Printable Futoshiki guide
An empty grid with the right signs on it has exactly one answer.
That is the fact the whole puzzle type rests on, and it is why most of the sheets this page prints have one number on them or none at all. Taking numbers away is not what makes a Futoshiki hard, though. What makes it hard is which deduction you have to find once they are gone, and that is the thing this builder controls.
What you can make on this page
Packs of one to six Futoshiki puzzles from 4x4 to 7x7, printed one or two to a page on A4 or US Letter, with answer keys two or four to a page. Name and date fields, a rules page, large print, and an ink-saving grid are optional. Nothing is fetched from a server: the puzzles are generated, solved, and checked in your own browser, so pressing New puzzles gives you a set nobody has seen.
The two controls that change the sheet are grid size and difficulty, and they are genuinely separate. A 7x7 is not harder than a 5x5, it is longer. An Expert 5x5 asks more of you than an Easy 7x7 and takes less of your afternoon.
Difficulty is the deduction you have to find, not the clues we took away
A clue-count definition of difficulty falls apart on this puzzle immediately, because two 5x5 grids with one number on each can be twenty seconds apart or twenty minutes apart depending on where the signs sit. So the levels here are named after a solver's toolkit instead. Every puzzle is put through a solver that owns five deductions and nothing else, ordered by how much work they are to spot, and a puzzle's level is the hardest one it forces.
| Level | Hardest deduction it forces | What that means at the table | Numbers given | Signs printed | Deduction rounds |
|---|---|---|---|---|---|
| Easy | Signs beside a filled cell | A sign next to a cell you have already filled: the neighbour is above or below that number. | 4.2 of 25 | 6.0 of 40 | 12 |
| Medium | Runs of signs | A run of signs pointing the same way. Three cells rising means the first is at most the size minus two. | 0.2 of 25 | 10.0 of 40 | 14 |
| Hard | Pairs and triples | Two cells in a row that can only hold the same two numbers, which locks those numbers out of the rest of the row. | 1.2 of 25 | 8.5 of 40 | 16 |
| Expert | Testing a candidate | Pencil a number in, follow it until it breaks a rule, and rub it out knowing it cannot be right. | 1.0 of 25 | 9.0 of 40 | 19 |
Measured from 24 generated 5x5 sheets at build time, not typed in. Every one of them was also proved unique by an exhaustive search that shares none of the solver’s reasoning.
Notice what happens between Easy and Medium. The number of clues barely moves, and the sheet looks the same from across the room. What changes is that the Easy grid can be picked apart by looking at one sign at a time, and the Medium grid cannot: somewhere on it there is a run of signs that has to be read as a run before anything can be written down.
The same size, three levels apart
Why the sheets are nearly empty
The builder removes numbers before it removes signs, and it keeps removing until one more would break the solve. Both halves of that matter. Removing until it breaks is what guarantees every clue left on the page is load-bearing. Removing numbers first is a deliberate choice about what kind of puzzle comes out: strip the signs first instead and you get a Latin square with a few arrows attached, which solves like a weak Sudoku.
The result is a sheet that looks alarming and is not. On the 5x5 sheets measured above, Medium, Hard and Expert average between 0.2 and 1.2 given numbers, and plenty of them have none at all. Easy keeps more on purpose, because somebody meeting the puzzle for the first time needs a place to put their pencil.
Two rules that keep the sheet honest
Every puzzle can be finished without guessing. This is the rule that matters most on paper and the one most printable Futoshiki packs quietly skip. A generator that only checks its puzzle has one answer can hand you a grid where the single answer is reachable only by trying something and backtracking, which is tolerable on a screen with an undo button and miserable in pen at a kitchen table. Here the puzzle is solved by a program restricted to deductions a person can follow, and if that program gets stuck the puzzle is thrown away before you see it. Uniqueness then comes free: no step ever discards a number that could have been right, so a solve that fills the grid has also proved nothing else fits.
Every clue printed is needed. Put any one of them back on the sheet after the fact and the puzzle would still work; take any one away and it stops being solvable. That is checked the blunt way, by removing each number and each sign in turn and confirming the solver fails. A spare sign is not harmless: it is a solver spending two minutes looking for the deduction it must be there for.
How to solve one, and where the signs actually help
- Find the longest run of signs pointing the same way. A run of 3 cells has to hold 3 different numbers in increasing order, so its smallest end is at most 3 and its largest end is at least 3.
