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Difference between revisions of "2018 AMC 10B Problems/Problem 8"

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==Problem==
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== Problem ==
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Sara makes a staircase out of toothpicks as shown:
  
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Sara makes a staircase out of toothpicks as shown:<asy>
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<asy>
 
size(150);
 
size(150);
 
defaultpen(linewidth(0.8));
 
defaultpen(linewidth(0.8));
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filldraw(shift((j,i))*v,black);
 
filldraw(shift((j,i))*v,black);
 
}
 
}
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}</asy>
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}
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</asy>
 +
 
 
This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
 
This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
  
 
<math>\textbf{(A)}\ 10\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 30</math>
 
<math>\textbf{(A)}\ 10\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 30</math>
  
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== Solution ==
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== Solutions ==
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=== Solution 1 ===
 
A staircase with <math>n</math> steps contains <math>4 + 6 + 8 + ... + 2n + 2</math> toothpicks. This can be rewritten as <math>(n+1)(n+2) -2</math>.  
 
A staircase with <math>n</math> steps contains <math>4 + 6 + 8 + ... + 2n + 2</math> toothpicks. This can be rewritten as <math>(n+1)(n+2) -2</math>.  
  
Line 25: Line 28:
 
So, <math>(n+1)(n+2) = 182.</math>  
 
So, <math>(n+1)(n+2) = 182.</math>  
  
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Inspection could tell us that <math>13 * 14 = 182</math>, so the answer is <math>13 - 1 = \boxed {(C) 12}</math>
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Inspection could tell us that <math>13 \cdot 14 = 182</math>, so the answer is <math>13 - 1 = \boxed {(C) 12}</math>
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== Solution 2 ==
 
  
 +
=== Solution 2 ===
 
Layer <math>1</math>: <math>4</math> steps
 
Layer <math>1</math>: <math>4</math> steps
  
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By: Soccer_JAMS
 
By: Soccer_JAMS
  
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== Solution 3 ==
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=== Solution 3 ===
−
 
 
 
We can find a function that gives us the number of toothpicks for every layer. Using finite difference, we know that the degree must be <math>2</math> and the leading coefficient is <math>1</math>. The function is <math>f(n)=n^2+3n</math> where <math>n</math> is the layer and <math>f(n)</math> is the number of toothpicks.
 
We can find a function that gives us the number of toothpicks for every layer. Using finite difference, we know that the degree must be <math>2</math> and the leading coefficient is <math>1</math>. The function is <math>f(n)=n^2+3n</math> where <math>n</math> is the layer and <math>f(n)</math> is the number of toothpicks.
  
Line 54: Line 55:
 
~Zeric Hang
 
~Zeric Hang
  
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== Solution 4 ==
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=== Solution 4 ===
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Notice that the number of toothpicks can be found by adding all the horizontal and all the vertical toothpicks. We can see that for the case of 3 steps, there are <math>2(3+3+2+1)=18</math> toothpicks. Thus, the equation is <math>2S + 2(1+2+3...+S)=180</math> with <math>S</math> being the number of steps. Solving, we get <math>S = 12</math>, or <math>\boxed {(C) 12}</math>.
 
Notice that the number of toothpicks can be found by adding all the horizontal and all the vertical toothpicks. We can see that for the case of 3 steps, there are <math>2(3+3+2+1)=18</math> toothpicks. Thus, the equation is <math>2S + 2(1+2+3...+S)=180</math> with <math>S</math> being the number of steps. Solving, we get <math>S = 12</math>, or <math>\boxed {(C) 12}</math>.
 
-liu4505
 
-liu4505
  
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== Solution 5... Not a solution! Just an observation. ==
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=== Solution 5 General Formula ===
 +
There are <math>\frac{n(n+1)}{2}</math> squares. Each has <math>4</math> toothpick sides. To remove overlap, note that there are <math>4n</math> perimeter toothpicks. <math>\frac{\frac{n(n+1)}{2}\cdot 4-4n}{2}</math> is the number of overlapped toothpicks
 +
Add <math>4n</math> to get the perimeter (non-overlapping). Formula is <math>\text{number of toothpicks} = n^2+3n</math> Then you can "guess" or factor (also guessing) to get the answer <math>\boxed{\text{(C) }12}</math>.
 +
~bjc
  
 +
=== Not a solution! Just an observation. ===
 
If you are trying to look for a pattern, you can see that the first column is made of 4 toothpicks. The second one is made from 2 squares: 3 toothpicks for the first square and 4 for the second. The third one is made up of 3 squares: 3 toothpicks for the first and second one, and 4 for the third one. The pattern continues like that. So for the first one, you have 0 "3 toothpick squares" and 1 "4 toothpick squares". The second is 1 to 1. The third is 2:1. And the amount of three toothpick squares increase by one every column.
 
