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标题: Concave n-gon [打印本页]

作者: yma16 时间: 2005-8-17 19:33
标题: Concave n-gon

1 Can you find a formula to compute the max number of angles that are bigger than 180 degree in concave n-gon? For example, when n=4, the answer is 1; when n=5 or 6, the answer is 2.

2 In a concave n-gon, no 3 points are colinear. Is it sometimes possible to use the vertices of this n-gon to form a new n-gon such that the number of angles that are bigger than 180 degree is different from the old n-gon?

Note: the angles are inner angles which is the default meaning.

作者: husonghu 时间: 2005-8-18 05:32
标题: yma16, there is something I can't understand .....（图）

In your question #1, you said, "when n=5 or 6, the answer is 2". Why when n=6, the max number of angles that are bigger than 180 degree is not 3? Seems we can draw a concave 6-gon with the max number of angles that are bigger than 180 degree being 3 easily, see below (I don't know if my plot can be shown or not):

作者: yma16 时间: 2005-8-18 17:21
标题: 回复：yma16, there is something I can't understand ..

Sorry, that is my mistake. The formula is n-3.

For part 2, you can try points (2,1), (-2,1), (1,0), (-1,0), (2,-1), and (-2, -1). These points can give you 2 angles > 180 degree or 1 angles > 180 degree.

How come I cannot see the picture? It is a red x.

Thank you for your interest.

作者: 寒潭清 时间: 2005-8-18 17:47
标题: 这题正在思考中...我也看不到图片.[:)]

这题正在思考中...我也看不到图片.

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