Brain Teaser

Question

Regular hexagon patio blocks of area 1 are used to outline the garden shown, with n=5 blocks on each side placed edge to edge.

If n=202, then what is the area of the garden enclosed by the path?


Answer

When n=5, the blocks enclose 37 hexagons of equal area to each of the tiles of area 1, and so the area of the garden is 37.

It turns out that the number of hexagons in a hexagonal arrangement is always the difference between two consecutive cubes, as shown in the visual below, where the blocks that are added to a 3x3x3 cube to make a 4x4x4 cube are in a hexagonal arrangement (which shows that the hexagonal number 37 is 43-33=37).

This pattern can be extended for any hexagonal arrangement, which means that a hexagonal garden with a side of s is made up of s3-(s-1)3 blocks. Since the length of the side of the garden is one less than n, the hexagonal area enclosed by n blocks each side is A=(n-1)3-(n-2)3.

Therefore, when n=202, A=(202-2)3-(202-1)3=120601.


  • Problem by Harsh Shrivastava
  • Solution by David Vreken
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