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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.
David Wilson / dwilson@cae.wisc.edu