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The players then take turns drawing vertical or horizontal lines between two adjacent dots. If a player completes one or two boxes (of size 1x1) by drawing a line, s/he places his or her initial in the box(es) and then must place one other line. His or her turn continues until s/he places a line that does not complete a box. The winner is whoever ends up with the most boxes.
For information on the Dots-and-Boxes game, see "Dots-and-Boxes for Experts" by Ian Stewart in Scientific American (v. 284, no. 1, January 2001) pp. 102-103. The article is based on The Dots-and-Boxes Game by Elwyn Berlekamp (A. K. Peters, 2000).
The following sample game is from page 4 of the book. The analysis is done using the Dots-and-Boxes Analysis Programs.
+ -3 + -3 + -3 + -3 -3 -3 -3 The numbers indicate the result of a move with best play. Negative numbers indicate + -3 + -3 + -3 + the second player to move wins. Here, the the second player can win by 3 boxes no -3 -3 -3 -3 matter what the first player does. + -3 + -3 + -3 + Let A be the first player and B be the -3 -3 -3 -3 second player. + -3 + -3 + -3 + Player A moves. + -1 + -3 + -1 + -1 -1 -1 -1 Good move. A center move gives the player B the greatest opportunity to make a mistake. + -3 + -1 + -3 + -1 -1 -1 -1 + -1 #-----# -1 + -1 -3 -3 -1 + -1 + -1 + -1 + Player B moves. + -3 + -3 + -3 + -3 -3 -3 -3 Now player B wins by 1 box with best play. + -3 + -3 + -3 + -3 -1 -1 -3 + -3 +-----+ -3 + -3 -3 -3 -3 + -3 #-----# -3 + Player A moves. + -1 + -1 + -1 + -1 -1 -1 -1 + -1 + -3 + -1 + Now, with exactly the right move, player B wins by 3 with best play. -1 -1 -1 -1 #-----#-----+ -1 + -1 -1 -1 -1 + -1 +-----+ -1 + Player B moves. + -1 + -1 + -1 + -1 -1 -1 -1 Now player B wins by 1 with best play. + -1 + -3 + -1 + -1 -1 -1 -3 #-----+-----+ -1 + | | -5 -5 -1 | # -5 +-----+ -1 + Player A moves. + -1 # -1 + -1 + | -1 | 1 -1 A good move. | + 1 # -1 + -1 + -1 -1 -1 -1 +-----+-----+ 1 + | | -1 1 1 | + 1 +-----+ 1 + Player B moves. + -1 + -1 + -1 + | -1 | -1 -1 Now, player A wins by 1 with best play. | + -1 + -1 + -1 + -1 -1 -1 -1 +-----+-----#-----# | | -5 -5 1 | + -5 +-----+ 1 + Player A moves. + 1 + 1 + 1 + + -1 + -3 + -1 + | | 1 | 1 1 Now, player B can win -1 | -1 -1 | by 1 box by immediately | + 1 + 1 #-----# giving away two boxes. + -1 + -1 +-----+ Result at right with 1 1 1 1 player A to move after -3 -1 -3 -3 taking the two boxes. +-----+-----+-----+ +-----+-----+-----+ | | | | | -1 1 1 | A | A | -1 | | | | + 1 +-----+ 1 + +-----+-----+ -1 + Player B moves. + 1 + 1 + 1 + | 1 | 1 1 Now, player A wins by 1 with best play. | + 1 + -1 +-----+ 1 -1 -3 -3 +-----+-----+-----+ | | -5 -5 -5 | + -5 +-----#-----# Player A moves. + 3 + 3 + 3 + | 3 | 3 3 | # 3 + 1 +-----+ Good move. | | 3 3 3 | #-----+-----+-----+ | | 1 1 1 | + 1 +-----+-----+ Player B moves. + 1 + 1 + 1 + | 1 | 1 1 Good move. | + -5 #-----#-----+ | | -5 -5 -5 | +-----+-----+-----+ | | -5 -5 -5 | + -5 +-----+-----+ Player A moves. #-----# 3 + 1 + | 3 | 3 1 Good move. | + 3 +-----+-----+ | | 3 3 3 | +-----+-----+-----+ | | 1 1 1 | + 1 +-----+-----+ Player B moves. +-----+ -1 + -1 # | | -5 | 1 | Good move. | | + -5 +-----+-----# | | -5 -5 -5 | +-----+-----+-----+ | | -5 -5 -5 | + -5 +-----+-----+ Player A moves. +-----+ 1 # 1 + | | | 5 | | | | | | + 5 +-----#-----+ The only winning move. | | 5 5 5 | +-----+-----+-----+ | | 5 5 5 | + 5 +-----+-----+ Player B moves. +-----#-----# 1 + | | | 3 | B | | | | | + 3 +-----+-----+ | | 3 3 3 | +-----+-----+-----+ | | 3 3 3 | + 3 +-----+-----+ Player B moves again. +-----+-----#-----# | | | 1 | B | B | | | | + 1 +-----+-----+ | | 1 1 1 | +-----+-----+-----+ | | 1 1 1 | + 1 +-----+-----+ Player B moves again. +-----+-----+-----+ | | | -9 | B | B | | | | + -9 +-----+-----+ A move like this, which gives the opponent | the option make a sacrifice, is called | -9 -9 -9 a loony move. In this situation, all of | player B's possible moves were loony. +-----+-----#-----+ | | | -3 | 1 | | + -1 +-----#-----+ Player A moves. +-----+-----+-----+ | | | -7 | B | B | | | | + -7 +-----+-----+ If player A captures the two boxes at the | lower left he must then give up the other | -7 -7 -7 4 boxes. By instead making a sacrifice, | player A wins. +-----+-----+-----# | | | | -3 | A | | | | + 1 +-----+-----# Player A moves again. +-----+-----+-----+ | | | 5 | B | B | | | | + 5 +-----+-----+ | | 5 5 5 | +-----+-----+-----+ | | | | 1 | A | | | | #-----#-----+-----+ Player B moves. +-----+-----+-----+ | | | 1 | B | B | | | | + 1 +-----+-----+ | | 1 1 1 | +-----#-----+-----+ | | | | | B | B | A | | | | | +-----#-----+-----+ Player B moves again giving up the remaining boxes.
Translations of this page: Kazakh, Norwegian.
David Wilson / dwilson@cae.wisc.edu