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Difference between revisions of "2003 AMC 10A Problems/Problem 23"

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<math> \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 </math>
 
<math> \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 </math>
  
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== Solution 1 ==
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== Solution ==
 
There are <math>1+3+5+...+2003=1002^{2}=1004004</math> small equilateral triangles.  
 
There are <math>1+3+5+...+2003=1002^{2}=1004004</math> small equilateral triangles.  
  
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Therefore the total number of toothpicks is <math>1504503+3006=1,507,509 \Rightarrow C</math>
 
Therefore the total number of toothpicks is <math>1504503+3006=1,507,509 \Rightarrow C</math>
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== Solution 2 ==
 
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Notice that for every large equilateral triangle made of smaller equilateral triangles, the first small equilateral triangle needs 3 toothpicks, and every other triangle is built from that first triangle and needs only 2 toothpicks. Thus the answer is 3 plus some multiple of 2 so it must be odd. <math>C</math> is the only odd answer choice.
 
  
 
== See Also ==
 
== See Also ==

Revision as of 19:01, 4 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 $1504503+3006=1,507,509 \Rightarrow C$

See Also

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