2018 AMC 10B Problems/Problem 8
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 | |
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All AMC 10 Problems and Solutions |
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