How to play Futoshiki

Futoshiki gives you almost no numbers — just a web of ‹ and › signs. Reading what those signs imply about the extremes is the whole art. Here are the rules and the techniques.

The rules

• Fill the N×N grid so each row and column contains the numbers 1 to N exactly once (like Sudoku without boxes).
• Every inequality sign between two neighboring cells must hold. The point of the sign aims at the smaller number.
• Some cells may be given at the start (locked).

Controls

• Tap a cell, then a number on the pad. Tapping the same number again erases it.
• On desktop, the keyboard works: digits enter, Backspace erases, arrow keys move around.
• Rule-breaking cells outline in red if you pause; the selected cell highlights its row and column.

Solving strategies

1. Chase the chains. Signs chain into ladders: in a ‹ b ‹ c ‹ d, cell a is at most N−3 and d at least 4. Long ladders often pin cells to a single value at once.
2. Place the extremes. A 1 can never sit on the open (larger) side of a sign; an N can never sit on the point side. Scan each row and column for the only legal home of its 1 and its N — this move alone cracks most 5×5 boards.
3. Shrink ranges. Think of each cell as a range. A cell that must be less than a neighbor loses its top value; one that must be greater loses its bottom. Repeat and ranges collapse to singletons.
4. Use uniqueness. When only two cells in a row can hold a value, a single adjacent sign usually rules one of them out.
5. Cascade. Every number you place re-tightens all four neighboring signs. Re-scan around each entry before moving on.

A worked example

A 5×5 row shows a ‹ b ‹ c with no numbers. From the chain, a ≤ 3, b is 2–4, and c ≥ 3. Now suppose the column tells us c can't be 5 (the 5 is already placed there) — then c is 3 or 4. If c were 3, the chain would force b = 2 and a = 1; a's column already has a 1, contradiction — so c = 4, b is 2 or 3, and a shrinks to 1 or 2. One more sign anywhere in that neighborhood settles all three. That's Futoshiki: ranges narrowing until only the truth is left.

Common mistakes

• Misreading the sign. The point always aims at the smaller number — vertical signs included (the point can aim up or down).
• Ignoring un-signed cells. Row and column uniqueness applies everywhere, not just where signs are.
• Forgetting the extremes. If you're stuck, ask "where can the 1 go in this row? where can the N?" — the answer is usually forced.
• Guessing. Never needed: every board has a logical next step and exactly one solution.

Frequently asked questions

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