Orchard Planting Problem/Classic Form/12

Classic Problem

$12$ trees are to be planted in an orchard so as to make the number of rows of $3$ trees the largest number possible.

That is:

$12$ points are to be configured in the plane so that the number of straight lines that can be drawn through exactly $3$ of these points is maximised.

Solution

The number of rows is $19$.

$3$ of the points are at infinity.

One of the $19$ rows is also at infinity, and passes through each of those $3$ points.

$\blacksquare$

Sources

• 1995: N.J.A. Sloane and Simon Plouffe: The Encyclopedia of Integer Sequences

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $19$