- Look for a sign pointing into a wall of one. A cell on the small end of a < can never be the largest number, and a cell on the large end can never be 1. On a 5x5 that is two numbers gone from each of them before you start.
- Pencil the survivors into the corner of each cell. This is the step people skip and then wonder why the puzzle feels like guesswork.
- Scan each row and column for a number with only one home left. This is the Sudoku half of the puzzle and it does most of the finishing.
- When that dries up, look for two cells in one line holding the same two pencil marks. Those two numbers belong to those two cells, whichever way round, and they are gone from the rest of the line.
A 4x4 is too small to be hard, and the builder says so
Offering four difficulty levels at four grid sizes suggests sixteen genuinely different sheets, and that is not what comes out. The harder deductions need room. Pairs and triples only bite when a line is long enough for two cells to be tied together while the rest stay open, and on a 4x4 there is usually a cheaper route to the same answer, so a puzzle asked to be Hard comes back solvable by the Medium toolkit instead.
| Grid | Cells | Sign positions | Numbers given at Hard | Signs printed at Hard | Hard sheets that really need pairs |
|---|---|---|---|---|---|
| 4 x 4 | 16 | 24 | 0.0 | 5.3 | 75% |
| 5 x 5 | 25 | 40 | 0.0 | 10.3 | 100% |
| 6 x 6 | 36 | 60 | 0.8 | 18.3 | 100% |
| 7 x 7 | 49 | 84 | 2.0 | 23.0 | 100% |
Four generated sheets per size at build time. The last column is a small sample and it moves around; a longer run of a hundred and forty-four sheets put 4x4 Hard at about a third, 5x5 at about nine in ten, and 6x6 and 7x7 at every sheet.
The useful response to that is not to hide the combination but to report it, so the builder counts how many sheets in your pack came out easier than the level you asked for and tells you underneath the preview. Expert is unaffected, because following a candidate to a contradiction is available on any size of grid.
The three-line Latin square we threw away
Underneath every Futoshiki is a Latin square, and there is a famous three-line way to make one: fill the grid with (row + column) mod size, then shuffle the rows, the columns and the numbers. It is quick, it always works, and an earlier version of this puzzle on another site used it. It also produces a tiny, lopsided corner of the possible grids, and the giveaway is an intercalate: a 2x2 pattern of two numbers arrangeda b over b a, which is the smallest ambiguity a Latin square can contain and precisely the local knot the signs have to untie.
| 5x5 grids built by | Mean intercalates | Grids with at least one | What that does to the puzzle |
|---|---|---|---|
| The shift construction | 0.00 | 0% | The knot never occurs at odd sizes, so a whole family of deductions never comes up. |
| Backtracking with a shuffled order | 3.37 | 84% | The knot turns up on most grids, and the signs above it have to resolve it. |
600 grids of each kind, counted at build time. The zero in the first row is not a rounding artefact: at odd sizes the shift construction cannot produce an intercalate at all, which is a two-line arithmetic result and a good example of a generator quietly deciding what your puzzles feel like.
This page builds its grids by filling cells one at a time with the candidate order shuffled at every step. That is not a uniformly random Latin square either, and it would be dishonest to claim it is, but it reaches the whole space rather than one family of it.
Best printable Futoshiki settings
| Use case | Recommended settings | Why it works |
|---|---|---|
| First Futoshiki, ages 8 to 10 | 4x4, Easy, 2 puzzles, rules page on | Four numbers to place and a sheet that never needs more than one sign read at a time. The rules page is worth printing once and keeping. |
| Maths lesson on inequality signs | 5x5, Easy or Medium, 4 puzzles, 2 per page, name and date on | The greater-than sign stops being notation and starts being the thing that solves the problem, which is the lesson. |
| Care home or large-print activity | 4x4 or 5x5, Easy, large print, 1 per page | Large print widens the cells and the sign lanes together, so the chevrons stay readable rather than just the numbers getting bigger. |
| An adult puzzle for the evening | 6x6, Hard, 2 puzzles, answers on | Big enough that pairs and triples genuinely come up, short enough to finish in a sitting. |
| Puzzle-club sheet | 7x7, Expert, 1 per page, ink saving on | One grid to a page, no numbers to speak of, and at least one place where a candidate has to be followed until it breaks. |
| Substitute-teacher pack | 5x5, Medium, 6 puzzles, 2 per page, rules page on | Three sheets of work with the method on the front, and every puzzle in the pack is a different grid. |
Printing choices that matter
- Two per page is the default because a Futoshiki grid is small. One per page is worth it at 7x7, or with large print, where the sign lanes need the room more than the cells do.
