Puzzle guide
How to solve Futoshiki
Futoshiki is a number grid with little signs between the cells, just like the < and > you know from school. Every row and every column holds each number once, and the signs tell you which neighbour is bigger. Here is how to use them, step by step.
The rules
- Fill every cell with a number from 1 up to the size of the puzzle: 1 to 4 on a 4×4 puzzle, 1 to 9 on a 9×9.
- Every number appears exactly once in each row and each column.
- Follow the signs: the point of a sign faces the smaller number, like 2 < 3. Between cells above each other it works the same way, up and down.
- Numbers that are printed at the start stay as they are.
A solved puzzle
Your first moves
- Look at the signs first. The cell on the point side can never hold the largest number, and the cell on the open side can never hold 1.
- Follow chains: two or three signs in a row tell you a lot more than one.
- Fill a row or column that has only one empty cell left.
- Use notes for the numbers a cell can still hold, and cross them off as the board fills.
Techniques, step by step
The last number in a row
This row already holds 4, 2 and 3. Every number appears once in a row, so the last cell must be the one that is missing: 1. The same works for columns, and for a cell whose row and column together already use every other number.
A sign cuts off both ends
The point of the sign faces the left cell, so the left cell is smaller. On a 4×4 puzzle it can be 1, 2 or 3, but never 4, because something has to be bigger than it. The right cell can be 2, 3 or 4, but never 1. That is one sign, and it already takes away two numbers.
Follow a chain of signs
Here three signs run through a whole row of four: the first cell is smaller than the second, which is smaller than the third, which is smaller than the fourth. Four different numbers that climb all the way can only be 1, 2, 3, 4. Shorter chains help too: in a chain of three cells on a 4×4 puzzle, the smallest is 1 or 2 and the largest is 3 or 4.
Only one place for a number
Where can the 4 go in the top row? Not in the first two cells: each is smaller than its right neighbour, so neither can be the largest. Not in the last cell either, because its column already has a 4. Only the third cell is left, so the 4 goes there. Asking where a number can go is often quicker than asking what a cell can hold.
Pairs
Two cells in this row can only hold 1 or 2. Between them they will use up both, one each. So no other cell of the row can be 1 or 2, and the other two cells are left with 3 and 4. Groups of three work the same way: three cells that share three numbers take those numbers away from the rest of the row or column.
Numbers with only two homes
The opposite of a pair: if two numbers can only go into the same two cells of a row, those two cells hold those numbers and nothing else. Cross every other note out of them. This is the idea the hard boards need, after singles, signs and pairs have run out.
Common mistakes
- Reading a sign the wrong way round. The point always faces the smaller number. 3 > 1 is right, 1 > 3 is not.
- Forgetting the signs between rows. The signs between cells above each other count just as much as the ones in a row.
- Checking only the row. A number must also be new in its column.
- Old notes. When you place a number, take it out of the notes in its row and column, or the notes will mislead you.
Questions and answers
Ready to play?
Read the signs, follow the chains, look for the one place a number can go, and use pairs when you are stuck. That is how every Futoshiki is solved.