2011 Putnam A1

Define a *growing spiral* in the plane to be a sequence
of points with integer coordinates $P_0 = (0,0), P_1, \dots, P_n$ such
that $n \geq 2$ and:

\item the directed line segments $P_0 P_1, P_1 P_2, \dots, P_{n-1} P_n$

are in the successive coordinate directions east (for $P_0 P_1$), north,

west, south, east, etc.;

\item the lengths of these line segments are positive and strictly

increasing.

[Picture omitted.] How many of the points $(x,y)$ with

integer coordinates $0\leq x\leq 2011, 0\leq y\leq 2011$ *cannot*

be the last point, $P_n$ of any growing spiral?

将平面中的*生长螺旋*定义为序列

具有整数坐标的点 $P_0 = (0,0), P_1, \dots, P_n$ 这样

$n \geq 2$ 和：

\item 有向线段 $P_0 P_1, P_1 P_2, \dots, P_{n-1} P_n$
位于连续的坐标方向东($P_0 P_1$)、北、
西、南、东等；
\item 这些线段的长度是正数并且严格
增加。
[图片省略。] 多少个点 $(x,y)$ 与
整数坐标 $0\leq x\leq 2011, 0\leq y\leq 2011$ *不能*
是任何增长螺旋的最后一个点，$P_n$？