Dots and boxes

The board is a grid of dots, and each move is one horizontal or vertical line between two adjacent dots, never a diagonal. A box is the square between four dots. The player who draws its fourth side closes it, the box takes that player's colour and initial, and the same player draws again. Taking a box is never compulsory, so a player may leave it and draw somewhere else.

Playing on screen works as it does on paper. A tap or click next to the gap between two dots draws that line, while a tap in the middle of a box picks nothing. Against the computer you draw in blue and the computer in orange. One line can be the fourth side of two boxes at once, and it still earns a single extra turn. The opening board of 5 by 5 boxes has 36 dots and 60 lines, and every one of them is drawn before the game ends. When one player holds more than half the boxes, the game says the result is decided and play continues until the last box is closed.

Who starts is a setting with three choices: you, the computer, or each in turn from game to game. When you start, your first tap draws the first line. When the computer starts, or when the boxes carry numbers, a Start button begins the game instead.

The folded panel Rules and help changes the rules before a game begins. They stay fixed while it runs, and New game unlocks them again.

• Closing a box means drawing another line is ticked by default, which is the usual rule. Unticked, the turn passes after every line, a variant some players prefer.
• Outer edge already drawn draws the frame of the board before the first line, another variant played on paper.
• Numbers in the boxes turns on the classroom version with two dice, described with the printable sheets below.
• Mark the lines that hand over a box shows every line that would give a box its third side as a dashed red line.

The computer opponent plays at three difficulty levels. Each follows fixed habits designed for this site, since no published standard defines a level for this game.

• Easy closes a box it can close three times in four (75%). Otherwise it looks for a line that hands nothing over only half the time (50%) and draws any line the rest, so it often gives boxes away.
• Medium closes every box it can and never draws a third side while a safe move is left. When every line gives something away, it gives the fewest boxes. It never declines boxes, so it never keeps control of the chains at the end.
• Hard plays like medium early on. Once 12 or fewer safe lines are left, it searches the rest of the game exactly, within about half a second a move. It counts chains and declines boxes when that pays.

In tests run for this site on 3 by 3 and 4 by 4 boxes, hard beats medium in most games. It is not unbeatable. On big boards early in a game it relies on rules of thumb, and only the endgame is searched exactly.

Three buttons sit under the board. Undo takes back your last line together with the computer's reply, though not once the game has ended. Hint draws the line the hard level would choose as a dashed blue line. New game clears the board.

Your record against the computer counts games won, drawn and lost for each level and board size, with your widest win and the last 10 games. It is kept in this browser's local storage and never sent anywhere, and Clear my record deletes it.

In the two-player game, both players share one phone, tablet or computer and hand it over when the turn passes, a style called pass and play. The board, the rules and the options work as they do against the computer.

Each player may type a name of up to 14 characters. Closed boxes then show each player's initial: one letter, two letters when both names begin with the same one, and 1 and 2 if those still match. Names are never stored and never go into the page's link.

The first move goes to player 1, player 2, or the two alternately, game by game. Undo takes back the last line, whoever drew it. Two-player games stay out of the record against the computer.

No second device is needed. As with the who am I game, everything happens on the one screen passed round the table, and the game sends nothing over the network while it is played.

A game starts on 5 by 5 boxes, which is 6 by 6 dots and 60 lines, and any board from 2 by 2 to 8 by 8 boxes can be set.

Sizes here count boxes, as the settings do, while players on paper often count dots. Boxes across and Boxes down are set separately, which gives 49 board sizes.

A draw needs an even total of boxes. The default board has 25, so one player always wins, while the 16 boxes of a 4 by 4 board can end 8 to 8. Daniel Allcock of the University of Texas names 5 by 5 as the typical board and notes that boards odd in both directions rule out a tie.

Computers have solved only small boards. Joseph Barker and Richard Korf showed in 2012 that 3 by 3 boxes is a win for the second player by 3 boxes with best play. Their largest, 4 by 5 boxes, is a tie with best play and took a search of 10 days on one computer. The result of 5 by 5 with best play is not known, so nobody can yet say which side wins it.

The first trick is to avoid drawing the third side of a box, because the next player closes it and moves again. A line that gives no box its third side is a safe line, and early in a game both players draw such lines.

Ticking Mark the lines that hand over a box in Rules and help dashes in red every line that would add a third side. That way the board itself shows which moves give boxes away. Medium and hard never draw one of those while any other choice remains.

Harmless moves run out. On a crowded board every remaining line gives the other player at least one box, and the open boxes have joined into chains and loops. From then on the game turns on who must open them, as the section on chains and loops explains.

