MathDB

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.

[asy] size(250);int i,j; real r=sqrt(3); for(i=0; i<6; i=i+1) { for(j=0; j<4; j=j+1) { draw(shift(((j*r)*dir(60*i+150)).x, ((j*r)*dir(60*i+150)).y)*shift((4r*dir(60i+30)).x,(4r*dir(60i+30)).y)*polygon(6)); }}[/asy] If

n=202,

then the area of the garden enclosed by the path, not including the path itself, is

m(\sqrt{3}/2)

square units, where

m

is a positive integer. Find the remainder when

m

is divided by

1000.

Problem Statement