Difference between revisions of "2018 AMC 10B Problems/Problem 8"
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Inspection could tell us that <math>13 * 14 = 182</math>, so the answer is <math>13 - 1 = \boxed {(C) 12}</math> | Inspection could tell us that <math>13 * 14 = 182</math>, so the answer is <math>13 - 1 = \boxed {(C) 12}</math> | ||
+ | == Solution 2 == | ||
+ | |||
+ | Layer 1: <math>4</math> steps | ||
+ | |||
+ | Layer 1,2: <math>10</math> steps | ||
+ | |||
+ | Layer 1,2,3: <math>18</math> steps | ||
+ | |||
+ | Layer 1,2,3,4: <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: | ||
+ | |||
+ | <math> 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180 </math> | ||
+ | |||
+ | From this we see that the solution is indeed <math>\boxed {(C) 12}</math> | ||
==See Also== | ==See Also== |
Revision as of 16:52, 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 1: steps
Layer 1,2: steps
Layer 1,2,3: steps
Layer 1,2,3,4: 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 |
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