Editing Dots and boxes (section)

== Strategy ==
[[File:Dots-and-boxes.svg|upright=1.35|thumb|Example game of Dots and Boxes on a 2×2 square board. The second player ("B") plays a rotated mirror image of the first player's moves, hoping to divide the board into two pieces and tie the game. But the first player ("A") makes a ''sacrifice'' at move 7 and B accepts the sacrifice, getting one box. However, B must now add another line, and so B connects the center dot to the center-right dot, causing the remaining unscored boxes to be joined in a ''chain'' (shown at the end of move 8). With A's next move, A gets all three of them and ends the game, winning 3–1.]]
[[File:dots-and-boxes-chains.png|thumb|upright=1.35|The "double-cross" strategy: faced with position 1, a novice player would create position 2 and lose. An experienced player would create position 3 and win.]]
For most novice players, the game begins with a phase of more-or-less randomly connecting dots, where the only strategy is to avoid adding the third side to any box. This continues until all the remaining (potential) boxes are joined into ''chains'' – groups of one or more adjacent boxes in which any move gives all the boxes in the chain to the opponent. At this point, players typically take all available boxes, then ''open'' the smallest available chain to their opponent. For example, a novice player faced with a situation like position 1 in the diagram on the right, in which some boxes can be captured, may take all the boxes in the chain, resulting in position 2. But with their last move, they have to open the next, larger chain, and the novice loses the game.<ref name="ww"/><ref name="west">{{Citation
 | last = West
 | first = Julian
 | contribution = Championship-level play of dots-and-boxes
 | editor-last = Nowakowski
 | editor-first = Richard
 | title = Games of No Chance
 | pages = 79–84
 | publisher = MSRI Publications
 | place = Berkeley
 | year = 1996
 | contribution-url = http://library.msri.org/books/Book29/files/westboxes.pdf }}.</ref>

A more experienced player faced with position 1 will instead play the ''double-cross strategy'', taking all but 2 of the boxes in the chain and leaving position 3. The opponent will take these two boxes and then be forced to open the next chain. By achieving position 3, player A wins. The same double-cross strategy applies no matter how many long chains there are: a player using this strategy will take all but two boxes in each chain and take all the boxes in the last chain. If the chains are long enough, then this player will win.

The next level of strategic complexity, between experts who would both use the double-cross strategy (if they were allowed to), is a battle for control: an expert player tries to force their opponent to open the first long chain, because the player who first opens a long chain usually loses.<ref name="ww"/><ref name="west" /> Against a player who does not understand the concept of a sacrifice, the expert simply has to make the correct number of sacrifices to encourage the opponent to hand them the first chain long enough to ensure a win. If the other player also sacrifices, the expert has to additionally manipulate the number of available sacrifices through earlier play.

In [[combinatorial game theory]], Dots and Boxes is an [[impartial game]] and many positions can be analyzed using [[Sprague–Grundy theorem|Sprague–Grundy theory]]. However, Dots and Boxes lacks the [[normal play convention]] of most impartial games (where the last player to move wins), which complicates the analysis considerably.<ref name="ww"/><ref name="west" />