2003 AMC 10A Problems/Problem 23

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

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