My brain and coconuts
In Surat Thani, Thailand, last week I met a coconut farmer with ten acres of land. He told me that each palm tree needs to be at least five meters from every other palm tree to grow as fast as possible. After only three years it will start to bear coconuts.
Later I was wondering how many palm trees he could have on his land. Obviously it would depend on the shape of the land, so I looked for the minimum and maximum number of trees.
What do you think?
October 28th, 2008 at 6:16 pm
The (unattainable) maximum comes from an arbitrarily long plantation, narrow enough that the area is 10 acres, with trees planted along it at 5-metre intervals.
Minimum? I supposed that the plantation was rectangular, and that the farmer planted the trees in rows parallel to whichever edge of the plantation let more trees fit, in a “triangular” arrangement (i.e., where, except for some trees near the edges of the plantation, each tree, together with its two nearest neighbours in a neighbouring row, are the corners of an equilateral triangle). Then if the plantation is 199m by 203.36m, the farmer plants 46 rows, parallel to the long sides, with 42 trees per row, a total of 1932. Alternatively, if the plantation is 167m by 242.33m, the farmer plants 56 rows, parallel to the short sides, alternating 35 and 34 trees per row, a total of 1932.
Mind you, these solutions could be cooked (non-rectangular plantations in which 1932 trees don’t fit) or busted (alternative ways of planting 1933 or more trees in the above shapes).
October 28th, 2008 at 6:55 pm
Here’s one way to cook them. The trees are planted in equally spaced rows and are equally spaced in each row, as before, but the spacing in each row might be greater than 5 m, so the spacing between rows might be less than the 5*sqrt(3)/2 m which is necessary if the spacing in each row is 5 m. This might allow more rows and thus more trees.
Then if the plantation is 159m by 254.52m, the farmer plants 59 rows, parallel to the short sides, with 32 trees per row, a total of 1888. Alternatively, if the plantation is 147m by 275.3m, the farmer plants 64 rows, parallel to the short sides, alternating 30 and 29 trees per row, a total of 1888.