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Difference between revisions of "2003 AMC 10A Problems/Problem 23"
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== Problem ==
 
== Problem ==
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A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have <math>3</math> rows of small congruent equilateral triangles, with <math>5</math> 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 <math>2003</math> small equilateral triangles?  
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A large [[equilateral triangle]] is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have <math>3</math> rows of small congruent equilateral triangles, with <math>5</math> 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 <math>2003</math> small equilateral triangles?  
  
 
[[Image:2003amc10a23.gif]]
 
[[Image:2003amc10a23.gif]]
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So, the number of toothpicks on the inside of the large equilateral triangle is <math>\frac{10040004\cdot3-3006}{2}=1504503</math>
 
So, the number of toothpicks on the inside of the large equilateral triangle is <math>\frac{10040004\cdot3-3006}{2}=1504503</math>
  
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Therefore the total number of toothpicks is <math>1504503+3006=1,507,509 \Rightarrow C</math>
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Therefore the total number of toothpicks is <math>1504503+3006=1,507,509 \Rightarrow \mathrm{(C)}</math>
  
 
== See Also ==
 
== See Also ==
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*[[2003 AMC 10A Problems/Problem 24|Next Problem]]
 
*[[2003 AMC 10A Problems/Problem 24|Next Problem]]
  
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[[Category:Introductory Algebra Problems]]
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[[Category:Introductory Combinatorics Problems]]

Revision as of 16:01, 6 February 2007

Problem

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $3$ rows of small congruent equilateral triangles, with $5$ 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 $2003$ small equilateral triangles?

2003amc10a23.gif

$\mathrm{(A) \ } 1,004,004 \qquad \mathrm{(B) \ } 1,005,006 \qquad \mathrm{(C) \ } 1,507,509 \qquad \mathrm{(D) \ } 3,015,018 \qquad \mathrm{(E) \ } 6,021,018$

Solution

There are $1+3+5+...+2003=1002^{2}=1004004$ small equilateral triangles.

Each small equilateral triangle needs $3$ toothpicks to make it.

But, each toothpick that isn't one of the $1002\cdot3=3006$ toothpicks on the outside of the large equilateral triangle is a side for $2$ small equilateral triangles.

So, the number of toothpicks on the inside of the large equilateral triangle is $\frac{10040004\cdot3-3006}{2}=1504503$

Therefore the total number of toothpicks is $1504503+3006=1,507,509 \Rightarrow \mathrm{(C)}$

See Also

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