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.
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 the number of ways.
假如整点正六边形存在,一定有边长最小的一个,记作
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,然而我无能为力啊= =