Difference between revisions of "2003 AMC 10A Problems/Problem 23"
(added pic) |
(→Solution) |
||
| Line 6: | Line 6: | ||
<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> | ||
| − | == Solution == | + | == Solution 1 == |
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. | ||
| Line 16: | Line 16: | ||
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> | ||
| + | == Solution 2 == | ||
| + | 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 == | ||
*[[2003 AMC 10A Problems]] | *[[2003 AMC 10A Problems]] | ||
Revision as of 18:57, 4 February 2007
Contents
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 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
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. 
is the only odd answer choice.
