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Difference between revisions of "2015 AMC 10B Problems/Problem 24"
 
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==Problem:==
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==Problem==
 
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin <math>p_0=(0,0)</math> facing to the east and walks one unit, arriving at <math>p_1=(1,0)</math>. For <math>n=1,2,3,\dots</math>, right after arriving at the point <math>p_n</math>, if Aaron can turn <math>90^\circ</math> left and walk one unit to an unvisited point <math>p_{n+1}</math>, he does that. Otherwise, he walks one unit straight ahead to reach <math>p_{n+1}</math>. Thus the sequence of points continues <math>p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)</math>, and so on in a counterclockwise spiral pattern. What is <math>p_{2015}</math>?
 
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin <math>p_0=(0,0)</math> facing to the east and walks one unit, arriving at <math>p_1=(1,0)</math>. For <math>n=1,2,3,\dots</math>, right after arriving at the point <math>p_n</math>, if Aaron can turn <math>90^\circ</math> left and walk one unit to an unvisited point <math>p_{n+1}</math>, he does that. Otherwise, he walks one unit straight ahead to reach <math>p_{n+1}</math>. Thus the sequence of points continues <math>p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)</math>, and so on in a counterclockwise spiral pattern. What is <math>p_{2015}</math>?
  
 
<math> \textbf{(A) } (-22,-13)\qquad\textbf{(B) } (-13,-22)\qquad\textbf{(C) } (-13,22)\qquad\textbf{(D) } (13,-22)\qquad\textbf{(E) } (22,-13) </math>
 
<math> \textbf{(A) } (-22,-13)\qquad\textbf{(B) } (-13,-22)\qquad\textbf{(C) } (-13,22)\qquad\textbf{(D) } (13,-22)\qquad\textbf{(E) } (22,-13) </math>
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==Solution:==
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 +
==Solution 1==
 +
The first thing we would do is track Aaron's footsteps:
 +
 
 +
He starts by taking <math>1</math> step East and <math>1</math> step North, ending at <math>(1,1)</math> after <math>2</math> steps and about to head West.
 +
 
 +
Then he takes <math>2</math> steps West and <math>2</math> steps South, ending at <math>(-1,-1</math>) after <math>2+4</math> steps, and about to head East.
 +
 
 +
Then he takes <math>3</math> steps East and <math>3</math> steps North, ending at <math>(2,2)</math> after <math>2+4+6</math> steps, and about to head West.
 +
 
 +
Then he takes <math>4</math> steps West and <math>4</math> steps South, ending at <math>(-2,-2)</math> after <math>2+4+6+8</math> steps, and about to head East.
 +
 
 +
From this pattern, we can notice that for any integer <math>k \ge 1</math> he's at <math>(-k, -k)</math> after <math>2 + 4 + 6 + ... + 4k</math> steps, and about to head East.  There are <math>2k</math> terms in the sum, with an average value of <math>(2 + 4k)/2 = 2k + 1</math>, so:
 +
 
 +
<cmath>2 + 4 + 6 + ... + 4k = 2k(2k + 1)</cmath>
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 +
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>.
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 +
==Solution 2==
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We are given that Aaron starts at <math>(0, 0)</math>, and we note that his net steps follow the pattern of <math>+1</math> in the <math>x</math>-direction, <math>+1</math> in the <math>y</math>-direction, <math>-2</math> in the <math>x</math>-direction, <math>-2</math> in the <math>y</math>-direction, <math>+3</math> in the <math>x</math>-direction, <math>+3</math> in the <math>y</math>-direction, and so on, where we add odd and subtract even.
 +
 
 +
We want <math>2 + 4 + 6 + 8 + ... + 2n = 2015</math>, but it does not work out cleanly. Instead, we get that <math>2 + 4 + 6 + ... + 2(44) = 1980</math>, which means that there are <math>35</math> extra steps past adding <math>-44</math> in the <math>x</math>-direction (and the final number we add in the <math>y</math>-direction is <math>-44</math>).
 +
 
 +
So <math>p_{2015} = (0+1-2+3-4+5...-44+35, 0+1-2+3-4+5...-44)</math>.
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 +
We can group <math>1-2+3-4+5...-44</math> as <math>(1-2)+(3-4)+(5-6)+...+(43-44) = -22</math>.
 +
 
