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Kids Learning Softwares and games.

THE GAMES

The following games have been selected on the basis of being both important mathematics problems, and being fun to play. Each addresses a different mathematical concept while at the same time all being related to graphs and their properties

Introduction
This game is based on a simple idea: draw a map or any picture as complicated as you wish and color each region using the fewest possible number of colors, the only requirement being that regions sharing a common border must receive different colors.


Introduction
This game deals with the general concept of a graph and its chromatic number. If you draw a bunch of circles, which we call vertices, and join some of them by lines, which we call edges, you get what is called a graph. If you now color the vertices in such a way that those joined by an edge receive different colors, then the smallest number of colors that can be used is called the chromatic number of the graph. The goal of this game is for you to devise ingenious methods to get as close as possible to this number.

The concept of a graph is very important in mathematics and industries. Think of an airline company tracing routes; cities are thought of as vertices and routes as edges. Or think of computers as vertices and edges being the direct internet connections between them. As another example, suppose you have several fish, thought of as vertices, and join two by an edge if the two fish have a chance to eat each other; then we get a graph and the smallest number of aquariums needed to hold these fish without any danger of two fish eating each other is nothing else but the chromatic number of the graph, the number of aquariums corresponding to the number of colors used, right?

   
Introduction
This third game in the Colorful Mathematics series also deals with graphs, but presents a variation on the concept of the previous game.

To review, a graph is a group of circles ("vertices") some of them connected by lines ("edges"). This time, the edges must be colored in such a way that those connected to a common vertex receive different colors. The smallest number of colors that can be used is now called the edge chromatic number of a graph. The goal of this game is for you to devise ingenious methods to try to get as close as possible to this number both with the sample graphs provided and any graph created by the player.

The concept of a graph is very important in mathematics and industries. For a simple example, imagine that in a hockey league, a few games remain to be played until the end of the season, each team playing at most once each day and at most once against any other team. Now think of teams as vertices and connect two of them by an edge if they have a game remaining to play against each other. Then we get a graph and the smallest number of days required to complete the season is nothing else but the edge chromatic number of the graph, the number of days left corresponding to the number of colors used, right?

Introduction
This game is another variation on the concept of a graph and its chromatic number, but you will now play against the computer! If you draw a bunch of circles, called vertices, and if you join some of them with lines, called edges, you get what is called a graph. You are now ready for the game.

First, you must choose the number of colors allowed; then, you can color the vertices alternately with the computer in such a way that two vertices connected by an edge receive different colors. You win the game if you succeed to color the entire graph with the number of colors fixed at the beginning, otherwise the computer wins. It is certainly easy to win by choosing a large number of colors at the beginning, but the goal of the game is for you to devise ingenious methods to find the smallest possible number of colors for which you can always win, regardless of the opponent. This number is called the two-player chromatic number of the graph.

The concept of a graph is very important in mathematics and industries. Imagine, for example, that in a pet shop you buy several fish, that we represent as vertices, and join two of them by an edge, if one fish might eat the other. You want to buy enough aquariums to hold them all without some fish eating others. The clerk is helping you put the fish in the aquariums but is actually trying to force you to buy as many aquariums as possible. This will happen, for example, if the clerk put a big fish in each of the aquariums, and a small fish remains to be placed. The smallest number of aquariums required so that the clerk is unable to force you to buy another one, regardless of his strategy, is the two-player chromatic number of the graph; the number of aquariums corresponding to the fixed number of colors.

Introduction
This fifth game in the Colorful Mathematics series introduces another important property of graphs, that of a dominating set.

Starting with any graph, one color is available to select some of the vertices so that every vertex is either colored or connected by an edge to a colored one; this then, is a dominating set.The goal is to find the smallest number of vertices required to accomplish this, or in other words, the smallest size of a dominating set. This number is called the dominating number of the graph and you will have to create ingenious methods to get as close as possible to this number.

This property of a graph is important in mathematics and industries. As a simple example, think of street corners as vertices, and join two of them by an edge if there is a road between them. On how many street corners should ice cream stands be placed so that nobody should have to go further than one block to get an ice cream? The smallest such number is nothing else but the dominating number of the graph, where ice cream stands should be placed on the street corners from the dominating set, right?

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