2015 AMC 10A Problems/Problem 3
Contents
[hide]Problem
Ann made a -step staircase using toothpicks as shown in the figure. How many toothpicks does she need to add to complete a -step staircase?
Solution
We can see that a -step staircase requires toothpicks and a -step staircase requires toothpicks. Thus, to go from a -step to -step staircase, additional toothpicks are needed and to go from a -step to -step staircase, additional toothpicks are needed. Applying this pattern, to go from a -step to -step staircase, additional toothpicks are needed and to go from a -step to -step staircase, additional toothpicks are needed. Our answer is
Solution 2
Alternatively, we can see with the -step staircase has toothpicks. Generalizing, we see that a staircase with steps has toothpicks. So, for steps, we have toothpicks. So our answer is or .
Solution 3
If one is too lazy to derive a formula for the number of picks needed for a given number of steps, one can simply see that to get to steps, we add two blobs that have three picks each (the top and the right), and two more blobs that have two blocks each to form the steps. This adds picks. Then, to get to steps, we add two more edge blobs with picks each and more blobs that have two picks each. We add more for a total increase of
~Technodoggo
Video Solution (CREATIVE THINKING)
https://youtu.be/StaCPwJ9zSU
~Education, the Study of Everything
Video Solution
~savannahsolver
See Also
2015 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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All AMC 10 Problems and Solutions |
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