Difference between revisions of "2018 AMC 10B Problems/Problem 8"
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− | Sara makes a staircase out of toothpicks as shown:<asy> | + | == Problem == |
+ | Sara makes a staircase out of toothpicks as shown: | ||
+ | |||
+ | <asy> | ||
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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> | ||
− | == Solution == | + | == Solutions == |
− | + | === 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 23: | Line 28: | ||
So, <math>(n+1)(n+2) = 182.</math> | So, <math>(n+1)(n+2) = 182.</math> | ||
− | Inspection could tell us that <math>13 | + | Inspection could tell us that <math>13 \cdot 14 = 182</math>, so the answer is <math>13 - 1 = \boxed {(C) 12}</math> |
− | |||
− | |||
+ | === Solution 2 === | ||
Layer <math>1</math>: <math>4</math> steps | Layer <math>1</math>: <math>4</math> steps | ||
Line 39: | Line 43: | ||
<math> 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180 </math> | <math> 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180 </math> | ||
− | From this we see that the solution is | + | From this we see that the solution is <math>\boxed {(C) 12}</math> |
By: Soccer_JAMS | 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>. | ||
+ | |||
+ | ~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 <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 | ||
+ | |||
+ | === 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. | ||
+ | |||
+ | The list is as follow for the number of toothpicks used... | ||
+ | <math>4</math>,<math>4+3</math>,<math>4+6</math>,<math>4+9</math>, 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 == | ||
{{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 14:25, 18 July 2024
Contents
Problem
Sara makes a staircase out of toothpicks as shown:
This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
Solutions
Solution 1
A staircase with steps contains
toothpicks. This can be rewritten as
.
So,
So,
Inspection could tell us that , so the answer is
Solution 2
Layer :
steps
Layer :
steps
Layer :
steps
Layer :
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 . Using this pattern:
From this we see that the solution is
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 and the leading coefficient is
. The function is
where
is the layer and
is the number of toothpicks.
We have to solve for when
. Factor to get
. The roots are
and
. Clearly
is impossible so the answer is
.
~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 toothpicks. Thus, the equation is
with
being the number of steps. Solving, we get
, or
.
-liu4505
Solution 5 General Formula
There are squares. Each has
toothpick sides. To remove overlap, note that there are
perimeter toothpicks.
is the number of overlapped toothpicks
Add
to get the perimeter (non-overlapping). Formula is
Then you can "guess" or factor (also guessing) to get the answer
.
~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...
,
,
,
, and so on.
4, 7, 10, 13, 16, 19, ...
- Flutterfly
Video Solution (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
Video Solution
See Also
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 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.