Research and Implementation of Dots-and-Boxes Chessboard Based on the Game Strings-and-Coins - Advanced Research on Computer Education, Simulation and Modeling - page 16

z Edges correspond with strings
The initial chess board of Dots-and-Boxes has no connected edges at all, but in
Stings-and-Coins chess board, each coin is connected with four strings, we can con-
sider the edges of the Dots-and-Boxes game as strings, but it is seen that placing an
edge in the Dots game position separates two boxes, so this has exactly the effect of
cutting a string in the corresponding Strings-and-Coins position. Similarly each coin
has a degree property which represents the count of strings connected to it.
Each edge in Dots will has the corresponding position in Strings, and the followed
illustration Fig.3 shows how the edges correspond with strings.
Fig. 3. The corresponding relationship of edges and strings
In Fig.3, the left one is a box in Dots and it can have four edges a, b, c and d while the
right one is a coin with four strings a, b, c and d connected to it in Strings game. When
placing the edge a in the left Dots box, it will has the same effect in the right coin by
cutting the corresponding string, when all of the four edges of one box are connected, it
represents that all of the four strings will be cut from the coin, and Fig.4 illustrates the
coin's corresponding state while setting an edge a in the left box, we will see that in
Strings the string of this coin has been removed.
Fig. 4. The corresponding relationship when placing one edge in Dots
z The corresponding relationship of chains and cycles
Fig.5 shows that a chain has been made, and from what we talked above we can easily
know how to get the corresponding graph in Strings-and-Coins game. The way is just to
cut the corresponding strings and we will get the right chessboard description like
what Fig.6 shows. We can see that the representation of chain in Strings-and-Coins is
clearer and it is vivid for us to judge if there's a chain exists or distinguish the
different coins. Similarly, Fig.7 shows the corresponding relationship of cycles in
two games.