 +
Thus <math>p_{2015} = \boxed{\textbf{(D)}\; (13, -22)}</math>.
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==Solution 3==
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Looking at his steps, we see that he walks in a spiral shape. At the <math>8</math>th step, he is on the bottom right corner of the <math>3\times 3</math> square centered on the origin. On the <math>24</math>th step, he is on the bottom right corner of the <math>5\times 5</math> square centered at the origin. It seems that the <math>p_{n^2-1}</math> is the bottom right corner of the <math>n\times n</math> square. This makes sense since, after <math>n^2-1</math>, he has been on <math>n^2</math> dots, including the point <math>p_0</math>. Also, this is only for odd <math>n</math>, because starting with the <math>1\times 1</math> square, we can only add one extra set of dots to each side, so we cannot get even <math>n</math>. Since <math>45^2=2025</math>, <math>p_{2024}</math> is the bottom right corner of the <math>45\times 45</math> square. This point is <math>\frac{45-1}{2}=22</math> over to the right, and therefore <math>22</math> down, so <math>p_{2024}=(22, -22)</math>. Since <math>p_{2024}</math> is <math>9</math> ahead of <math>p_{2015}</math>, we go back <math>9</math> spaces to <math>\boxed{\textbf{(D)}\; (13, -22)}</math>.
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==Solution 4 (similar to 3)==
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We call <math>p_0</math> the first point, <math>p_1</math> 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 <math>3^2=9,</math> he is at <math>(1,-1)</math>; at step <math>5^2=25,</math> he is at <math>(2,-2)</math>; and so on. We see that at step <math>n^2</math> where <math>n</math> is an odd number, he is at <math>(x,-x)</math> where <math>x=\dfrac{n-1}2.</math> The closest <math>n^2</math> to <math>2016</math> (we want <math>p_{2015},</math> which is the <math>2016</math>th step) is <math>45^2=2025,</math> so we let <math>n=45.</math> On the <math>45^2=2025</math>th step, <math>x=\dfrac{45-1}2=22,</math> so he is at <math>(22,-22).</math> We see that he approached <math>(22,-22)</math> from the left, so backtrack <math>2025-2016=9</math> steps to the left (i.e. subtracting <math>9</math> from the x-coordinate). Thus, we have <math>(22-9,-22)=\boxed{\textbf{(D)}~(13,-22).}</math>
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~Technodoggo
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==Solution 5 (Similar to Solution 1)==
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The first thing is to see the amount of footsteps at points <math>(k,k)</math>
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We see that at <math>(1,1)</math> he has taken 2 footsteps.
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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 <math>2k+2k+(2k+1)+(2k+1)</math> steps = <math>8k+2</math> steps
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We can find the total steps taken to get to a point <math>(k+1,k+1)</math>
 +
 
 +
<math>2+10+18... +8k+2</math> = <math>(4k+2)(k+1</math>) works as a formula
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We find that to get to <math>(22,22)</math> this is 2+10+18...+ (4(21)+2)(21)= 1892. We then simply go <math>88</math> steps to get to <math>(-22,-22)</math>
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and <math>35</math> more to get to the 2015th step at <math>(13,-22)</math> or answer D
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==Solution 6==
 +
If we graph <math>p_n</math> for a few values of <math>n</math>, we begin to see a particular pattern for <math>p_n</math> when <math>n</math> is a perfect square. For example, <math>p_0=(0,0)</math>, <math>p_4=(-1,1)</math>, <math>p_{16}=(2,-2)</math>, and so on. Similarly, <math>p_1=(1,0)</math>, <math>p_9=(2,-1)</math>, <math>p_{25}=(3,-2)</math>, and so on. Using this, we can generate a pattern for even and odd squares: <cmath>p_{(2n)^2}=(-n, n)</cmath> <cmath>p_{(2n+1)^2}=(n+1,-n)</cmath>
 +
 
 +
We notice that <math>p_{2015}</math> is quite close to <math>p_{2025}=p_{45^2}</math>. By the same pattern for odd squares, <math>p_{2025}=(23,-22)</math>. <math>p_{2015}</math> is just <math>10</math> units to the left of <math>p_{2025}</math>. Therefore, <math>p_{2015}=\boxed{\textbf{(D) }(13,-22)}</math>.
 +
 
 +
~[https://artofproblemsolving.com/wiki/index.php/User:Sid2012 sid2012]
  
 
==See Also==
 
==See Also==
−
{{AMC10 box|year=2015|ab=B|num-b=19|num-a=21}}
+
{{AMC10 box|year=2015|ab=B|num-b=23|num-a=25}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 22:36, 4 November 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 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

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 $(k+1,k+1)$

$2+10+18... +8k+2$ = $(4k+2)(k+1$) works as a formula

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 $(-22,-22)$

and $35$ more to get to the 2015th step at $(13,-22)$ or answer D

Solution 6

If we graph $p_n$ for a few values of $n$, we begin to see a particular pattern for $p_n$ when $n$ is a perfect square. For example, $p_0=(0,0)$, $p_4=(-1,1)$, $p_{16}=(2,-2)$, and so on. Similarly, $p_1=(1,0)$, $p_9=(2,-1)$, $p_{25}=(3,-2)$, and so on. Using this, we can generate a pattern for even and odd squares: \[p_{(2n)^2}=(-n, n)\] \[p_{(2n+1)^2}=(n+1,-n)\]

We notice that $p_{2015}$ is quite close to $p_{2025}=p_{45^2}$. By the same pattern for odd squares, $p_{2025}=(23,-22)$. $p_{2015}$ is just $10$ units to the left of $p_{2025}$. Therefore, $p_{2015}=\boxed{\textbf{(D) }(13,-22)}$.

~sid2012

See Also

2015 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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