Cultivating New Solutions for the Orchard-Planting Problem
Some trees are planted in an orchard. What is the maximum possible number of distinct lines of three trees? In his 1821 book Rational Amusement for Winter Evenings, J. Jackson put it this way:
Fain would I plant a grove in rows
But how must I its form compose
With three trees in each row;
To have as many rows as trees;
Now tell me, artists, if you please:
'Tis all I want to know.
Those familiar with tic-tac-toe, three-in-a-row might wonder how difficult this problem could be, but it’s actually been looked at by some of the most prominent mathematicians of the past and present. This essay presents many new solutions that haven’t been seen before, shows a general method for finding more solutions and points out where current best solutions are improvable.