2002 AIME II Problems/Problem 4
Patio blocks that are hexagons unit on a side are used to outline a garden by placing the blocks edge to edge with on each side. The diagram indicates the path of blocks around the garden when .
If , then the area of the garden enclosed by the path, not including the path itself, is square units, where is a positive integer. Find the remainder when is divided by .
When , the path of blocks has blocks total in it. When , there is just one lonely block. Thus, the area of the garden enclosed by the path when is
where is the area of one block. Then, because is equal to the sum of the first integers:
Since , the area of the garden is
, Remainder .
Note that this is just the definition for a centered hexagonal number, and directly applying the formula for term is . Applying this for as we want the inner area gives . Then continue as above.
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