Medium 5x5
Tap a cell, then a number. The × key clears a cell. Turn on Notes to pencil in the numbers a cell could still be. With a keyboard, focus the board, move with the arrow keys and type a number. Backspace clears and N switches notes on and off.
A repeated number or a broken sign turns red straight away. Hint marks the cells it's talking about and says why, and Apply fills the cell in.
The rules
- On a 5x5 board, every row and every column holds the numbers 1 to 5, each one once. A 4x4 board uses 1 to 4, a 6x6 board 1 to 6.
- A sign between two cells must be true. The narrow end points at the smaller number, so 2 < 4 and 5 > 3 are fine.
- Signs between a cell and the one below it work the same way, turned on their side.
That's a Latin square with a few clues, like sudoku without the boxes. The signs are what make it its own puzzle: one sign tells you a little, but a run of them tells you a lot.
Where the boards come from
The boards are made in your browser by the generator from my Futoshiki app, with the same three levels. It fills a random grid, adds signs that agree with it, then takes numbers away one by one, and after that takes away the signs it doesn't need. Each time, a solver that only uses the techniques below has to finish the board, or the number or sign goes back. A separate search then checks there's only one answer.
Easy 4x4 boards need at least one sign step. Medium 5x5 boards can need chains. Hard 6x6 boards always need at least one pair. Hint uses the same solver, so it shows the simplest step that works from where you are.
The five techniques, simplest first
Rows count from the top and columns from the left. In the pictures, the cells a step leans on have a gold outline, the cell it fills is shaded with the answer shown faintly, and fixed numbers sit on darker cells. Every picture is a real position from a real board, and the site checks each step again before it goes up.
1. Single: the only number left
Look at a cell's row and column together. If they already hold every number but one, that's the number. It's sudoku's oldest trick, and on 4x4 boards it does a lot of the work.
2. Hidden single: the only place left
Turn it around. Pick a number a row still needs and check each empty cell. If the number already sits in the columns of all but one of them, it goes in that one.
3. Sign: what one sign rules out
A cell that's bigger than its neighbour can never be 1, and a cell that's smaller can never be the top number. When the neighbour is filled, the sign does more: bigger than a 3 means 4 or more.
4. Chain: follow the signs
Signs that point the same way add up. If a cell is bigger than a cell that's bigger than a 1, it has to be at least 3. Count the steps down to the smallest filled number, or to an empty end, which counts as at least 1.
Then let the row and column finish the job.
Chains are the heart of the puzzle. On a 5x5 board, a cell three steps above an empty cell is at least 4, and three steps below one is at most 2. Look for long runs of signs first on every new board.
5. Pair: two cells, two numbers
Sometimes two cells in one row can each only hold the same two numbers. Then those two numbers belong to them, and no other cell in the row can use either. That often settles a third cell straight away.
Pairs need notes. Once you're stuck on a hard board, pencil in what each cell in a row could still be, after the signs. Two cells with the same two notes are your pair.
A routine for any board
- Mark the ends of the chains.
A cell on the open side of a sign can't be 1, and a cell at the point can't be the top number. Long chains set real bounds.
- Fill the singles.
Go through the rows and columns with the most numbers in them first.
- Hunt the hidden singles.
For each row, ask where its missing numbers can go. The top and bottom numbers are easiest, because the signs rule them out of so many cells.
- Redo the chains.
Every number you place can tighten a chain that runs through it.
- Notes and pairs last.
Only on the hard boards, and only for the row or column you're stuck on.
Mistakes that cost the most time
Reading the sign backwards. The open side faces the bigger number, the point faces the smaller one. If you're not sure, say it out loud: this cell is bigger than that one.
Forgetting the vertical signs. Signs between rows count just as much as signs in a row, and they're easier to skip on a phone screen.
Guessing on a 6x6. A wrong guess can survive for a long time on a big board. If no step is left, a Hint will show the one you missed.
Questions
What does futoshiki mean?
It's Japanese for inequality, which is what the signs between the cells are about. You'll also see it called unequal or more or less.
Does every board have only one answer?
Yes. The generator only keeps a board the step by step solver can finish, and a full search confirms there's no second answer.
Is it like sudoku?
Close. Each row and column holds every number once, like sudoku, but there are no boxes, and the signs do the job that extra numbers would. If you like sudoku, singles and pairs will feel familiar, and chains are the new part.
Do I ever need to guess?
No. Every board here can be solved with the five techniques above.
Is there a version for my phone?
Yes. My Futoshiki app for Android has 300 boards from 4x4 to 6x6, a daily puzzle, notes, and hints that explain every step. There are no lives, and it works offline.
When a board looks empty, find the longest run of signs and count along it. That's almost always where the first number is hiding.
Futoshiki
300 boards from 4x4 to 6x6, a daily puzzle, notes, and hints that explain every step.