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Reasoning4 min read

Dot Path strategy: how to solve one-line puzzles

Tips for Dot Path and other one-line puzzles, where you draw a path that visits every dot exactly once: find the forced moves, start from the ends, and avoid cutting off a region.

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The rules

Draw one line through neighboring dots, up, down, left or right, so every blue dot is visited exactly once. Red dots are off limits. In math this kind of route is called a Hamiltonian path.

Find the forced moves

Key idea

A dot with only one free neighbor must be an end of the line. A dot with exactly two free neighbors must be passed straight through them. Mark these first and much of the board solves itself.

Start from an end

If you can find a dot that must be an end, start there. If a board has a ringed dot, the line has to begin on it.

Don’t cut the board in two

Before sweeping along an edge, check you won’t leave a pocket of dots you can’t get back to. If the remaining dots split into two separate groups, step back.

Use parity on bigger boards

Color the grid like a chessboard. Each step moves to the other color, so a line through an odd number of dots starts and ends on the same color. If one color has two more dots than the other, there is no line at all, which helps you spot a dead end early.

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