Question

Patio blocks that are regular 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 $n = 5.$

If $n = 202,$ then the area of the garden enclosed by the path, not including the path itself, is $m\left(\sqrt{3}/2\right)$ square units, where $m$ is a positive integer. Find the remainder when $m$ is divided by $1000.$

Answer

Solution:

The garden enclosed by the path is itself a hexagonal arrangement of unit hexagons with $n - 1$ on each side. Counting from the center outward in rings of $6, 12, \ldots$ hexagons, it contains $1 + 6 + 12 + \cdots + 6(n-2) = 1 + 3(n-2)(n-1)$ blocks, which for $n = 202$ is $1 + 3 \cdot 200 \cdot 201 = 120601.$

Each unit hexagon consists of $6$ equilateral triangles of side $1,$ so its area is $6 \cdot \frac{\sqrt{3}}{4} = 3 \cdot \frac{\sqrt{3}}{2}.$ The garden's area is therefore $3 \cdot 120601 = 361803$ times $\frac{\sqrt{3}}{2},$ so $m = 361803,$ and the remainder upon division by $1000$ is $803.$

2002 AIME II Problem 4

Solution:

Problem 4 in Other Years