Define a growing spiral in the plane to be a sequence of points with integer coordinates P0=(0,0),P1,…,Pn such that n≥2 and:• The directed line segments P0P1,P1P2,…,Pn−1Pn are in successive coordinate directions east (for P0P1), 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≤x≤2011,0≤y≤2011 cannot be the last point, Pn, of any growing spiral? Putnamanalytic geometrycollege contests