为何没有格点正三角形、正五边形和正六边形

遇到一道题  hdu 5365

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Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 65536/65536 K (Java/Others)
Total Submission(s): 1081    Accepted Submission(s): 478

Problem Description

AFA is a girl who like runing.Today,he download an app about runing .The app can record the trace of her runing.AFA will start runing in the park.There are many chairs in the park,and AFA will start his runing in a chair and end in this chair.Between two chairs,she running in a line.she want the the trace can be a regular triangle or a square or a regular pentagon or a regular hexagon.
Please tell her how many ways can her find.
Two ways are same if the set of chair that they contains are same.

 

Input

There are multiply case.
In each case,there is a integer n(1 < = n < = 20)in a line.
In next n lines,there are two integers xi,yi(0 < = xi,yi < 9) in each line.

 

Output

Output the number of ways.

 

Sample Input

4 0 0 0 1 1 0 1 1

 

Sample Output

1

看到一个博客说地球人都知道不能构造出整点正3.5.6边形；
然而，我想了想，发现一个惊人的事实；
原来我居然是外星人！

于是我找了找，发现一个特别优雅的证明：



作者：德安城
链接：https://www.zhihu.com/question/25304120/answer/30445478
来源：知乎

假如整点正六边形存在，一定有边长最小的一个，记作 .
以 为中心，将 逆时针旋转90度，得到 。显然 也是整点。类似定义 ~ ，它们也都是整点。
如你所见， 是一个更小的整点正六边形，矛盾。
由此也可说明整点正三角形不存在。因为只要有整点正三角形，就一定有整点正六边形。
值得注意的是，这样的证明是可以推广的（ 时）。以下是五边形的情形。

另外，由此也可以证明有理数坐标的正边形不存在。因为假如存在有理数坐标的正边形，取所有这些横纵坐标分母的最小公倍数，将所有坐标均扩大倍，就得到了一个整点正边形，这与上面的证明矛盾。

数学真是特别优美的东西啊= =