Difference between revisions of "2018 AMC 10B Problems/Problem 8"
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== Solution == | == Solution == | ||
| − | A staircase with <math>n</math> steps contains <math>4 + 6 + 8 + ... + 2n | + | 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> | So, <math>(n+1)(n+2) - 2 = 180</math> | ||
Revision as of 13:52, 20 February 2018
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]](http://latex.artofproblemsolving.com/6/4/d/64dae9d5742d72398d8f4ea4711a8da0029f8bac.png)
This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
Solution
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 indeed 
By: Soccer_JAMS
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. 
