Numberlink / Flow Puzzle

Join the pairs, fill the board

Two 1s, two 2s, two 3s, scattered on a grid. Draw one unbroken path between each matching pair.

Two constraints make it a puzzle rather than a doodle. Paths may not cross or share a square, and in the standard version every square on the board must end up covered by exactly one path. That second rule is what gives Numberlink its tight, no-loose-ends quality: if you have squares left over at the end, some route needs to take a longer way round.

Every board is generated from a complete valid solution first, so the endpoints you are looking at can always be joined exactly as intended.

The rules, stated fully

• One path per pair. Each numbered endpoint joins to the cell carrying the same number, with a single unbroken line and nothing else.
• Edges only. Never on a diagonal.
• No sharing. Two paths may not occupy the same square, and no path may cross itself.
• Fill everything. Every square on the board ends up covered by exactly one path, and a finished grid has no empty cells left anywhere in it.

How to solve one

Corners first. A number in a corner has only two ways out, so the first few steps of its path decide themselves, and tracing those forced moves carves up the easy regions before you have made a single choice.

Then look for pinch points. A line of empty squares squeezed between two finished paths has to belong to whichever pair can still legally reach it, and often only one can.

The habit that separates good solvers from frustrated ones is checking the fill rule constantly. If completing a path would seal off a square that no other route could ever reach, that path is wrong, however neat it looks. Think of the board as a budget: every square gets spent exactly once, so a route that hugs a wall too tightly starves its neighbour. The elegant solution is very often the curviest one.

Size and difficulty pull in different directions

Grid size runs 5×5 to 9×9, six by default. A 5×5 is a couple of minutes. A 9×9 needs genuine forward planning.

Difficulty changes how many numbered pairs are on the board, and the effect is not what people expect. Fewer pairs means long, sweeping routes with plenty of room, which is easier. More pairs breaks the board into short interlocking journeys that are much harder to fit together without one blocking another.

So a small grid on easy is the right start for a young child, and a large grid on hard is a proper sit-down for somebody who wants one.

Printing a set

Set the size and difficulty and the preview updates as you go. Generate New reshuffles; a typed seed brings the same board back, which is what you want when the whole class needs identical grids to talk about afterwards.

The answer key adds a second page with every pair joined in colour, so marking is a glance rather than a trace. Name and date fields for class sets. A4 or US Letter.