Dots and Boxes

The board starts as a square grid of dots with nothing drawn. On your turn you draw one line joining two dots that sit next to each other, across or down — never diagonally, and never over a line that is already there. The boards on this page run from 4 × 4 dots (3 × 3 boxes and 24 possible lines) up to 7 × 7 dots (6 × 6 boxes and 84 lines).

Drawing the fourth side of a box claims it, and here it fills with your colour. Claiming a box also obliges you to draw again at once: your turn only ends with a line that closes nothing. One line can close two boxes when it is the shared side of two three-sided boxes, and then both are yours. When the last line goes in — and the last line always closes a box — the player with more boxes wins.

Because that box is then one line from finished, and it is your opponent who draws next. Most of the opening is about avoiding exactly this. A safe line leaves every box it touches with at most two sides drawn, so it gives nothing away, and for the first part of the game both players draw safe lines while the board fills with two-sided boxes.

Those two-sided boxes join up into corridors. A run of them with both ends open is a chain, and a run that closes back on itself is a loop. Once the safe lines are gone, every line left on the board opens one of them, and whoever opens a chain hands all of it to the other player, who can take it box by box.

Chains of three boxes or more are long chains, and the endgame is fought over how many of them there are. Count the lines on a finished board two ways and a fixed identity falls out: the number of turns in a game equals the number of dots plus the number of double-crosses — single lines that close two boxes at once. The test suite checks that identity in every game it plays, and it turns the question of who must open the first long chain into a question of odd and even.

So the fight happens while safe lines remain: lengthening a chain, splitting one in two or joining two into one changes the count, and the player who gets the parity right will be the one taking the long chains at the end. Loops do not count. Keeping control of a loop costs two double-crosses, which leaves the parity where it was.

By refusing the last two boxes. When your opponent opens a long chain, take every box but the last two, then draw the line at the far end instead of taking them. Your opponent takes the pair with one line and must then open the next chain for you. This all-but-two trick is called double-dealing, and it is how control is kept. In a loop the same trick costs four boxes, because the last four are handed back as two pairs.

Worked through on a 4 × 4 board split into four chains of four: the player in control takes 2 and hands back 2 on each of the first three chains, then takes all 4 of the last one, and wins 10–6. Grab all four boxes of the first chain instead and control changes hands; against good play the game then ends 8–8. The Hard computer on this page makes exactly that choice — the test suite builds this board, opens a chain and checks that Hard takes two boxes and then draws the line at the far end.

The gap between the levels is measured, not asserted. In 40 seeded games in the test suite — ten on each board, with each side drawing first in half of them — Hard beat Medium in all 40, and Medium beat Easy in 38. None of the levels throws a die: when two lines are equally good the choice is drawn from the game's own number, so a game replays line for line from that number.

The French mathematician Édouard Lucas published it in 1889 as La Pipopipette, and it has been played in the margins of exercise books ever since. Its theory came much later: Elwyn Berlekamp, a co-author of Winning Ways for Your Mathematical Plays, gave the game a book of its own in 2000, The Dots and Boxes Game: Sophisticated Child's Play, and the long chain rule and double-dealing described above are his terms.

Computers have since solved small boards outright. In 2012 Joseph Barker and Richard Korf solved the 4 × 5 board and showed that with perfect play it ends in a tie. For a simpler game of lines on a grid, try tic tac toe; for another game decided by what you leave your opponent, try Four in a Row.