## Christmas Trees

#### Problem

When Christmas trees are planted they should stand at least 2 metres away from one another whilst growing.

What is the maximum number of trees that can be planted in one square kilometre?

#### Solution

The best arrangement is a triangular matrix:

By applying the Pythagorean Theorem to the triangle, $2^2 = d^2 + 1^2$, hence the smallest allowable distance between rows, $d=\sqrt{3}$ metres.

Assuming that we can plant a tree in the top left corner of the square then it is possible to plant 501 trees across the top row. The next row will contain 500 trees, then the pattern repeats: 501, 500, 501, ... .

As $\dfrac{1000}{\sqrt{3}} = 577.35...$ we will be able to plant 577 complete rows. Therefore we can plant $\dfrac{576}{2} = 288$ blocks of $501 + 500 = 1001$ trees and a final row of 501 trees. That is, a maximum of $288 \times 1001 + 501 = 288789$ trees can be planted in a square kilometre.

An alternative approach can be used to closely approximate this answer. It can be seen that each tree occupies the same area as two equilateral triangles.

That is, each tree occupies $2\sqrt{3} m^2$. As $1 km^2 = 1000 \times 1000 = 1000000 m^2$, the maximum number of trees that can be planted is approximately $\dfrac{1000000}{2\sqrt{3}} \approx 288673$ trees.

If the trees were planted in a square matrix, what is the maximum number of trees that could be planted in 1 km^{2}?

Can you explain why a triangular arrangement is the most efficient method of planting trees?