Difference between revisions of "2018 AMC 10B Problems/Problem 8"
Soccer jams (talk | contribs) (→Solution) |
Soccer jams (talk | contribs) (→Solution 2) |
||
Line 27: | Line 27: | ||
== Solution 2 == | == Solution 2 == | ||
− | Layer 1: <math>4</math> steps | + | Layer <math>1</math>: <math>4</math> steps |
− | Layer 1 | + | Layer <math>1 and 2</math>: <math>10</math> steps |
− | Layer 1,2 | + | Layer <math>1,2 and 3</math>: <math>18</math> steps |
− | Layer 1,2,3 | + | Layer <math>1,2,3 and 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 2. Using this pattern: | 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: |
Revision as of 16:55, 16 February 2018
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?
Solution
A staircase with steps contains
steps. 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 2. Using this pattern:
From this we see that the solution is indeed
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.