2002 AIME II Problems/Problem 4

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Problem

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

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 $1000$.

Solution 1

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 $\boxed{803}$.

Solution 2

Note that this is just the definition for a centered hexagonal number, and the formula for $(n-1)^{th}$ term is $3n(n+1)+1$. Applying this for $200$ as we want the inner area gives $120601$. Then continue as above.

See also

2002 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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