Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.
[AIME 2002 II Problem 4.gif](https://artofproblemsolving.com/wiki/index.php/File:AIME_2002_II_Problem_4.gif)
If $n=202$, then the area of the garden enclosed by the path, not including the path itself, is $m\left(\sqrt3/2\right)$ square units, where $m$ is a positive integer. Find the remainder when $m$ is divided by $1000$.
When $n>1$, the path of blocks has $6(n-1)$ blocks total in it. When $n=1$, there is just one lonely block. Thus, the area of the garden enclosed by the path when $n=202$ is
\[(1+6+12+18+\cdots +1200)A=(1+6(1+2+3...+200))A\],
where $A$ is the area of one block. Then, because $n(n+1)/2$ is equal to the sum of the first $n$ integers:
\[(1+6(1+2+3...+200))=(1+6((200)(201)/2))A=120601A\].
Since $A=\dfrac{3\sqrt{3}}{2}$, the area of the garden is
\[120601\cdot \dfrac{3\sqrt{3}}{2}=\dfrac{361803\sqrt{3}}{2}\].
$m=361803$, $\dfrac{m}{1000}=361$ Remainder $\boxed{803}$.