Difference between revisions of "2003 AMC 10A Problems/Problem 23"
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===Solution 3=== | ===Solution 3=== | ||
| − | Experimenting a bit we find that the number of toothpicks needs for a triangle with <math>1</math>, <math>2</math> and <math>3</math> rows is <math>1\cdot{3}</math>, <math>3\cdot{3}</math> and <math>6\cdot{3}</math> respectively. Since <math>1</math>, <math>3</math> and <math>6</math> are triangular numbers we know that depending on how many rows there are in the triangle, the number we multiply by <math>3</math> to find total no.toothpicks is the corresponding triangular number. Since the triangle in question has <math>2n-1=2003\implies{n=1002}</math> rows, we can use <math>\frac{n(n+1)}{2}</math> to find the triangular number for that row and multiply by <math>3</math>, hence finding the total no.toothpicks; this is just <math>\frac{3\cdot{1002}\cdot{1003}}{2}=3\cdot{512}\cdot{1003}=\boxed{ | + | Experimenting a bit we find that the number of toothpicks needs for a triangle with <math>1</math>, <math>2</math> and <math>3</math> rows is <math>1\cdot{3}</math>, <math>3\cdot{3}</math> and <math>6\cdot{3}</math> respectively. Since <math>1</math>, <math>3</math> and <math>6</math> are triangular numbers we know that depending on how many rows there are in the triangle, the number we multiply by <math>3</math> to find total no.toothpicks is the corresponding triangular number. Since the triangle in question has <math>2n-1=2003\implies{n=1002}</math> rows, we can use <math>\frac{n(n+1)}{2}</math> to find the triangular number for that row and multiply by <math>3</math>, hence finding the total no.toothpicks; this is just <math>\frac{3\cdot{1002}\cdot{1003}}{2}=3\cdot{512}\cdot{1003}=\boxed{\mathrm{(C)}\ 1,507,509}</math>. |
== See Also == | == See Also == | ||
Revision as of 19:30, 27 December 2012
Problem
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have 
rows of small congruent equilateral triangles, with 
small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of 
small equilateral triangles?
Solution
Solution 1
There are 
small equilateral triangles.
Each small equilateral triangle needs toothpicks to make it.
But, each toothpick that isn't one of the 
toothpicks on the outside of the large equilateral triangle is a side for 
small equilateral triangles.
So, the number of toothpicks on the inside of the large equilateral triangle is 
Therefore the total number of toothpicks is 
Solution 2
We see that the bottom row of small triangles is formed from 
downward-facing triangles and 
upward-facing triangles. Since each downward-facing triangle uses three distinct toothpicks, and since the total number of downward-facing triangles is 
, we have that the total number of toothpicks is 
Solution 3
Experimenting a bit we find that the number of toothpicks needs for a triangle with 
, and rows is 
, 
and 
respectively. Since , and 
are triangular numbers we know that depending on how many rows there are in the triangle, the number we multiply by to find total no.toothpicks is the corresponding triangular number. Since the triangle in question has 
rows, we can use 
to find the triangular number for that row and multiply by , hence finding the total no.toothpicks; this is just 
.
See Also
| 2003 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 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 | ||
