Difference between revisions of "2015 AMC 10B Problems/Problem 24"

Revision as of 21:44, 26 September 2024

Problem

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

$\textbf{(A) } (-22,-13)\qquad\textbf{(B) } (-13,-22)\qquad\textbf{(C) } (-13,22)\qquad\textbf{(D) } (13,-22)\qquad\textbf{(E) } (22,-13)$

Solution 1

The first thing we would do is track Aaron's footsteps:

He starts by taking $1$ step East and $1$ step North, ending at $(1,1)$ after $2$ steps and about to head West.

Then he takes $2$ steps West and $2$ steps South, ending at $(-1,-1$ ) after $2+4$ steps, and about to head East.

Then he takes $3$ steps East and $3$ steps North, ending at $(2,2)$ after $2+4+6$ steps, and about to head West.

Then he takes $4$ steps West and $4$ steps South, ending at $(-2,-2)$ after $2+4+6+8$ steps, and about to head East.

From this pattern, we can notice that for any integer $k \ge 1$ he's at $(-k, -k)$ after $2 + 4 + 6 + ... + 4k$ steps, and about to head East. There are $2k$ terms in the sum, with an average value of $(2 + 4k)/2 = 2k + 1$ , so:

$2 + 4 + 6 + ... + 4k = 2k(2k + 1)$

If we substitute $k = 22$ into the equation: $44(45) = 1980 < 2015$ . So he has $35$ moves to go. This makes him end up at $(-22+35,-22) = (13,-22) \implies \boxed{\textbf{(D)} (13, -22)}$ .

Solution 5 (Similar to Solution 1)

The first thing is to see the amount of footsteps at points $(k,k)$

We see that at $(1,1)$ he has taken 2 footsteps.

To get to $(2,2)$ , Aaron takes $2$ steps West, $2$ steps South, $3$ steps East, and $3$ steps North.

We can think about this in the following way. For Aaron to get from $(k,k)$ to $(k+1,k+1)$ , he takes 2k+2k+(2k+1)+(2k+1) steps = 8k+2 steps

We can find the total steps taken to get to a point $(k+1,k+1) as 2+10+18... +8k+2 = (4k+2)(k+1).

We find that to get to (22,22) this is 2+10+18...+ (4(21)+2)(21)= 1892. We then simply go 88 steps to get to$ (Error compiling LaTeX. Unknown error_msg)(-22,-22) $and 35 more to get to the 2015th step at$ (13,-22)$ or answer D

Solution 2

We are given that Aaron starts at $(0, 0)$ , and we note that his net steps follow the pattern of $+1$ in the $x$ -direction, $+1$ in the $y$ -direction, $-2$ in the $x$ -direction, $-2$ in the $y$ -direction, $+3$ in the $x$ -direction, $+3$ in the $y$ -direction, and so on, where we add odd and subtract even.

We want $2 + 4 + 6 + 8 + ... + 2n = 2015$ , but it does not work out cleanly. Instead, we get that $2 + 4 + 6 + ... + 2(44) = 1980$ , which means that there are $35$ extra steps past adding $-44$ in the $x$ -direction (and the final number we add in the $y$ -direction is $-44$ ).

So $p_{2015} = (0+1-2+3-4+5...-44+35, 0+1-2+3-4+5...-44)$ .

We can group $1-2+3-4+5...-44$ as $(1-2)+(3-4)+(5-6)+...+(43-44) = -22$ .

Thus $p_{2015} = \boxed{\textbf{(D)}\; (13, -22)}$ .

Solution 3

Looking at his steps, we see that he walks in a spiral shape. At the $8$ th step, he is on the bottom right corner of the $3\times 3$ square centered on the origin. On the $24$ th step, he is on the bottom right corner of the $5\times 5$ square centered at the origin. It seems that the $p_{n^2-1}$ is the bottom right corner of the $n\times n$ square. This makes sense since, after $n^2-1$ , he has been on $n^2$ dots, including the point $p_0$ . Also, this is only for odd $n$ , because starting with the $1\times 1$ square, we can only add one extra set of dots to each side, so we cannot get even $n$ . Since $45^2=2025$ , $p_{2024}$ is the bottom right corner of the $45\times 45$ square. This point is $\frac{45-1}{2}=22$ over to the right, and therefore $22$ down, so $p_{2024}=(22, -22)$ . Since $p_{2024}$ is $9$ ahead of $p_{2015}$ , we go back $9$ spaces to $\boxed{\textbf{(D)}\; (13, -22)}$ .

Solution 4 (similar to 3)

We call $p_0$ the first point, $p_1$ the second, and so on. As in Solution 3, we see that he walks in a counterclockwise spiral. We see that his path is traced out by a series of squares with odd-length sides that contain each other. At step $3^2=9,$ he is at $(1,-1)$ ; at step $5^2=25,$ he is at $(2,-2)$ ; and so on. We see that at step $n^2$ where $n$ is an odd number, he is at $(x,-x)$ where $x=\dfrac{n-1}2.$ The closest $n^2$ to $2016$ (we want $p_{2015},$ which is the $2016$ th step) is $45^2=2025,$ so we let $n=45.$ On the $45^2=2025$ th step, $x=\dfrac{45-1}2=22,$ so he is at $(22,-22).$ We see that he approached $(22,-22)$ from the left, so backtrack $2025-2016=9$ steps to the left (i.e. subtracting $9$ from the x-coordinate). Thus, we have $(22-9,-22)=\boxed{\textbf{(D)}~(13,-22).}$ ~Technodoggo

@@ Line 20: / Line 20: @@
 If we substitute <math>k = 22</math> into the equation:  <math>44(45) = 1980 < 2015</math>.  So he has <math>35</math> moves to go.  This makes him end up at <math>(-22+35,-22) = (13,-22) \implies \boxed{\textbf{(D)} (13, -22)}</math>.
+==Solution 5 (Similar to Solution 1)==
+The first thing is to see the amount of footsteps at points <math>(k,k)</math>
+We see that at <math>(1,1)</math> he has taken 2 footsteps.
+To get to <math>(2,2)</math>, Aaron takes <math>2</math> steps West, <math>2</math> steps South, <math>3</math> steps East, and <math>3</math> steps North.
+We can think about this in the following way. For Aaron to get from <math>(k,k)</math> to <math>(k+1,k+1)</math>, he takes 2k+2k+(2k+1)+(2k+1) steps = 8k+2 steps
+We can find the total steps taken to get to a point <math>(k+1,k+1) as 2+10+18... +8k+2 = (4k+2)(k+1).
+We find that to get to (22,22) this is 2+10+18...+ (4(21)+2)(21)= 1892. We then simply go 88 steps to get to </math>(-22,-22)<math> and 35 more to get to the 2015th step at </math>(13,-22)$ or answer D
 ==Solution 2==

2015 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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