# Rules of Dots-and-Boxes

## Unsolved Form

The game is played between two players on a rectangular array of dots, such as the following 4x3 grid.
. . . .
. . . .
. . . .

The players take turns in making legal moves, where a legal move consists of joining two horizontally or vertically adjacent dots which were previously unjoined. A player may not "pass" on a move, that is, each move must involve the joining of two dots. In the above example, the players might make the following moves:
._. . . ._. . .
. . . . . . ! .
. . . . . . . .
Player 1 => Player 2

When a player completes the fourth side of a unit square (box) he/she scores a point and must then make another move (ie: joining two more dots). The following sequence of mid-game moves illustrates this.
Player 1 draws a vertical line to leave the following array.
._._. .
!_._. . Score - Player 1 : 0
. . . . Player 2 : 0
Player 2 completes a box with a vertical line, scoring a point.
._._. .
!_!_. . Score - Player 1 : 0
. . . . Player 2 : 1
Player 2 moves again since he completed a box with the
last move and scores another point.
._._. .
!_!_! . Score - Player 1 : 0
. . . . Player 2 : 2
Player 2 moves yet again but does not complete a box.
._._._.
!_!_! . Score - Player 1 : 0
. . . . Player 2 : 2

A player who can complete a box is not obliged to do so.
The game ends when there are no legal moves available, ie: there are no unjoined vertically adjacent or horizontal dots left. The winner is the player to have completed the most boxes (scored the most points). Since there may be an even number of boxes in the array of points, it is possible for the result to be a tie.

Reference:
"Combinatorial Games"
Proceedings of Symposia in Applied Mathematics
Vol 43, 1991

## Solved Form

This form of the game is the same as the unsolved form with the exception that a player who can complete a box must do so. This restriction simplifies plays considerably, and Holladay (see below) solved this form of the game in 1966.
Reference:
"A NOTE ON THE GAME OF DOTS"
J. C. Holladay
American Mathematical Monthly
Vol 73, pp.717-720 (September 1966)

L. Weaver 09/95