Dots and Boxes

Dots and Boxes (also called Squares, La Pipopipette in French) is the pencil-and-paper game played by schoolchildren for over a century. Two players take turns drawing a line between adjacent dots on a grid; complete a box (4 lines) and you own it (and get another turn). Most boxes wins. Deceptively simple, mathematically deep — and used as a benchmark for game-tree search algorithms.

How to play

• Grid of dots (here 4×4 up to 7×7 dots; 5×5 by default). Players take turns drawing a single line between two adjacent dots — click between two dots to draw it.
• Play against the CPU (Easy, Medium or Hard) or a friend on the same screen. You are blue and move first; the CPU is red.
• If your line completes a box (4 surrounding lines), you own that box — mark it.
• You get another turn after completing a box.
• Game ends when all lines are drawn.
• Most boxes wins.

Beginner strategy

• Avoid third side of a box. Three sides = opponent completes the box.
• Don’t be forced to give boxes. If all moves give the opponent a box, you’ve lost the “chain.”
• Don’t take obvious boxes early. Saving early boxes can be strategic.
• Look for chain endings. The last box of a chain belongs to you.
• Mid-game strategy. Avoid drawing the third side of any box.

Intermediate strategy: long chains and short chains

Endgame Dots and Boxes is dominated by “chains” — connected sequences of boxes. The player FORCED to start a chain LOSES that chain. So the strategy becomes “force the opponent to start chains” by making sacrificial moves yourself.

• The double-cross. Sometimes giving up a box deliberately (leaving 2 boxes for opponent) forces them to start a much larger chain.
• Count chains. If there are 4 chains and you’re 4-down, who starts each chain matters.
• The long chain rule. Berlekamp’s rule links the number of dots and the number of long chains to which player ends up forced to open one.
• Half-hearted greed. Don’t take all available boxes. Strategic abandonment beats greed.

Advanced strategy

• Calculate chain control. Pre-compute who’s forced into each chain.
• The “loose” chain technique. A 4-box chain may be better than a 3-box chain.
• Computer-perfect Dots and Boxes. Exhaustive computer searches have solved small boards perfectly, up to 4×5 boxes.
• Endgame practice. Most close games are decided in the last 10 moves.

Dots and Boxes history

• 1889 — Edouard Lucas: French mathematician describes the game (“La Pipopipette”).
• 1982 — Winning Ways: Berlekamp, Conway and Guy analyse the game in their book on combinatorial games.
• 2000 — Berlekamp’s analysis: Elwyn Berlekamp publishes “The Dots and Boxes Game” — definitive mathematical treatment.
• 2012 — computer analysis: Barker and Korf solve the 4×5-box board (a tie with perfect play).
• Modern: remains a classroom puzzle and online multiplayer game.

FAQ

• Q: Standard grid size? A: Usually 5×5 dots (= 4×4 boxes). Larger grids exist.
• Q: Best strategy? A: Avoid drawing third sides. Force opponent into long chains. Use double-cross technique.
• Q: Is the game solved? A: Small boards are computationally solved, up to 4×5 boxes (which includes the 5×5-dot board). Larger boards are not.
• Q: Why “Dots and Boxes”? A: Connect the DOTS to form BOXES — the literal mechanic.

For other 2-player games, try Tic-Tac-Toe, Connect Four, Checkers, or Hex.