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Then it would de- scribe the information about the board game we choose and how to combine the game with before-mentioned three graph theories. At last, we would account for the advantage of combining with these three graph theories and the game specifically. 1 Introduction For computer sciences, graph theories are important. There are many computer sci- ence’s concepts relate with graph theory, and many researchers try to combine these science to games. For example, we can connect the concept of network and Prim’s minimum spanning tree [1], or link the graph and the concept of searching [2]. Therefore, learning and teaching graph theory much more efficiently aid learning computer sciences. About some knowledge in computer science, using graph theories assist in teaching is useful. It helps to describe some virtual concepts, like network connection. Some computer science, often combine with graph theories in class re- cently. There many kinds of concepts of graph theories, but we contact the concept of minimum spanning tree mostly. And, we choose minimum spanning tree theories as our target. Accordingly, how to make student to learn graph theories more efficiently and boost the interest in learning for them is administer to learn computer sciences. In order to increase the interest in learning and absorbing knowledge, using board game to help learning is a good method. Learning graph theories well help to learn computer sciences. By the reason, we combine the board game Ticket to Ride with three graph theories Dijkstra’s, Prim’s, and Kruskal’s minimum spanning tree. Except some saving rules, we change and create some rules in the game. By the new board game, students can understand the graph theories more efficiently and can get interest in learning. 276 W C. Chang, Y D. Chiu, and M F. Li 2 Related Work In the early centenary, games were just played for fun. The games, especially board games are. With the time goes, computers’ knowledge and technologies are develop- ing. So, computer science was becoming an important science. Accent on the materi- ality of board games, many researches tried to combine teaching computer sciences with playing board games. For example, Andre J. Henney and Johnson I. Agbinya used board game to explain the idea about mobile connection [3]. And Steve Goschnick and Sandrine Balbo linked programming to board games [4]. In these researchers’ concepts, the most access to combine with computer sciences and board game is – shows the network or graph theory on board game. For example, Peter Komisarczuk and Ian Welch used board game to teach internet engineering [5]. Darren Lim shows the graph theory on the board game [1]. Using games can help teaching computer sciences much more efficiently. Especially teaching graph theory or network concepts on board games. It can use the materiality of board games to describe the virtual of graph theories and network well. In this paper, we try to explain the three of ideals of graph theories - Dijkstra’s, Prim’s, and Kruskal’s minimum spanning tree on the board game Ticket to Ride. In the follow sections, we will describe these: 1. The concept of minimum spanning tree theories: Dijkstra’s, Prim’s, and Kruskal’s minimum spanning tree (section 2). 2. Related work (section 2). 3. The information about board game Ticket to Ride (section 3). 3. The design of combining graph theories and the board game Ticket to Ride (section 4). 4. Conclusion and evaluation about the board game in helping learning. 3 Brief Describe the Three Minimum Spanning Tree Theories We Use Between many connected nodes, they are undirected graph. Spanning trees means all of the nodes connection. If we give the weight to every path from node to another node and sum of all paths’ weight, the minimum spanning tree algorithms try to find the least sum [1, 6]. 3.1 Dijkstra’s Minimum Spanning Tree It describes a concept of minimum spanning tree. The theory can be use in teaching net- work, like the paths of router and router [7]. It goes from a start node, than it will begin to run follow steps: 1.Finding the connected node. 2. Calculating the sum of start node to next node. 3. Choosing the least sum of node as start point next round and avoiding ac- count for loop. It will cuckoo the three steps until finishing the spanning tree [1, 8]. 3.2 Prim’s Minimum Spanning Tree Similar to Dijkstra’s minimum spanning tree, the theory has a start node too. It can also use the same knowledge area in network. After choosing a start node, it finishes the Learning Kruskal's Algorithm, Prim's Algorithm and Dijkstra's Algorithm by Board Game 277 minimum spanning tree by repeating the follow steps: 1.Finding the connected nodes. 2. Choosing the least weight of paths to next node, and avoiding account for loop [1, 9]. After repeating the two steps, a Prim’s minimum spanning tree is created. 3.3 Kruskal’s Minimum Spanning Tree Unlike upper two theories of minimum spanning tree, the theory needn’t any start node. The point of the theory are choosing the least paths and avoiding account for the loop. It finishes the minimum spanning tree by repeating two steps: 1.Choosing the least weight path. 2. Connecting the nodes and avoiding the loop [1, 10]. The concept can be used in network, too. Fig. 1. Ticket to Ride Portugal Edition 4 What Is “Ticket to Ride” This is a board game; it contains the city name and tracks from one city to another. The game has many kinds of maps like Europe, South America, and Portugal etc. An ex- ample is shown in Fig.1. The main ideas of this game are: 1.Players can choose the mission about connects one city to another, and try to finish the mission to get points. 2. Players have to use their source in hand much more efficiently. 3. There are game plans in the game; players try to interrupt others finish their missions expect for finishing their missions. During playing the game, players get points by finishing missions or building the 278 W C. Chang, Y D. Chiu, and M F. Li longest tracks. Player who gets the highest points is the winner. Altogether, this is a game needed chicanery. 5 Design of Combining Minimum Spanning Tree Theories and Ticket to Ride In the section we fixed some rules based on original rules, and created a new edition of board game Ticket to Ride. The new edition of Ticket to ride is combined with the concept of board game Ticket to Ride and three minimum spanning tree theories: Dijkstra’s, Prim’s, and Kruskal’s minimum spanning tree. 5.1 Necessary Properties of the Game In this game, we need these things: (1) Railway map (We use Portugal railway map as an example in Fig.1.) (2) Each player has 45 railway carriages with its special color. (3) Starting Point Card (It presents the start point of each player.) (4) Ticket Card (It indicates the link from Starting point station to Destination sta- tion, and the score after the player completed the route.) (5) Knowledge Card (It identifies which algorithm has to apply on the railway sta- tions and the completed score. The related algorithms are Dijkstra’s algorithm which is applied in shortest path algorithm, Prim’s algorithm and Kruskal’s al- gorithm which are used in minimum spanning tree. (6) Railway Card (Each section railway of the map has a specific color which is composed of white, black, red, yellow, orange, blue, purple, green and full-color.) (7) The Number of each color card (white, black, red, yellow, orange, blue, purple, green) are thirty five, and the number of full-color is fifteen. 5.2 The Limitation of This Game There are the limitations of this game: (1) Players range :2-3 players (2) End conditions of single round game: a. When one of the players ran out of the 45 railway carriages b. When one of the players cannot put his/her railway carriages c. When one of the players reached scores over 150 (3) Completed conditions of the entire game: (A game consists of two single round games.) a. When one of the players reached scores over 300 b. When one of the players completed all the three knowledge card missions which involved Dijkstra, Prim and Kruskal algorithms. The winner has to complete each knowledge card. . Accordingly, how to make student to learn graph theories more efficiently and boost the interest in learning for them is administer to learn computer sciences. In order to increase the interest in. 7. Galarneau, L.: The elearning edge: leveraging interactive technologies in the design of en- gaging, effective learning experiences. In: Proceedings of e-Fest, Wellington, New Zea- land (2005). network and Prim’s minimum spanning tree [1], or link the graph and the concept of searching [2]. Therefore, learning and teaching graph theory much more efficiently aid learning computer sciences.