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

Revision as of 17:12, 23 October 2021

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 $13 * 14 = 182$ , 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

Not a Solution! Just an observation.

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

https://youtu.be/FbUEFq85jGE

~savannahsolver

@@ Line 1: / Line 1: @@
-Sara makes a staircase out of toothpicks as shown:<asy>
+== Problem ==
+Sara makes a staircase out of toothpicks as shown:
+<asy>
 size(150);
 defaultpen(linewidth(0.8));
@@ Line 10: / Line 13: @@
 filldraw(shift((j,i))*v,black);
 }
-}</asy>
+}
+</asy>
 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>
-== Solution ==
+== Solutions ==
+=== Solution 1 ===
-A staircase with <math>n</math> steps contains <math>4 + 6 + 8 + ... + 2n - 2</math> steps. 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>.
 So, <math>(n+1)(n+2) - 2 = 180</math>
@@ Line 25: / Line 30: @@
 Inspection could tell us that <math>13 * 14 = 182</math>, so the answer is <math>13 - 1 = \boxed {(C) 12}</math>
-== Solution 2 ==
+=== Solution 2 ===
+Layer <math>1</math>: <math>4</math> steps
+Layer <math>1,2</math>: <math>10</math> steps
+Layer <math>1,2,3</math>: <math>18</math> steps
+Layer <math>1,2,3,4</math>: <math>28</math> 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 <math>2</math>. Using this pattern:
+<math> 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180 </math>
+From this we see that the solution is <math>\boxed {(C) 12}</math>
+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 <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 have to solve for <math>n</math> when <math>n^2+3n=180\Rightarrow n^2+3n-180=0</math>. Factor to get <math>(n-12)(n+15)</math>. The roots are <math>12</math> and <math>-15</math>. Clearly <math>-15</math> is impossible so the answer is <math>\boxed {(C) 12}</math>.
-Layer 1: <math>4</math> steps
+~Zeric Hang
-Layer 1,2: <math>10</math> steps
+=== Not a Solution! Just an observation. ===
+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
-Layer 1,2,3: <math>18</math> steps
+=== 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
-Layer 1,2,3,4: <math>28</math> steps
+=== 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.
-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:
+The list is as follow for the number of toothpicks used...
+,4+3,4+6,4+9, and so on.
+, 7, 10, 13, 16, 19, ...
-<math> 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180 </math>
+- Flutterfly
-From this we see that the solution is indeed <math>\boxed {(C) 12}</math>
+=== Video Solution ===
+https://youtu.be/FbUEFq85jGE
-==See Also==
+~savannahsolver
+== See Also ==
 {{AMC10 box|year=2018|ab=B|num-b=7|num-a=9}}
 {{MAA Notice}}

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