Chains, loops and control
Beginners grab every box in reach. Strong players do something that looks odd: they give boxes away on purpose. This page walks through why, one step at a time.
Safe lines and the third side
In the opening nearly every line is harmless: it hands the opponent nothing. A line that puts the third side on a box is a present, because the box is taken and the taker moves again. So both sides keep drawing harmless lines while any exist.
When the last harmless line is gone, the grid has fallen apart into pieces: chains, corridors of boxes open at both ends, and loops, rings closed on themselves. The player to move now has to open one of them.
Who has control
When a player opens a long chain, the other takes all of it except the last two boxes, then draws the line that leaves those two for the opponent. This is called double-dealing. The opponent takes the two boxes and then must open the next chain.
The player who keeps doing this is said to have control. They give away two boxes per chain (four per loop) and take everything else. On a board with several long chains, control is worth far more than a few early boxes.
The long chain rule
Whether you get control depends on how many long chains (three boxes or more) exist when the safe lines run out. On a board with an odd number of dots, like the 4×4 boxes game with its 25 dots, the first player wants an odd number of long chains and the second player an even number. On a 3×3 or 5×5 boxes board the 16 or 36 dots reverse that wish.
During the opening, think about whether to join two short chains into a long one or to cut a long chain in two. That is what the fight is really about.
When to stop giving
Control is not free. If the last components are short chains of one or two boxes, or loops that cost four, holding control can cost more than it brings. Then simply take everything and open the smallest piece.
The dots puzzles here are exactly these decisions: one line keeps the right balance, while the greedy move of taking all you can falls short.