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Futoshiki Rules: How to Play (With Examples)

Futoshiki is a logic puzzle on a square grid. Fill it so every row and every column holds each number once, 1 to 5 on a 5×5, and every sign between two squares holds: the smaller number on the pointed side, the larger on the open side. The name is Japanese for “not equal”.

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The rules

Fill every square with a number from 1 up to the size of the grid, so that:

  1. Every row and every column holds each number once. On a 5×5, that’s 1, 2, 3, 4 and 5 in every row and every column.
  2. Every sign holds. A sign between two squares says which is the smaller: the point faces the smaller number, the open side the larger.

Some puzzles start with a few numbers filled in. Many start with none: the signs alone are enough.

Reading the signs

The signs read just as they do in maths. Along a row, 1 < 3 means 1 is less than 3, and 3 > 2 means 3 is more than 2. Down a column, the same sign is turned on its side: the point still faces the smaller number.

Reading the signs A row of four squares: 1, less than 3, greater than 2, less than 4. Each sign points at the smaller number and opens towards the larger. 1 3 2 4
1 < 3 > 2 < 4: every sign points at the smaller number.

What breaks a rule

The same number twice in a line. Two 3s in the top row:

The same number twice in a row: not allowed A 4 by 4 grid with two 3s in the top row, ringed in red. 3 1 3
Two 3s in one row (ringed): not allowed.

A sign that doesn’t hold. The sign says the first square is the smaller, but 4 isn’t less than 2:

A sign that doesn't hold: not allowed A 4 by 4 grid whose top row starts 4, then a less-than sign, then 2. Four isn't less than two, so the sign is broken, shown in red. 4 2
4 < 2 is false, so the sign turns red: not allowed.

In the game, a repeated number turns red, and so does a sign that doesn’t hold.

A puzzle from the game, solved

This is the first level in the game: Easy level 1, a 5×5 with three numbers given and five signs.

Easy level 1 from the game A 5 by 5 Futoshiki with three clues, a 2 in row 1 column 1, a 1 in row 2 column 3 and a 3 in row 4 column 2, and five signs. 2 1 3 1 2 3 4 5 1 2 3 4 5
Easy level 1, before you start. Rows and columns are numbered from the top left.

It takes 22 steps. Here are the first three, all from the signs.

Step 1: less than 2

The square in row 1, column 2 is less than the 2 beside it. The only number less than 2 is 1.

Step 1: less than 2 The square in row 1, column 2 is less than the 2 beside it, so it can only be 1. 2 1 1 3 1 2 3 4 5 1 2 3 4 5
Less than 2: it’s 1.

Step 2: less than 3, and not 2

The square in row 4, column 1 is less than the 3 beside it, so it’s 1 or 2. Column 1 already has its 2, in row 1. So it’s 1.

Step 2: less than 3, and not 2 The square in row 4, column 1 is less than the 3 beside it, so it is 1 or 2. Column 1 already has its 2, in row 1, so it is 1. 2 1 1 1 3 1 2 3 4 5 1 2 3 4 5
Less than 3 rules out 3, 4 and 5; column 1’s 2 rules out 2.

Step 3: a chain of two signs

On the right-hand side, three squares are joined by two signs: row 4, column 4 is less than row 4, column 5, which is less than row 3, column 5. Row 4 already has a 1 and a 3, so the smallest of the three is at least 2. The middle one is bigger than that and can’t be 3, so it’s at least 4. The top one is bigger again: 5.

Step 3: a chain of two signs Row 4, column 5 is greater than row 4, column 4, and row 3, column 5 is greater again. Row 4 already has a 1 and a 3, so the smallest of the three is at least 2, the middle one at least 4, and the top one 5. 2 1 1 5 1 3 1 2 3 4 5 1 2 3 4 5
A chain of two signs: the top of it can only be 5.

The rest

From there, most of it is counting what each row and column already has. In all, it takes 5 steps from the signs, 2 where a number has only one place left in a row, and 15 where a square has only one number left.

Easy level 1, solved The finished grid. Rows: 2 1 4 5 3; 5 4 1 3 2; 4 2 3 1 5; 1 3 5 2 4; 3 5 2 4 1. 2 1 4 5 3 5 4 1 3 2 4 2 3 1 5 1 3 5 2 4 3 5 2 4 1 1 2 3 4 5 1 2 3 4 5
Easy level 1, solved.

Where to start in any puzzle

Start with the 1s and the top number. The larger side of a sign is never 1, and the smaller side is never the top number (5 on a 5×5). Look along each row and column for the one square that isn’t ruled out. On every one of the game’s 150 levels, the first move comes from the signs, and on 135 of them it places a 1 or the top number.

Don’t wait for clues. 48 of the game’s levels start with no numbers at all: 28 of the 50 Easy ones, 16 Medium and 4 Hard. The signs are enough.

Common mix-ups

  • There are no boxes. Unlike Sudoku, only rows and columns have to hold each number once.
  • A sign only links the two squares it sits between. It says nothing about squares further along.
  • Signs that point the same way make a chain. In a row of four, three signs all pointing the same way can only be 1 < 2 < 3 < 4.

Next steps

Questions

What does Futoshiki mean?

It’s Japanese for “not equal”, from the signs between the squares.

Which way round do the signs go?

The point faces the smaller number and the open side faces the larger, just as in maths: 2 < 4, and 5 > 3. Down a column, the sign is turned on its side, but it works the same way.

Does every puzzle have one solution?

Yes. Every puzzle on PeerPuzzles has exactly one solution, and can be finished by logic alone, without guessing.

Is it like Sudoku?

Partly. Every row and column holds each number once, as in Sudoku, but there are no boxes, and the signs do the work that clues do in Sudoku. Many Futoshiki puzzles start with no numbers at all.

What size should I start with?

5×5. Every Easy level in the game can be finished with the signs and the two simplest ideas: a square with one number left, and a number with one place left.