Futoshiki Puzzle Maker
Printable Futoshiki sheets with a checked single answer, four sizes, and the count of how much the signs are really worth.
How to solve it
Fill the grid so every row and every column contains each number exactly once, and every arrow between two squares points from the smaller number to the larger. The signs are the quickest way in: a square on the big end of a chain of three arrows cannot be small, and in a 5 × 5 a cell with two arrows pointing into it has to be a 4 or a 5. Work the extremes first, then let the row-and-column rule finish the job.
⭐Even ALL the signs are not enough
Suppose you drew every sign a board can hold — an arrow between every pair of neighbouring squares, telling you the complete relative order of the entire grid. That is the most information signs can possibly carry, and it feels like it must determine the answer. It does not, and the point where it stops working is early.
These are counts rather than estimates — every Latin square of each size was enumerated and grouped by its pattern of signs:
| Grid | Latin squares | Distinct sign patterns | Settled by signs alone | Biggest ambiguous group |
|---|---|---|---|---|
| 3 × 3 | 12 | 12 | 12 (100.00%) | none — all distinct |
| 4 × 4 | 576 | 418 | 350 (60.76%) | 17 squares |
| 5 × 5 | 161,280 | 115,262 | 82,148 (50.94%) | 16 squares |
A 3 × 3 is settled every time. A 4 × 4 already fails 39% of the time, and a 5 × 5 comes out settled on only 50.9% of boards — barely better than a coin, with every possible arrow drawn. The reason is in the third column: there are simply fewer distinct patterns of signs than there are squares, so some squares have to share one.
And the sharing is not a near-miss between two lookalikes. On a 4 × 4 there is one pattern of signs shared by 17 different Latin squares — seventeen boards on which every neighbouring pair compares exactly the same way, and no arrow anywhere that could tell them apart. That is why a Futoshiki always prints some numbers as well: the signs narrow the field, and the numbers pick from what is left.
The share also falls steadily as the grid grows — 100%, then 61%, then 50.9% — so this gets worse with size rather than better, which is the opposite of the intuition that a bigger grid has more arrows to work with. It does have more arrows; it has disproportionately more squares.
Two ways of asking, one answer
There are two quite different ways to ask whether the signs settle a board. You can enumerate every Latin square, group them by their pattern of signs, and call a square settled when nothing else shares its pattern. Or you can hand the solver a board with no numbers at all and every sign drawn, and ask how many ways it can be completed. The first is bookkeeping; the second is a search. They share no code.
Run across the whole of 4 × 4 — all 576 squares — they agree on every single one. That is the check worth having, because the table above is only as trustworthy as the thing that produced it, and the enumerator also reproduces the published count of Latin squares exactly at all three sizes.
⚠️Worth saying plainly: an earlier version of this page was going to claim that a revealed number is worth "exactly three" inequality signs, measured from twelve random 5 × 5 boards. A second implementation measured 1.29, and checking other sizes showed the figure wandering rather than holding. Two implementations disagreeing by more than double means neither number is publishable, so that claim was dropped and the enumeration above took its place — it counts every case instead of sampling, so it cannot be a fluke of which boards were drawn.
How to use
- Pick a grid size, a difficulty and how many puzzles you want on the sheet.
- Press New sheet — every board is checked for a single answer before it appears.
- Print it; the controls and the notes are left off the page.
- Toggle Show solutions when you want the completed grids revealed.
Frequently asked questions
What is Futoshiki?
A Latin square puzzle with inequality signs. Fill the grid so every row and column holds each number exactly once, and every arrow between two squares points from the smaller number to the larger. The name is Japanese for inequality. A few numbers are usually printed as well, because the signs alone are rarely enough.
How do you solve a Futoshiki?
Start with the extremes. A square at the big end of a chain of arrows cannot be small, and in a 5 by 5 a cell with two arrows pointing into it has to be a 4 or a 5. Fix those, then let the row-and-column rule cascade. Working the chains first and the grid second is far faster than the other way round.
Why do Futoshiki puzzles also print numbers?
Because the signs cannot do it alone above a 3 by 3, and this is measurable rather than a matter of taste. Enumerating every Latin square and drawing every possible arrow, a 3 by 3 is settled every time, a 4 by 4 only 60.76 percent of the time, and a 5 by 5 only 50.94 percent. The numbers pick from what the arrows leave behind.
Can two different grids have the same arrows?
Yes, and more than two. On a 4 by 4 there is one pattern of signs shared by seventeen different Latin squares — seventeen grids where every neighbouring pair compares the same way, with no arrow anywhere that could tell them apart. That is why a complete set of arrows is not a complete description of the answer.
Does a bigger grid make the signs more useful?
The opposite, which is the surprising part. The share of grids settled by their signs alone falls from 100 percent at 3 by 3 to about 61 percent at 4 by 4 and 51 percent at 5 by 5. A bigger grid does have more arrows, but it has disproportionately more possible grids, so the arrows fall further behind.
Does every puzzle here have one answer?
Yes, and each board is checked before it appears. The solver counts answers and stops at two, since two is all a uniqueness check needs to find, and only single-answer boards are printed. It is cross-checked against a completely separate method that simply enumerates every Latin square and filters the list.
What do the difficulty settings change?
How far the generator strips the board back. Gentle leaves plenty of both numbers and arrows, Hard takes away everything it can while keeping one answer. Both numbers and signs are removed in a single shuffled pass, which is what makes the result look like a published Futoshiki rather than all of one and none of the other.
What size should I choose?
4 by 4 is a couple of minutes and a good way to learn the chain logic. 5 by 5 is the classic size and the best starting point. 6 by 6 and 7 by 7 are proper sittings where the arrow chains get long enough to carry most of the solve.
Can I print these?
Yes, that is the point of the tool. The Print button hides the controls and the explanation and leaves the sheet. You can put one, two or four puzzles on a page, and no puzzle is split across a page break. Turn the solutions on first if you want them printed underneath.
Is Futoshiki related to Sudoku?
Both are Latin square puzzles — fill a grid so no number repeats in a row or column. Sudoku adds boxes; Futoshiki adds inequalities instead, and drops the boxes. The inequality is a much weaker constraint than a box, which is exactly why Futoshiki grids are usually small and why they still need some numbers given.
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