Numberlink Puzzles
Join each pair of matching numbers with a path. Paths never share a square, and every square in the grid ends up used. Tap between two squares to link them. Sixty grids, each with one answer and none needing a guess.
Also published as nanbarinku and arukone.
Related Practice
What Is A Numberlink Puzzle?
Numberlink hands you a grid with pairs of matching numbers scattered through it, and asks you to join each pair with a path. Paths run between neighbouring squares — up, down, left or right, never diagonally — and no two of them may share a square. The fourth rule is the one that turns it into a real puzzle: every square in the grid has to be used by the time you finish. Nothing is left over.
Why "Use Every Square" Is The Important Rule
Without it, most numberlink grids would have dozens of answers — you could route a path almost any way round and leave the spare squares empty. Insisting that nothing is left over removes nearly all of those alternatives at a stroke, because every leftover square would need somewhere to belong. It is also what makes the puzzle solvable by reasoning rather than by trying routes: once you know that a blank square must be used, you know exactly how many links it takes.
The Whole Method Is Counting Links
| Square | Links it takes | Why | Useful case |
|---|---|---|---|
| Numbered | Exactly 1 | A path ends there | A number in a corner has two directions at most |
| Blank | Exactly 2 | A path passes through it, and it cannot be left out | A blank corner has only two neighbours, so it uses both |
That is the entire technique. Blank corners are forced immediately. A numbered square with all but one neighbour blocked is forced. And each link you draw changes the count on two more squares, which usually forces something else.
Two joins you can always refuse. If one piece of path already runs to a 3 and another already runs to a 5, they can never be joined — the result would have mismatched ends. And joining two squares that are already connected would close a ring, which has no ends at all and so can never be one of the paths. Both refusals settle more of a numberlink than any positive deduction does.
Why These Grids Have Exactly One Answer
There is a specific reason, and it is built into how the puzzles are made rather than checked afterwards. In the answer to every grid here, no path ever touches itself: two squares of the same path are never neighbours unless they are consecutive along it. That matters because a path that doubles back beside itself can almost always be re-routed through those same squares to give a second, different answer. Refusing it while the paths are being laid out removes most ambiguity before anything is checked — and every grid is then verified twice anyway, once by a solver restricted to ordinary deduction and once by a program that walks the paths square by square.
More Pairs Is Easier, Not Harder
This runs against intuition, so it is worth saying plainly: a grid with many pairs is gentler than the same grid with few. Every numbered square is a path end, and a path end is a square whose link count is pinned to one — so more numbers means more certainty on the board before you have drawn anything. Fewer pairs means long winding paths and far less pinned down. That is the difficulty axis here, and it shows up in the generator too: on a seven-by-seven, grids with up to sixteen pairs are found in about twenty milliseconds, while the same grid limited to twelve takes nearly ten times as long, because far more candidate layouts fail the no-guessing check.
Numberlink Puzzles Frequently Asked Questions
What is a numberlink puzzle?
A grid with pairs of matching numbers. Join each pair with a path between neighbouring squares. No two paths share a square, and every square must be used when you finish.
Why does every square have to be used?
Because without it the puzzles would have dozens of answers. Insisting nothing is left over kills almost all the alternative routes. It is the standard rule for this type, and it is what makes a single answer possible.
Can paths go diagonally?
No — only up, down, left and right, one square at a time. Paths may bend as often as they like but never cut corners.
Can a path pass through a numbered square?
Never — not even one belonging to another pair. Every numbered square is a path end, so exactly one link touches it. A number in a corner often has only one possible direction.
How do you start a numberlink?
Count links. A numbered square takes exactly one; a blank square takes exactly two, because a path passes through it. A blank corner has only two neighbours, so it must use both.
Why can I not join two different numbers?
Because a path joins the two squares with the same number. Joining a piece that reaches a 3 to one that reaches a 5 would leave a path with mismatched ends. Ruling those out is one of the most productive moves.
What is the trick to making these have one answer?
The paths never touch themselves. A path that doubles back beside itself can usually be re-routed through those squares, which is where a second answer hides. Every puzzle here is built from paths that never do it.
Does every puzzle have one answer and no guessing?
Yes to both. A solver restricted to counting links and refusing illegal joins must finish it, so every step is forced and the answer is the only one — and a separate path-walking program confirms it independently.
Is numberlink the same as Flow Free?
Essentially yes — Flow Free is numberlink on a phone, with coloured dots instead of numbered squares. Both give you pairs of endpoints to join with paths that never cross, and in both a finished board uses every square. Numberlink is the older name; Nikoli published it in Japan as Nanbarinku, and as Arukone when the clues are pairs of letters rather than numbers.