MathDB

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\ge 2

and:
• The directed line segments

P_0P_1,P_1P_2,\dots,P_{n-1}P_n

are in successive coordinate directions east (for

P_0P_1

), north, west, south, east, etc.
• The lengths of these line segments are positive and strictly increasing.
\begin{picture}(200,180)
\put(20,100){\line(1,0){160}} \put(100,10){\line(0,1){170}}
\put(0,97){West} \put(180,97){East} \put(90,0){South} \put(90,180){North}
\put(100,100){\circle{1}}\put(100,100){\circle{2}}\put(100,100){\circle{3}} \put(115,100){\circle{1}}\put(115,100){\circle{2}}\put(115,100){\circle{3}} \put(115,130){\circle{1}}\put(115,130){\circle{2}}\put(115,130){\circle{3}} \put(40,130){\circle{1}}\put(40,130){\circle{2}}\put(40,130){\circle{3}} \put(40,20){\circle{1}}\put(40,20){\circle{2}}\put(40,20){\circle{3}} \put(170,20){\circle{1}}\put(170,20){\circle{2}}\put(170,20){\circle{3}}
\multiput(100,99.5)(0,.5){3}{\line(1,0){15}} \multiput(114.5,100)(.5,0){3}{\line(0,1){30}} \multiput(40,129.5)(0,.5){3}{\line(1,0){75}} \multiput(39.5,20)(.5,0){3}{\line(0,1){110}} \multiput(40,19.5)(0,.5){3}{\line(1,0){130}}
\put(102,90){P0} \put(117,90){P1} \put(117,132){P2} \put(28,132){P3} \put(30,10){P4} \put(172,10){P5}
\end{picture}
How many of the points

(x,y)

with integer coordinates

0\le x\le 2011,0\le y\le 2011

cannot be the last point,

P_n,

of any growing spiral?

Problem Statement