Play Numberlink Online
Connect each pair of matching numbers with a continuous path. Paths cannot cross or overlap, and every cell must be filled. A classic flow puzzle — pure logic, no guessing.
Created by Brian Hamilton
Drag from a number to connect pairs — fill every cell
How to play Numberlink
Join each pair of matching numbers with a single path.
- Paths run horizontally and vertically, never diagonally.
- Paths may not cross or share a square with each other.
- Every square must be used — no square is left empty.
- Every puzzle has exactly one solution.
Controls: drag from one number to its partner to draw a path. Drag back along a path to erase it.
Start where there is no choice
Numberlink looks like it should be solved by drawing the obvious short connections first. That is usually the slowest route. The productive opening is to find endpoints that have almost no freedom, and the grid's own boundary is what creates them.
An endpoint in the middle of the board has four ways to leave. One against an edge has three. One in a corner has two — and if either of those is blocked, its entire first move is forced.
Measured over 72 generated boards, 8% of endpoints sit in a corner and 37% along an edge — so nearly half of all endpoints start out more constrained than they look. Sweep those before touching anything in the open middle.
Every square must be used, and that is a solving tool
The rule that paths must cover the whole grid is usually read as a difficulty setting. It is better understood as the strongest constraint you have.
It means no square may be stranded. If a square can only ever be reached by one path, that path must go there — even when a shorter route between its endpoints exists. And if a square has only two free neighbours, any path passing through it must enter by one and leave by the other, which forces two moves at once.
The same rule kills most of your instincts about efficiency. A direct line between two numbers is very often wrong, because it leaves squares behind that nothing else can reach.
| Length | 2 | 3 | 4 | 5 | 6–9 | 10+ |
|---|---|---|---|---|---|---|
| Share | 35% | 21% | 14% | 9% | 18% | 3% |
A third of all connections are just two adjacent squares, and the average is under four. The interesting work is in the handful of long ones, which have to wind around everything else to pick up the squares nobody else can reach.
A worked board
Notice how few turns the paths make — 0.8 on average. Numberlink paths are mostly straight runs with an occasional bend, so if you find yourself drawing something elaborate, it is probably wrong.
Where solvers get stuck
Connecting the easy pairs first. The two numbers sitting next to each other are the least informative thing on the board. Joining them commits squares you may need, and teaches you nothing.
Drawing the shortest route. Because every square must be covered, the direct line is frequently wrong. Ask instead which squares are hard to reach, and which path is going to have to collect them.
Ignoring dead ends. A square with only one free neighbour left cannot be passed through — only ended at. Since only endpoints end, such a square means something earlier is wrong. Spotting that early saves a lot of redrawing.
Two things you can check about these puzzles
Every puzzle has exactly one solution. This was previously not true and could not have been: the generator cut a snaking path into segments at random positions and shipped the result unchecked. Measured against an exhaustive counter, only 4 of 43 boards had a single solution and 7×7 was 0 of 19. Boards are now counted before release and retried until the count is exactly one. 72 boards across every size and difficulty were re-checked against a separate counter written from the rules alone; all 72 were unique.
More pairs means more certainty, not less. It is tempting to think a board with fewer numbers is a better puzzle. In Numberlink, fewer endpoints means the paths have more freedom to swap around each other, which is exactly what creates second solutions — at three pairs on a 5×5, not one board in the sample was unique. The pair counts are now set where uniqueness is actually achievable, and the grids stop at 9×9 because at 11×11 no setting produced a provable board in reasonable time.
Numberlink, Arukone and Flow
Numberlink is Nikoli's, also published as Arukone and Nanbarinku, and is the direct ancestor of the mobile game Flow Free. The commercial versions usually drop the fill-the-grid rule, which makes them far easier and much less constrained.
Deciding whether an arbitrary Numberlink board can be solved at all is NP-complete, proved by Kotsuma and Takenaga in 2010 — which is why generating one with a guaranteed single answer takes real work, and why the boards here stop at a modest size.
If you like drawing a route under constraints, Hidato is a single numbered chain rather than many separate ones, and Slitherlink asks for one closed loop instead.
More path puzzles
If you like joining things up under constraints, try these: