有幾個三角形？

How Many Triangles?

最近娛樂新聞中，有個圖形遊戲的問題，引發廣泛的關注，在電視節目中，藉由訪談，已有部分的解答。但是究竟有幾個三角形呢？其實，由不同的定義與原理審視，答案並不相同。本文之目的在提供以圖論為基礎的觀點所探討的成果。在空間資訊的應用中，「位相」（Topology）的分析，即是以此為發展基本之作業方法。故而，此一議題，應不只有遊戲的層面，還有實際上應用的潛勢。究竟有幾個三角形呢？由圖形遊戲的觀點，目前電視節目中訪談所得答案是24個。但是如果包涵零面積之三角形，應有46 個。而由位相理論看，答案則一個也沒有或只有一個。若將所有交點均視為節點（Node），所有的其間線段均視為「邊」（Link, Arc, Edge，連接、弧），則有5 個、或6 個。 There is a game of graphics attracted public attention and has been reported in TV news in March, 2015. Through interview with math teachers, a partial solution to this game was revealed in the news.But, “How many triangles are in the figure”? The resulting solution differs with the definition. Thiswriting intends to provide a view from graph theory. In the spatial analysis of Geographic InformationSystems (GIS), “topology” provides a suite of fundamental tool, which is largely based on graph theory.Subsequently, the subject of graphics is not only having entertainment meaning, but also potential forpractical application.“How many triangles are in the figure”? From the view of an entertaining game, the answer is 24as provided in the TV interview. But, if the triangles with zero area were counted, the answer wouldbe 46. From topological view, there is either no triangle or one. If all the intersections in the figureare taken as node and all line segments between them are taken as edge (link, arc), the answer wouldbe either 5 or 6.

There is a game of graphics attracted public attention and has been reported in TV news in March,2015. Through interview with math teachers, a partial solution to this game was revealed in the news.But, “How many triangles are in the figure”? The resulting solution differs with the definition. Thiswriting intends to provide a view from graph theory. In the spatial analysis of Geographic InformationSystems (GIS), “topology” provides a suite of fundamental tool, which is largely based on graph theory.Subsequently, the subject of graphics is not only having entertainment meaning, but also potential forpractical application.“How many triangles are in the figure”? From the view of an entertaining game, the answer is 24as provided in the TV interview. But, if the triangles with zero area were counted, the answer wouldbe 46. From topological view, there is either no triangle or one. If all the intersections in the figureare taken as node and all line segments between them are taken as edge (link, arc), the answer wouldbe either 5 or 6.