Numberlink Solver

Enter the numbered endpoints of a Numberlink or Flow-style puzzle and the tool finds the paths that join each pair without crossing, checks that there is only one way to draw them, and can reveal the paths one step at a time.

How to use

1. Choose "Rows" and "Columns" (3 to 15). To try the tool first, press "Small example 5×5", "Medium example 7×7" or "Flow-style 8×8".
2. Tap a cell and press a number on the keypad (1 to 20); each number goes in exactly two cells. "Clear" empties a cell. The message under the grid names a number that appears once or more than twice.
3. Leave "Fill every cell" on for newspaper puzzles and Flow-style apps, where every cell is used by a path. Turn it off for books that allow empty cells.
4. Press "Solve". The paths are drawn as coloured lines and the message says whether the answer is unique. "Hide answer" removes them, and "Hint: one cell" draws one more step of one path with the reason.
5. "Save as PNG" downloads the grid with the paths shown. "Clear all" empties the board.

How it works

• The solver grows paths from their endpoints one cell at a time, always continuing the path end with the fewest possible moves. After every move it checks that each unfinished pair can still reach its partner through free cells and, with "Fill every cell" on, that no free area is cut off. It keeps searching after the first answer to check uniqueness and stops after 10 seconds.
• Hints use two rules: a path end that can move to only one free cell, and, when every cell must be filled, a free cell with only two open neighbours, one of them a path end. When neither applies, the next step comes from the solution and the message says so.
• The board, the answer and the hints are kept in this browser only.

Numberlink rules

Join each pair of equal numbers with one line that moves horizontally or vertically from cell to cell. Lines do not cross, do not branch and do not share a cell. A line never runs alongside itself: if two of its cells touched, a shorter line would exist. In the common newspaper and app form every cell of the grid is used by some line.

Solving Numberlink by hand

Start at the edges and corners. An endpoint in a corner with one neighbour already taken has only one way out. Pairs that sit next to each other or close together along an edge are usually joined by the shortest route, which keeps the middle free for the longer lines.

With every cell filled, look for cells that would be left behind: an empty cell with only two free neighbours must be passed through by the line that reaches one of them, and an empty dead end must hold a line end, which is impossible, so the lines must avoid creating it.

Questions and answers

What if the solver finds more than one answer?

A well-made Numberlink has one answer. Several answers usually mean a number is in the wrong cell or the "Fill every cell" setting does not match the puzzle. One possible drawing is shown so that you can compare it with the paper.

Is this the same as Flow Free?

Flow-style app levels use the same rules with colours instead of numbers: join each pair and fill every cell. Give each colour its own number and keep "Fill every cell" on.

Why does it say there is no answer that fills every cell?

With "Fill every cell" on, every cell must be used. Some puzzle books leave cells empty; turn the setting off and solve again. If there is still no answer, check the positions of the numbers.

How large can the grid be?

Up to 15×15 with up to 20 pairs. Large open grids with few pairs take the longest; the search stops after 10 seconds.