Sugar Cane (1 Viewer)

[Sy123]

In order to grow 1 square unit of sugar cane, this 1 square unit of sugar cane must be next to 1 square unit of water.
A farm is established with the goal of finding the most efficient layout of these water and sugar cane units.

The efficiency of the farm is measured as a percentage of land utilised for growing.

i.e. a 60% efficient farm is a farm where 60% of tiles/units are sugar cane units.

Find the maximum efficiency for this sugar cane farm, and if possible draw a layout of this farm.



e.g. of the layout and efficiency

I have read figures of as high as 80% efficiency
However I want to see if someone can think of a mathematical explanation and possibly a higher efficiency.

 

[Obvious]

Sugar cane (o)
Water (w)
Nothing (x)

x o x
o w o
x o x

Tessellate that shape. It's impossible to go over eighty percent efficiency.

 

[Sy123]

Sugar cane (o)
Water (w)
Nothing (x)

x o x
o w o
x o x

Tessellate that shape. It's impossible to go over eighty percent efficiency.

That has only 44% efficiency, for each 9 blocks only 4 have sugar cane.

 

[Obvious]

That has only 44% efficiency, for each 9 blocks only 4 have sugar cane.

The x's are only for formatting, don't pay attention to them.

 

[Girls]

If left, right, up and down are the only directions allowed, then
__x
x o x
__x

is the only possible solution with 80% efficiency as someone has already said.

if diagonals etc. are permitted, then a circle with water in the middle and cane around the outside makes efficiency practically 100%.

 

[Sy123]

The x's are only for formatting, don't pay attention to them.

Ah yes I see what you mean by tessellate now, thank you that makes a lot of sense.