2003 AMC 10A Problem 23

Toothpicks are shared between neighboring triangles, so counting "3 per triangle" would count many twice. Find a way to give each toothpick to exactly one triangle.

1. Count the rows. In the picture the rows have $1$, $3$, $5$ small triangles: row $k$ has $2k - 1$. A bottom row of $2003$ triangles means $2k - 1 = 2003$, so there are $k = 1002$ rows.
2. Up and down triangles. In each row the triangles alternate pointing up and pointing down, starting and ending with up. Row $k$ has $k$ up-triangles and $k - 1$ down-triangles. (Row $3$ in the picture: $3$ up and $2$ down, which makes $5$.)
3. Each toothpick belongs to one up-triangle. Look at the picture. Every horizontal toothpick is the bottom edge of an up-triangle. Every slanted toothpick is the left or right side of an up-triangle. The down-triangles have no toothpicks of their own: their three sides are borrowed from the up-triangles around them. And no two up-triangles share a side.
4. So toothpicks $= 3 \times$ (number of up-triangles). Check on the picture: $1 + 2 + 3 = 6$ up-triangles, and $3 \times 6 = 18$ toothpicks, which matches counting them directly.
5. Count up-triangles for $1002$ rows.
$$1 + 2 + \cdots + 1002 = \frac{1002 \times 1003}{2} = 501 \times 1003 = 502503.$$
6. Toothpicks. $3 \times 502503 = 1507509$.

The answer is $\boxed{\textbf{(C)}\ 1,507,509}$.

When pieces are shared between shapes, look for a family of shapes that uses every piece exactly once. Here the up-pointing triangles do that, so the count becomes a single multiplication. Testing the rule on the small picture first catches the two usual mistakes: miscounting the rows and counting inside edges twice.

Count small cases and look for the pattern. With $1, 2, 3, 4$ rows you need $3, 9, 18, 30$ toothpicks, which is $3$ times $1, 3, 6, 10$ (the sums $1$, $1+2$, $1+2+3$, $1+2+3+4$). For $1002$ rows that is $3 \times 502503$. You do not even need to finish the multiplication: $3 \times 3$ ends in $9$, and only choice (C) ends in $9$.

Row $k$ from the top contains $2k - 1$ small triangles: $k$ of them point up and $k - 1$ point down. A base row of $2003 = 2 \cdot 1002 - 1$ triangles means there are $1002$ rows.

Look only at the upward-pointing triangles. Each toothpick in the figure is a side of exactly one of them: the horizontal toothpicks are the bottoms of upward triangles, and the slanted toothpicks are their left or right sides. (A downward triangle borrows all three of its sides from neighboring upward triangles.) So no toothpick is counted twice if we count three per upward triangle.

Number of upward triangles: $1 + 2 + \cdots + 1002 = \dfrac{1002 \cdot 1003}{2} = 501 \cdot 1003 = 502503$.

Toothpicks: $3 \cdot 502503 = 1507509$.

Check on the given figure ($3$ rows): $3 \cdot (1 + 2 + 3) = 18$ toothpicks, which matches a direct count.

The answer is $\boxed{\textbf{(C)}\ 1,507,509}$.