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

Revision as of 23:27, 19 January 2019

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?

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

Solution

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

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 $B$ . -liu4505

@@ Line 54: / Line 54: @@
 == 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 <math>2(3+3+2+1)=18</math> 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 B.
+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>B</math>.
 -liu4505

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