If you are trying to look for a pattern, you can see that the first column is made of 4 toothpicks. The second one is made from 2 squares: 3 toothpicks for the first square and 4 for the second. The third one is made up of 3 squares: 3 toothpicks for the first and second one, and 4 for the third one. The pattern continues like that. So for the first one, you have 0 "3 toothpick squares" and 1 "4 toothpick squares". The second is 1 to 1. The third is 2:1. And the amount of three toothpick squares increase by one every column.
  
 
The list is as follow for the number of toothpicks used...
 
The list is as follow for the number of toothpicks used...
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4,4+3,4+6,4+9, and so on.
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<math>4</math>,<math>4+3</math>,<math>4+6</math>,<math>4+9</math>, and so on.
 
4, 7, 10, 13, 16, 19, ...
 
4, 7, 10, 13, 16, 19, ...
  
 
- Flutterfly
 
- Flutterfly
  
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==See Also==
+
==Video Solution (HOW TO THINK CREATIVELY!!!)==
 +
https://youtu.be/8j0RvjRsjCc
 +
 
 +
~Education, the Study of Everything
 +
 
 +
 
 +
 
 +
=== Video Solution ===
 +
https://youtu.be/FbUEFq85jGE
  
 +
== See Also ==
 
{{AMC10 box|year=2018|ab=B|num-b=7|num-a=9}}
 
{{AMC10 box|year=2018|ab=B|num-b=7|num-a=9}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 13:25, 18 July 2024

Problem

Sara makes a staircase out of toothpicks as shown:

[asy] size(150); defaultpen(linewidth(0.8)); path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45); for(int i=0;i<=2;i=i+1) { for(int j=0;j<=3-i;j=j+1) { filldraw(shift((i,j))*h,black); filldraw(shift((j,i))*v,black); } } [/asy]

This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?

$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 30$

Solutions

Solution 1

A staircase with $n$ steps contains $4 + 6 + 8 + ... + 2n + 2$ toothpicks. This can be rewritten as $(n+1)(n+2) -2$.

So, $(n+1)(n+2) - 2 = 180$

So, $(n+1)(n+2) = 182.$

Inspection could tell us that $13 \cdot 14 = 182$, so the answer is $13 - 1 = \boxed {(C) 12}$

Solution 2

Layer $1$: $4$ steps

Layer $1,2$: $10$ steps

Layer $1,2,3$: $18$ steps

Layer $1,2,3,4$: $28$ steps

From inspection, we can see that with each increase in layer the difference in toothpicks between the current layer and the previous increases by $2$. Using this pattern:

$4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180$

From this we see that the solution is $\boxed {(C) 12}$

By: Soccer_JAMS

Solution 3

We can find a function that gives us the number of toothpicks for every layer. Using finite difference, we know that the degree must be $2$ and the leading coefficient is $1$. The function is $f(n)=n^2+3n$ where $n$ is the layer and $f(n)$ is the number of toothpicks.


We have to solve for $n$ when $n^2+3n=180\Rightarrow n^2+3n-180=0$. Factor to get $(n-12)(n+15)$. The roots are $12$ and $-15$. Clearly $-15$ is impossible so the answer is $\boxed {(C) 12}$.

~Zeric Hang

Solution 4

Notice that the number of toothpicks can be found by adding all the horizontal and all the vertical toothpicks. We can see that for the case of 3 steps, there are $2(3+3+2+1)=18$ toothpicks. Thus, the equation is $2S + 2(1+2+3...+S)=180$ with $S$ being the number of steps. Solving, we get $S = 12$, or $\boxed {(C) 12}$. -liu4505

Solution 5 General Formula

There are $\frac{n(n+1)}{2}$ squares. Each has $4$ toothpick sides. To remove overlap, note that there are $4n$ perimeter toothpicks. $\frac{\frac{n(n+1)}{2}\cdot 4-4n}{2}$ is the number of overlapped toothpicks Add $4n$ to get the perimeter (non-overlapping). Formula is $\text{number of toothpicks} = n^2+3n$ Then you can "guess" or factor (also guessing) to get the answer $\boxed{\text{(C) }12}$. ~bjc

Not a solution! Just an observation.

If you are trying to look for a pattern, you can see that the first column is made of 4 toothpicks. The second one is made from 2 squares: 3 toothpicks for the first square and 4 for the second. The third one is made up of 3 squares: 3 toothpicks for the first and second one, and 4 for the third one. The pattern continues like that. So for the first one, you have 0 "3 toothpick squares" and 1 "4 toothpick squares". The second is 1 to 1. The third is 2:1. And the amount of three toothpick squares increase by one every column.

The list is as follow for the number of toothpicks used... $4$,$4+3$,$4+6$,$4+9$, and so on. 4, 7, 10, 13, 16, 19, ...

- Flutterfly

Video Solution (HOW TO THINK CREATIVELY!!!)

https://youtu.be/8j0RvjRsjCc

~Education, the Study of Everything


Video Solution

https://youtu.be/FbUEFq85jGE

See Also

2018 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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All AMC 10 Problems and Solutions

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