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 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.

See Also

• 2003 AMC 10A Problems
• Previous Problem
• Next Problem