How to play Flow-style puzzles

Every puzzle on this site follows four rules, and every lesson uses the same small set of words to talk about them. This page covers both, shows a finished board, and then clears up the three ideas that slow most new players down.

The rules

1. Connect each pair of same-colored dots with one path.
2. A path moves between cells that share a side: up, down, left or right. It never moves diagonally.
3. Paths never cross, never branch and never share a cell. A path can't pass through a dot, including a dot of its own color partway along.
4. When you finish, every cell on the board is covered by some path.

Some Numberlink rule sets only ask you to connect the pairs. Here, and in the solver, a board counts as solved only when every cell is covered. That fourth rule does most of the work when you solve. A cell that no path can reach means a move was wrong, so empty cells tell you which moves to reject.

The words used on this site

• Dot or endpoint: one of the two fixed circles of a color. Each color has exactly two.
• Path: the line you draw for one color. A finished path runs from one dot to its partner.
• Cell: one square of the grid.
• Neighbor: a cell that shares a side with another cell. A corner cell has two neighbors, any other cell on the edge has three, and an inner cell has four.
• Path end or tip: the place a path can grow from. Before you draw anything, both dots of a color are path ends. Once you draw part of a path, its last cell is the tip.
• Open: an empty cell that a path could still move into.
• Wall or edge: the border of the board. Paths stop at it, except on boards with warps.
• Hole: a gap in the board. It isn't a cell, paths go around it, and it doesn't need to be filled.
• Joining or connecting: the last step of a color, when its two path ends meet.

The lessons describe positions as "row 2, column 3", counting from 1 at the top-left cell.

What a solved board looks like

Read the finished board color by color. Blue starts at its dot in row 1, column 5, steps left, then runs straight down column 4 and turns left into its other dot at row 5, column 3. Green leaves its dot at row 2, column 3, climbs to the top row, follows the wall around the top-left corner and comes down column 1 to row 4. Orange takes three cells. Nothing is left empty, nothing crosses, and no path splits in two.

Holes change the shape of the board but not the rules. On the board below, the dark squares at row 2, column 2 and row 4, column 4 are holes. Blue wraps around the first one, and Orange runs down column 5 beside the second.

One solution per puzzle

Every puzzle published on this site has exactly one solution. Before a puzzle goes up, an exact solver proves that no second answer exists, and a step-by-step solver confirms that a person can finish it by reasoning alone. You never need to guess. When you feel stuck, some cell or path end on the board can always be settled by one of the techniques in the lessons.

Uniqueness also lets you reason backward. A move that would leave two possible ways to finish can't be part of the answer. The uniqueness trick turns that into a practical rule.

Common beginner mistakes

Trying to keep paths short

Short paths feel efficient, but the board has to be full. Every cell a short path skips must be filled by some other color, and often no other color can reach it. A path is exactly as long as the cells around it demand.

Expecting paths to go straight

Paths wind. They follow walls, wrap around other dots and double back across the board. On the solved board above, Green's dots are only two rows and two columns apart, yet Green climbs to the top row first and uses seven cells. A route that looks like a detour is often the only way to cover the cells it passes.

Joining dots that touch

Two dots of the same color side by side look like they should connect at once. Sometimes they do. Often they don't.

Here is why Green can't join directly. The bottom-right corner, row 5, column 5, has only two neighbors, and one of them is Blue's dot at row 5, column 4. So Blue has to step out of its dot into that corner. Now look at row 5, column 3. Its neighbors are Green's dot above it, an empty cell to its left, and Blue's dot to its right, which is no longer usable because Blue has already left it. With only two usable neighbors, the path through that cell must come from Green's dot. Green leaves its dot downward and has to come back to its partner the long way.

The reasoning in that paragraph is the subject of Edges and corners. The safe version of joining is covered in Forced moves: two ends of one color connect for certain only when joining is the one thing either of them can do.

Where to go next

Work through your first puzzle, a 5×5 board solved one step at a time, then read the technique lessons in order. When you want to practice, every puzzle on the site has hints and a full walkthrough.