The Rules of Futoshiki

Three rules, and you will have absorbed two of them before the end of your first grid. Here is each one shown on a board rather than described.

Rules one and two: every number once along every line

In an n by n grid the numbers used are 1 to n and nothing else, and each of them appears exactly once in every row and exactly once in every column. A 6×6 uses 1 to 6; a 9×9 uses 1 to 9. There is no 0 and there is never a number larger than the side of the grid.

The pair of rules together is what mathematicians call a Latin square, and on its own it is not a puzzle: any grid can be finished in an enormous number of ways, and you could produce one by hand in a minute. The two rules are the frame rather than the content, which is worth knowing because it tells you where to look when you are stuck — never at the frame.

They do matter in one direction that beginners underuse. A row that is missing a single number is finished, and so is a column, and a value crossed out of five cells of a row leaves it with nowhere to go but the sixth. Half the moves in an easy grid are that sentence applied over and over, and people who read the signs beautifully and forget to sweep the rows are the ones who stall.

Illegal — the last two rows both put a 1 in the third column
1	2	3	4
2	1	4	3
3	4	1	2
4	3	1	2

Legal — every number once along every row and every column
1	2	3	4
2	1	4	3
3	4	1	2
4	3	2	1

Rule three: the sign points at the smaller number

Between some pairs of neighbouring cells — side by side or one above the other — a sign is printed. The wide end opens towards the larger number and the point is at the smaller one, exactly as the symbol does in ordinary arithmetic. Vertically the same symbol is drawn as a wedge, pointing up when the upper cell is the smaller of the two.

The sign relates those two cells and no others. A cell that is greater than its right-hand neighbour is not thereby greater than anything else in the row, and it certainly is not the largest number in the row. Reading a single sign as a statement about a whole line is the most expensive mistake available in this puzzle, because everything it then leads you to is wrong.

Where a sign is absent, nothing at all is being claimed. Two neighbouring cells with no sign between them are simply two cells whose order the puzzle has not told you — one is larger and you have to find out which from somewhere else. An empty gap is not a hint that the numbers are close together and it is not a hint that they are far apart.

Illegal — every row and column is fine, and both signs point the wrong way
4<∧	3	2	1
3	4	1	2
2	1	4	3
1	2	3	4

Legal — the corner sits under both its neighbours, as the signs demand
1<∧	2	3	4
2	1	4	3
3	4	1	2
4	3	2	1

What a grid with no numbers in it is telling you

A futoshiki that arrives completely empty looks like a trick the first time you meet one. It is not: it is the ordinary case at these sizes, and it means the signs alone are enough. Every grid on this site is checked before it is published to have exactly one finished arrangement satisfying all three rules, whether it carries five given numbers or none.

The way in is always the same. Look for the longest run of signs all pointing the same way. A run of k cells that ascends has to start at 1 or above and finish at k or above, and if it spans the whole line then the line is completely settled by the signs and nothing else — the smallest goes at the small end and they climb from there.

From that first foothold the ordinary sweeping takes over, and it is worth being clear that the signs never stop being useful. They are not an opening gambit you discard. Late in a grid, when a cell is down to two candidates, a single sign against a neighbour that is down to three is very often the thing that settles it.

Six signs, no numbers, and one possible answer
		>	∨
<		<
∨	<

The only arrangement those six signs allow
2	1	4>	3∨
3<	4	1<	2
4∨	2<	3	1
1	3	2	4

Five things beginners think the rules say

They do not say that a cell with a sign on both sides is somewhere in the middle of the range. A cell can be greater than one neighbour and less than another and still be a 2, provided the neighbour below it is a 1.

They do not say every pair of neighbours has a sign. Most pairs on most grids have none, and a grid dense with signs is usually an easier grid rather than a harder one, because each sign is another thing you are being told.

They say nothing about diagonals. Two cells touching at a corner are unrelated, however tempting the pattern between them looks.

They do not say the given numbers are placed to be helpful. The generator removes every given it can and keeps only the ones the reasoning genuinely needs, so a grid with four numbers in it is not a grid with four hints in it — it is a grid where four could not be removed.

And they do not permit guessing. Every grid published here was checked to be finishable by argument alone, so a grid that will not move is one where something has been missed rather than one that wants a coin flipped. That is a promise about the generator, and it is why the generator throws so much away.

Common questions

Which way does a futoshiki sign point?

The point is at the smaller number and the open side faces the larger one, the same way the symbol works in arithmetic. Drawn vertically it becomes a wedge whose point is still at the smaller cell.

What does it mean when there is no sign between two cells?

Nothing at all is being said about that pair. One of them is larger and the puzzle has simply not told you which, so the order has to come from somewhere else on the grid.

Can the same number appear twice in a futoshiki grid?

Yes — every number appears once per row and once per column, so in a 6×6 each of the six values appears six times in total. What it may never do is appear twice along the same line.