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Puzzle Quarry

Futoshiki

Choose a size and a level and a fresh grid is built in your browser, checked, and put on screen. No limit, no account, nothing uploaded.

Medium Time 0:00 Filled 0/36

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A medium 6×6 grid with 0 numbers given and 20 signs. Pick a cell, then a number — or press the pencil to write candidates.

Check this before you rely on it. This futoshiki generator is provided free and without warranty, and its results are not professional advice.

Three rules, and the third one is why you are here

An n by n grid is filled with the numbers 1 to n. Every row holds each number exactly once and so does every column, which by itself would be a Latin square and no puzzle at all. What makes it a puzzle is the third rule: some pairs of neighbouring cells have an inequality sign printed between them, and the number on the open side of the sign has to be the larger of the two.

That is the entire statement. There are no boxes, no regions, no diagonals and no arithmetic. A 5×5 grid holds twenty-five numbers drawn from five values, which means every value appears five times, and a beginner who expects the values to be scarce spends the first grid confused about why 3 keeps turning up.

Most grids here arrive with no numbers written in at all. That surprises people who have come from other number puzzles, where the given digits are the puzzle and the signs would be decoration. Here it is the other way round: the signs carry the grid, and at the smaller sizes the generator routinely finds that every single given number can be taken away and the puzzle still has exactly one answer.

Why this board has pencil marks when the others do not

Press a cell and then a number and it goes in. Press the pencil and the same numbers go in small, several to a cell, as possibilities rather than answers. The other two puzzles on this site have no such thing and do not need one, because in both of them a cell has two or three states and you can simply cycle through them.

Here a cell has up to nine, and — this is the part that matters — almost every deduction you make removes one possibility rather than settling a cell. You will work out that a cell cannot be a 4 a dozen times before you ever work out what it is. Somewhere to write that down is not a convenience feature, it is the notation the puzzle is played in, and a futoshiki board without it makes you rediscover the same elimination five times.

The keyboard does everything the pad does. Arrow keys move, a digit writes, shift and a digit pencils it instead, backspace clears the cell, and the P key puts the pencil down and picks it up again. The pad exists because a phone has no keyboard, not because it is the intended way in.

What the four levels actually measure

Not from the size. A 9×9 with plenty of signs is a long puzzle rather than a hard one — it asks the same three questions forty times over. The four levels name the reasoning a grid forces you into instead, and they are measured rather than estimated.

Easy is finished by reading the signs once and then crossing numbers out of rows and columns. Medium needs you to turn a question round at least once and ask where a particular value could live rather than what a particular cell could hold. Hard needs a sign read against what its neighbour can still be, rather than against the range 1 to n. Expert needs a set of cells weighed together — two cells that own two values between them, or two values that own two cells.

Every published grid is solved by a program that always applies the cheapest technique that fits, and the hardest one it was ever forced onto is the level printed on the grid. A grid built for one level that turns out to need another ships under the level it measured. That occasionally means asking for expert at 4×4 and being handed a hard grid, because sixteen cells do not always leave room for a set argument — an honest label beats a flattering one.

Common questions

Is futoshiki a kind of sudoku?

No. There are no boxes and no regions — only rows, columns and the inequality signs. The signs are what the puzzle is about, and no sudoku technique that depends on a nine-cell box applies here at all.

Can a futoshiki have no numbers filled in at the start?

Yes, and most of the small grids here do. The signs alone are frequently enough to pin exactly one answer, which is checked before any grid is shown.

What does the sign between two cells actually mean?

It compares only those two neighbouring cells, and it says which of the two holds the larger number. It says nothing about any other cell in the row or the column.

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