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Futoshiki

In Diem this puzzle is called Signs: the chevrons between the squares are the greater-than signs.

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Each number once per row and column. Every sign is true: it points to the smaller number.

Tap a square, then a number.

Use the arrow keys to move between squares. Type a number to fill a square; Backspace clears it.

Futoshiki rules

A solved 4 by 4 Futoshiki: each number once in every row and column, and every sign true.4213312424311342
  1. Every row holds each number once.
  2. Every column holds each number once.
  3. Every sign is true: its point faces the smaller number.

Futoshiki is a Latin square with clues in between. A 4 × 4 grid uses the numbers 1 to 4, a 5 × 5 grid 1 to 5. Some boards start with a few numbers in place; the hardest start with none, and the signs alone decide everything. In the solved grid above, the top-left 4 is bigger than the 3 below it, and in the third row 4 > 3 > 1.

How to solve Futoshiki

These are the ideas Diem's Hint button names, easiest first. A hint rings the square you can work out next and names the idea; it never shows the number.

1. Last Spot

The last column already has 2, 1 and 3, so its gap is 4.2134
A row or column with one gap takes the number it is missing. The last column here has 2, 1 and 3, so the bottom square is 4.

2. Cross Out

The 2s in the first and last columns leave the second row only one place for its 2.1222
Pick a number and a row, and cross out the squares whose column already has it. The two 2s rule out the first and last squares of the second row, and its 1 is taken, so its 2 can only go in the third square.

3. The ends of a sign

The sign says the ringed square is smaller than 2, so it is 1.221
The small end of a sign can never be the biggest number, and the big end can never be 1. Here the sign says the ringed square is smaller than 2, and only 1 is.

4. Only Fit

The ringed square sees 5 and 3 in its row and 2 and 1 in its column, so only 4 fits.25314
List what a square sees in its row and column. The ringed square sees 5 and 3 in its row and 2 and 1 in its column. Only 4 is left.

5. Lock: follow the chain

On an empty 5 by 5 board, a chain of signs forces the ringed square to be 2.2
Signs in a row make a chain, and a chain counts. On this empty board the ringed square is smaller than the square below it, which is smaller than the one to its left, which is smaller than the one above that: three bigger numbers, so it is at most 2. It is also bigger than the square above it, so at least 2. It is 2.

Hard boards here need chains, often long ones. The Diem app adds two more ideas in its later levels: Pairs (two squares in a line that can only hold the same two numbers) and No Room (a number that would leave another square with nothing left).

FAQ

How do you play Futoshiki?

Fill the grid so that every row and every column holds each number once (1 to 4 on a 4 × 4 grid, 1 to 5 on a 5 × 5 grid). Between some squares there is a sign, < or >, and every sign must be true: its point faces the smaller number. Tap a square, then a number.

What does Futoshiki mean?

Futoshiki is Japanese for "inequality", after the greater-than and less-than signs that make the puzzle.

What are good Futoshiki strategies?

Start with the ends of the signs: the small end of a sign can never be the biggest number, and the big end can never be 1. Then follow chains of signs: a square with three squares in a rising chain after it can be at most 2 on a 5 × 5 grid. Fill the rest with the rules of rows and columns, one square at a time.

Does every Futoshiki here have only one solution?

Yes. Each board is made by a generator that counts the solutions and keeps the board only when there is exactly one, and only when it can be solved step by step without guessing. You will never need a solver.

Can I play Futoshiki online without an account?

Yes. Play by touch, mouse or keyboard (arrow keys to move, numbers to fill). Your best time for each difficulty is kept in your browser only, and the clock stops whenever you leave the puzzle.

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