- Large print widens the lane between cells as well as the cells themselves. A bigger number with a cramped chevron beside it is the wrong half made bigger.
- Ink saving thins the cell outlines but leaves the signs at full weight, because a faint chevron is the one mark on this sheet you cannot guess from context.
- Answer keys default to two per page and grey out the numbers that were already printed, so you can see what you worked out rather than just what the grid holds.
- The rules page is one sheet and explains the pointing convention. Print it once for a class rather than once per pack.
Where this page is weaker than it looks
- Difficulty is a guarantee about the hardest deduction required, not about how long a sheet takes. An Expert 5x5 that needs one contradiction test can fall faster than a Medium 7x7 that needs thirty patient rounds.
- The 4x4 Hard problem above has no fix, only a warning. If you need a genuinely hard small sheet, Expert at 4x4 works; Hard often does not.
- A 7x7 Expert pack takes a second or two to build, because the contradiction test runs many times over while clues are being stripped. Everything else is close to instant.
- The solver knows five deductions. A human who spots a sixth will find some sheets easier than the label says, and there is no honest way for a generator to price that in.
- There is no symmetry control. Published Futoshiki grids sometimes place their signs symmetrically for looks; here the signs land wherever they are needed, which is better logic and a less tidy sheet.
How this page is made and reviewed
Print Puzzles is part of the Hamilton Digital Media Limited puzzle network. This page is written and reviewed by Brian Hamilton and Karan Hamilton. The engine behind it was checked with a temporary harness that generated sheets across every size and level and confirmed, for each one, that the grid is a Latin square, that every sign agrees with the answer, that the puzzle can be solved by deduction alone, that an independent exhaustive search finds exactly one answer, and that no clue on the sheet could be removed. Read more about how we make puzzle packs, or tell us about a sheet that does not work through the contact page.
Printable Futoshiki FAQ
What is a Futoshiki?
A grid where every row and every column holds each number once, exactly like a Sudoku without the boxes, plus greater-than and less-than signs between some pairs of neighbouring cells. The point of the sign is the narrow end: it always points at the smaller of the two cells. Futoshiki is Japanese for "not equal", and the signs are the whole puzzle, because most sheets start with barely any numbers on them at all.
Why are there almost no numbers filled in?
Because a Futoshiki does not need them. The builder strips numbers off first and only stops when the puzzle would no longer be solvable, which on a 5x5 usually leaves one or none. That is normal for this puzzle rather than a fault: an empty grid with the right signs on it has exactly one answer, and the signs are what you are meant to be reading. Easy sheets keep more numbers on purpose, because a beginner needs somewhere obvious to start.
Can I solve every sheet without guessing?
Yes, and that is checked rather than hoped for. Every puzzle is put through a solver that only makes deductions a person could follow, and if the solver gets stuck the puzzle is thrown away before you ever see it. Uniqueness comes free with that: because no step ever discards a number that could have been right, a solve that fills the grid has also proved no other answer exists.
What do the four difficulty levels actually change?
Which deduction you have to find, not how many clues were taken away. Easy needs only row and column scanning plus signs read against a cell you have already filled. Medium needs a run of signs read as a run. Hard needs pairs and triples inside a line. Expert needs at least one candidate followed until it breaks a rule. The number of clues follows from that rather than setting it.
Which grid size should I print?
5x5 is the standard and the one to start with. 4x4 suits younger children and first attempts, but be aware it is too small to be genuinely hard, and the builder says so when it happens. 6x6 is the sweet spot for an adult who wants a proper sit-down puzzle, and 7x7 is a long one: expect a full sheet to itself and fifteen minutes or more.
Do the printable Futoshiki puzzles come with answers?
Yes. Leave Answer keys selected and the PDF adds key pages after the worksheets, each grid filled in with the numbers that were already printed shown in grey so you can see at a glance what you worked out for yourself. The keys print two to a page to keep the pack short.
Is Futoshiki good for a classroom?
It is one of the better logic puzzles for maths lessons because the whole puzzle is inequality notation, which is a topic children meet on paper and rarely get to use for anything. A 4x4 or 5x5 sheet with name and date fields makes a starter that teaches the greater-than sign by needing it, and the no-guess guarantee means a pupil who is stuck is missing a deduction rather than out of luck.