Research and Implementation of Dots-and-Boxes Chessboard Based on the Game Strings-and-Coins - Advanced Research on Computer Education, Simulation and Modeling - page 15

are that each player connect adjacent dots on alternate turns, it is required that you
should only connect adjacent dots and should not make cater corners; the lines, which
are also called edges, are not belong to any side, and we only take boxes into account.
You can get a box by connecting the four adjacent dots with lines, and when you make
a box, you can put your name in it, and move again, when all boxes are formed, the
game is over, and the winner is the one who get the most boxes.
Strings-and-Coins is another game invented by Elwin Berlekamp and we can use it
to generalize Dots-and-Coins to general graphs. The rules of game Strings-and-Games
are simply, the players alternate cutting strings. When a player cuts all the edges sur-
rounding a coin, he takes the coin and moves again. The player with the most coins
wins the game.
Strings-and-Coins is played on the dual graph of Dots-and-Boxes, which means the
following: For any Dots-and-Boxes position, the corresponding Strings-and-Coins
game is constructed by considering the boxes as coins, and the edges of the
Dots-and-Boxes game as strings. It is seen that placing an edge in the Dots-and-Boxes
position separates two boxes, so this has exactly the effect of cutting a string in the
corresponding Strings-and-Coins position. From this point of view, the two games are
equivalent.

3 The New Description of Dots Chessboard Based on
Strings-and-Coins

The traditional description of the Dots chessboard is a 6*6 dots grid, and a 2*2 sub grid
is called a box, the nodes(i,j)and(k,l)are adjacent only when the expression |i-k|+|j-l|=1
exists, when the four edges of one box are all connected then the box will be got by the
player. Inspired by the equivalent game Strings-and-Coins, we can deprive the ideas
that providing another easier description of the board. Fig.1 shows the initial board in
traditional way, while Fig.2 shows the Strings-and-Coins chessboard.

Fig. 1. The traditional description Fig. 2. The Strings-and-Coins chessboard

The corresponding relationship of the two types of boards:

z Boxes correspond with coins
Fig.1 shows the initial Dots chess board of 6*6 type and we can get 5*5 boxes in this
Dots-and-Boxes board, while Fig.2 shows the description of Stings-and-Coins chess
board of 5*5 type. We can consider the boxes in Dots-and-Boxes as the coins in the
Strings-and-Coins, so we get the corresponding relationship of boxes and coins.