Question 24
Aaron the ant walks on the coordinate plane according to the following rules.
He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0).$
For $n=1,$ $2,$ $3,$ $\dots,$ right after arriving at the point $p_n,$ if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1},$ he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}.$ Thus the sequence of points continues $$
\begin{aligned} &p_2=(1,1),\; p_3=(0,1), \\ &p_4=(-1,1),\; p_5=(-1,0),\;\ldots \end{aligned}
$$ in a counterclockwise spiral pattern. What is $p_{2015}?$
He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0).$
For $n=1,$ $2,$ $3,$ $\dots,$ right after arriving at the point $p_n,$ if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1},$ he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}.$ Thus the sequence of points continues $$
\begin{aligned} &p_2=(1,1),\; p_3=(0,1), \\ &p_4=(-1,1),\; p_5=(-1,0),\;\ldots \end{aligned}
$$ in a counterclockwise spiral pattern. What is $p_{2015}?$