Question

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 $p_2=(1,1),$ $p_3=(0,1),$ $p_4=(-1,1),$ $p_5=(-1,0),$ $\vdots$ and so on in a counterclockwise spiral pattern. What is $p_{2015}?$

$(-22,-13)$

$(-13,-22)$

$(-13,22)$

$(13,-22)$

$(22,-13)$

Solution:

When Aaron reaches $(k,-k)$ , he has just completed the square spiral containing all grid points with coordinates between $-k$ and $k$ . Therefore $p_{(2k+1)^2-1}=(k,-k).$

With $k=22$ , this gives $p_{2024}=(22,-22)$ . Since $2024-2015=9$ , stepping backward along the bottom edge subtracts $9$ from the $x$ -coordinate, so $p_{2015}=(13,-22).$

Thus, the correct answer is D.

2015 AMC 10B Problem 24

Solution:

Problem 24 in Other Years