How to solve Futoshiki
Futoshiki is a logic puzzle from Japan whose name means "inequality". It was developed by Tamaki Seto and first published in 2001. In the west it has also appeared as "More or Less" and "Hutosiki".
Start here
The rules
Fill the grid so that every row and every column contains each number once, from 1 up to the size of the grid (1 to 9 in the 9×9 puzzles in our books). There are no boxes.
Some neighbouring squares have an inequality sign between them, and every sign must be true. The larger number is always on the open side of the sign and the smaller number on the pointed side. This holds whether the sign runs across a row (< or >) or up and down a column (^ or v).
The two numbers don't have to be consecutive: 9 > 1 is just as correct as 5 > 4.
Basic Futoshiki strategies
These seven techniques all start from the signs. They solve most easy and medium puzzles, and you'll use them on every hard one too.
A few terms first. A sign sits between two neighbouring cells, and the bigger number is always on its open side. A candidate is a number that could still go in a cell, the small pencil marks you write in it. A chain is a run of cells joined by signs that all point the same way, so each cell is bigger than the one before.
The diagrams use 5×5 grids so each step is easy to see. All our Futoshiki books use 9×9 grids, where every row and column holds 1 to 9. The reasoning is the same: only the largest number changes. Grey numbers are already placed, green marks the cells the step is about, blue marks the cells it compares them with, and red with a cross is a candidate you remove.
Pencil marks pay off quickly in Futoshiki. A sign often can't place a number on its own, but it can strike out two or three candidates, and the row or column finishes the job.
Reading the signs
Smallest and largest values
Seeing it. Look at the signs next to placed numbers. A cell that must be smaller than 2 can only be 1. A cell that must be bigger than the second-largest number can only be the largest.
Using it. Write the number in. In the diagram the green cell in row 2 is smaller than the 2 beside it, so it's 1. In row 4 the green cell is bigger than 4, and a 5×5 only goes up to 5, so it's 5. These are the quickest wins in a puzzle, so check every placed number's signs before anything else.
Reading the signs
Ruling out the extremes
Seeing it. A cell on the open side of any sign is bigger than something, so it can't be 1. A cell on the pointed side of any sign is smaller than something, so it can't hold the largest number.
Using it. Go along a row or column and ask where its 1 can go, then where its largest number can go. In the diagram, four of the five cells in row 3 are each bigger than a neighbour: one is bigger than the cell above it, two are bigger than the cell beside them, and one is bigger than the cell below. The only cell left for the 1 is in column 3.
Reading the signs
Ranges across a sign
Seeing it. Two cells joined by a sign limit each other. The smaller cell must sit below the biggest candidate left in the bigger cell. The bigger cell must sit above the smallest candidate left in the smaller one.
Using it. Trim both cells, then check again, because each cut can trim the other side further. In the diagram A < B. A has 2, 4 and 5, and B has 1, 3 and 4. B can be at most 4, so A must be under 4, which leaves only 2. A is at least 2, so B must be more than 2, and B loses its 1. The row or column often does half the work first: if a cell's row already holds 1 to 4, it must be the largest number, and the cell beside it can be no bigger.
Reading the signs
Bigger than two cells
Seeing it. A cell is bigger than two other cells in the same row or column. Those two can't be equal, so one of them is at least 2.
Using it. The cell must be at least 3: remove 1 and 2 from it. In the diagram the middle cell of row 2 is bigger than the cells on both sides. The same works the other way round: a cell smaller than two cells in its line can't be either of the two largest numbers. Check the two cells really share a line. If one is beside the cell and the other is above it, they could hold the same number, and you only know the cell is at least 2.
Chains
A chain as long as the grid
Seeing it. Follow the signs from cell to cell, always moving to the bigger number. The chain can turn corners between rows and columns. If it has as many cells as the grid is wide, there's no room to skip a number.
Using it. Fill the chain with 1 to the largest number, in order. In the diagram, five cells climb along row 1 and then down column 3. A 5×5 only has five numbers, so the chain is 1, 2, 3, 4, 5 and all five cells are solved at once.
Chains
Limits along a chain
Seeing it. Shorter chains don't give exact numbers, but each step up the chain adds at least one. The third cell from the bottom of a chain is at least 3, and the third from the top is at most the largest number minus 2.
Using it. Count each cell's place from both ends of the chain and cross off the numbers it can't reach. In the diagram, a three-cell chain in a 5×5 leaves the bottom cell 1 to 3, the middle 2 to 4 and the top 3 to 5. Combine these limits with the row and column, and a chain often solves one cell, then the rest follow.
Signs and lines together
Signs with rows and columns
Seeing it. A sign rarely places a number alone, but it removes one or two candidates. The cell's row and column remove others.
Using it. Put the two together. In the diagram the green cell's row already has 1 and 3, and its column has 4. It's on the pointed side of a sign, so it can't be 5. That leaves 2. After every placement, look at the signs around the new number. Each one is a fresh chance for the techniques above.
