key is a great way to see the pattern you missed.
Practice with a printable futoshiki
The fastest way to learn is a fresh puzzle. Generate one in seconds on the futoshiki generator — pick a difficulty and grid size, then print the blank grid or solve on screen with instant checking. Start on a 4×4 easy, where the signs do most of the work, then step up to 5×5 and 6×6 as the chains get longer and more numbers are hidden. If you enjoy the row-and-column logic, sudoku is a natural companion — and if you like the number reasoning, our kakuro guide has you covered.