2003 AMC 10A Problems/Problem 23
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?
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
We see that the bottom row of small triangles are formed from upward-facing triangles and downward-facing triangles. Since each upward-facing triangle uses three distinct toothpicks, and since the total number of upward-facing triangles is , we have that the total number of toothpicks is
Experimenting a bit we find that the number of toothpicks needs 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 .
In the final step of the problem, we know that the units digit of the answer must , so the only answer choice applicable must be . This saves you the time it takes to compute .
Alternatively, we can note that .~ dolphin7
We first see how many small equilateral triangles share a side with the bigger triangle, here we see there are such triangles in case of the perimeter being , after some observation, we find that if the perimeter is the no. of triangles is , we know that the perimeter when there are base triangles is, which is , so the no. of small triangles would be , hence the total no. of toothpicks is, which is . ~RMOAspirantFaraz
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