Copyright © G. P. Jelliss 2003
Problems of the planting of orchards have long been popular (According to H. E. Dudeney, John Jackson's Rational Amusements for Winter Evenings London 1821 had ten examples). The discussion of the basics of the subject which follows was first published as a series of Puzzle Questions and Answers in the Games and Puzzles Journal issues 13 - 16.
One to Four Trees
With one or two trees there is only one pattern of planting, but with three trees we can put them in a row (a line of 3, which we denote by 3/3) or in a triangle (lines of 2, which we denote by 3/2). With four trees there are three patterns when considered numerically (4/4, 4/3 and 4/2), but the last of these can exist in two topologically distinct forms: one in which one point is within the triangle formed by the other three, and one in which each tree is outside the triangle formed by the other three. If we try to deform the triangle form gradually into the quadrilateral form the central point has to cross one of the sides of the triangle—thus forming the 4/3 pattern—or else passes through a vertex, forming a simple triangle.
Five Trees
This leads to our first question in this series: How many different types of plantation are possible with five trees? There are two answers of course, (a) according to the number of lines and the numbers of trees in the lines, and (b) taking account of the topological considerations.
The answer is that there are 5 types of 5-tree plantations, in terms of lines and numbers of trees per line, but 12 types topologically different, as exemplified here. The seven 5/3 cases are subclassified as 5/3(2) and 5/3(1) according to the number of 3-unit lines.
Those who wish may pursue this count further to 6, 7 or more trees, but it becomes rapidly more difficult and less recreational, so we next begin to confine our attention to plantations with more regular properties.
Six Trees
How many different plantation patterns are possible of 6 trees with every tree in a line of at least three?
Answer, there are 14 cases. In the 3-line case, three trees occur at the vertices of a triangle and the third tree on each line can occur between the vertices or externally. The 4-line case, 6/4, is the first in which each tree belongs to two lines of three.
Seven Trees
How many different plantation patterns are possible of 7 trees with every tree in at least two lines of three or more?
The impossibility of forming a plantation of 7 trees in 7 lines of three is known in projective geometry as Fano's Axiom. The maximum for 7 trees is 6 lines of 3. If we label the seven points in the first diagram ABC (outer triangle) DEF (inner triangle) G (centre) and define seven lines to be ADB, BEC, CFA, AGE, BGF, CGD and DEF we get a 'finite projective geometry' in which Fano's axiom is contradicted, but many other projective properties remain true.
There are 6 ways of planting 7 trees in 5 lines of 3, with every tree in at least two lines.
No arrangement incorporating a line of 4 is possible since the other 3 points form a triangle whose sides can only pass through 3 of the 4 points on the other line, so one of the 4 points on this line has no other line passing through it.
Eight Trees
How many different plantation patterns are possible of 8 trees with every tree in at least two lines of three or more? The results are not unconnected to the 7 case.
Arrangements with a line of 4 are become possible with 8 trees. The 6 plantations with 2 lines of 4 correspond to the 6 plantations of type 7/3(5) in the diagram above.
To be continued ...