A chain is a row of boxes joined so that taking one opens the next, and a chain of 3 or more boxes is a long chain. A short chain holds one or two boxes. A loop is a chain that closes on itself, with at least 4 boxes.

Julian West, writing for the Mathematical Sciences Research Institute, explains why 3 is the line: the last two boxes of a long chain can always be declined. The player taking the chain takes all but those two and then draws the line at the far end of the chain. That is the double-dealing move, and in a loop it leaves the last four boxes.

The other player then takes the pair with one line, a double-cross, and must open the next chain. A player who keeps declining like this keeps control of the game. West describes that player as having control, and Allcock calls them the controller.

The hard level's own search gives exact results for three small endgames, in each of which the player to move must open something.

• Two chains of 3 boxes: whoever opens one loses 2 to 4. The other player takes one box, declines the last two and later takes the whole second chain. Taking all three at once would end 3 to 3.
• A chain of 3 and a single box: giving away the single box first wins 3 to 1, while opening the chain first loses.
• A loop of 4 and a chain of 3: opening the loop first loses only 3 to 4, and opening the chain first loses by more.

Giving boxes away on purpose is a sacrifice. The hard level makes them: it may give away a box or two to change who must open the first chain worth declining. It also hands over a pair of boxes by their middle line, so the pair cannot be declined.

Such endgames are still studied. Kevin Buzzard and Michael Ciere of Imperial College London wrote a paper on playing simple loony endgames optimally. In those positions, every move into a chain or loop lets the other player choose whether to decline.

On a board of 5 by 5 boxes, the first player wants an even number of long chains. Thomas Ferguson of UCLA states the long chain rule for any board. When the board is even in both directions, the starter wants an odd count, and otherwise an even one. Loops do not count.

West gives the same rule another way: the starter wants the dots plus those chains to add up to an even number. A 5 by 5 board has 36 dots, an even number, which leads to the same answer.

It is the rule Elwyn Berlekamp's book teaches. The Dots-and-Boxes Game: Sophisticated Child's Play appeared in 2000, after the game had a chapter of Winning Ways in 1982. The rule works because of the double-dealing move: whoever must open the first chain of three or more usually hands every later one to a player in control. The hard level plays for this count in the endgame, which is when its sacrifices begin.

Édouard Lucas described the game in print first in 1883, in Récréations mathématiques, tome II, as "le jeu de l'École Polytechnique". Lucas wrote that students of the École polytechnique had invented it a few years earlier, so he credits them rather than himself.

In his account it was played on squared paper. A closed square was marked with the player's initial, the player who closed it played on, and the winner was the first to hold more than half the squares.

On 5 May 1889 Lucas published it as "La Pipopipette, nouveau jeu de combinaisons". It was a boxed game of 36 pegs and 60 bars making 25 squares, dedicated to his former pupils. That is the same board this game starts on. "Pipo" was the students' short name for Polytechnique, and the text is reprinted in L'Arithmétique amusante (1895), Note III.

British searchers also type dots and squares, or simply boxes. Berlekamp's book also gives a chapter to a disguise called strings and coins. Each box is a coin, each line a string and the edge is the ground, so cutting a coin's last string pockets it.

Programs play it too. In 2005 BBC News reported that Deep Beige, written by a Nottingham student, had won gold for dots and boxes at the 10th Computer Olympiad.

Print a blank sheet puts 1, 2, 4 or 6 empty grids of the chosen board on one printable sheet, for play with pencils away from the screen.

NRICH, the University of Cambridge's maths enrichment site, sets the game for ages 7 to 11 as "Put Yourself in a Box". Its version needs two coloured pencils and dot paper, an initial goes in each closed box, and the player who closed it takes another turn.

Numbers in the boxes turns the board into a classroom version with dice. When the game starts, each box is dealt the result of its own throw of two dice, so common results fill more boxes than rare ones. Before every line the player throws two dice and adds or multiplies them, as chosen, and the computer throws for itself.

Only a side of an open box showing that result may be drawn, and a side shared by two boxes counts if either shows it. With no such side free, the turn passes. Closing a box earns another throw and another line, and every box counts one point whatever its number.

Adding gives 11 results from 2 to 12, and and 7 is the commonest at 6 throws in 36 (16.7%). Multiplying gives 18 results from 1 to 36. 6 and 12 come up 4 ways each, while 1, 9, 16, 25 and 36 come up once in 36.

This dice version is the site's own, because classroom sheets differ: some subtract, and some use dice that go up to ten. Printed sheets with numbers on get fresh numbers for every grid. The dice are thrown with the browser's cryptographic random generator only after the game starts, as in the Christmas dice game.