To solve Futoshiki, let the signs narrow each square, then fill in what’s certain. A sign’s larger side is never 1, its smaller side never the top number, and a chain of signs pointing the same way narrows every square along it. Once the signs have had their say, look for a square with one number left and a number with one place left in a row or column. On bigger grids, pairs and what-if finish the job. Six techniques cover every puzzle in the game.
New to Futoshiki? Read the Futoshiki rules first. This page assumes you know the two rules: each number once in every row and column, and every sign holds.
1. One number left
Look at what a square’s row and column already hold. When only one number is left that it could be, that’s its number.
2. One place left
Turn it round: pick a number a row or column still needs, and find where it can go. When there’s only one place left for it, it goes there.
3. Reading the signs
Every sign rules numbers out on both sides. The smaller side can’t be the top number, and the larger side can’t be 1.
It goes further than the ends. The smaller side must be less than the biggest number the larger side could still be, and the larger side more than the smallest the smaller side could be. Follow a chain of signs pointing the same way, and every square along it narrows.
A chain of three signs pointing the same way across a row of four settles all four squares: 1 < 2 < 3 < 4. The signs are where most puzzles start, and they keep paying off: whenever a square near a sign changes, look at the sign again.
4. Pairs
When two squares in a row or column can only be the same two numbers, they take those two numbers between them. No other square in that line can have either.
5. Hidden pairs
The same idea, turned round. When two numbers can only go in the same two squares of a line, those squares take them, so they can’t be anything else.
6. What if
When nothing else works, pick a square with few options and suppose it’s one of them. Follow what that forces through the signs and the lines. If a square runs out of numbers, or a sign can’t hold, the supposition was wrong: cross that number out.
How often each technique is needed
We solved every one of the game’s 150 levels by following the hints, which always use the simplest technique that works. This is how many times each technique was used, on average, per level:
| Technique | Easy 5×5 | Medium 6×6 | Hard 7×7 |
|---|---|---|---|
| One number left | 12.3 | 17.4 | 29.8 |
| One place left | 1.8 | 4.0 | 8.6 |
| Reading the signs | 10.4 | 13.5 | 13.4 |
| Pairs | 0 | 0.5 | 1.3 |
| Hidden pairs | 0 | 0.2 | 0.9 |
| What if | 0 | 0 | 10.7 |
The signs are needed on every level; one number left does the most work. Pairs and hidden pairs never come up on Easy, and appear on 15 and 10 of the 50 Medium levels. What if is only for Hard, where every level needs it, about 11 times each.
A 6×6 with no clues, solved in three stages
This is Medium level 43 from the game. It starts with no numbers at all, only 17 signs.
Stage 1: the 1s, then the signs everywhere
Column 4 needs a 1, but five of its six squares are on the larger side of a sign, and the larger side of a sign is never 1. So column 4’s 1 goes in the one square left, row 2. Column 1 works the same way: its 1 goes in row 6. Then the signs leave column 4 only one place for its 2, in row 3, just below the new 1.
That’s it for quick moves. Now work through every sign and write in what each square could still be. That rules out 32 numbers across the grid.
Stage 2: a hidden pair, then a pair
A hidden pair. In row 3, the 5 and the 6 can only go in columns 2 and 6. So those two squares take the 5 and the 6, and lose their other notes.
A pair. Now column 2 has two squares, in rows 3 and 4, that can only be 5 or 6. They take the 5 and the 6 between them, so no other square in column 2 can be either.
Stage 3: the rest
That breaks it open. Row 2’s 6 now has only one place left, in column 6, and from there it’s single after single. In all, the rest takes 24 squares with one number left, 4 numbers with one place left, and 5 more moves from the signs.
When you get stuck
- Look at every sign again. A sign next to a square that’s just changed often has something new to say.
- Find the 1s and the top numbers. Squares on the larger side of a sign can’t be 1; squares on the smaller side can’t be the top number.
- Fill in notes. Pairs and hidden pairs are easy to see in notes and almost impossible without them.
- Try what if on a square with two options. Follow it a few moves through the signs, and see whether anything runs out of room.
- Don’t guess. Every puzzle here can be finished by logic.