Orchard Planting Problem

Classic Problem

A given number $n$ of trees are to be planted in an orchard so as to make the number of rows of $k$ trees the largest number possible.

That is:

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

Classic Form

A given number $n$ of 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:

$n$ 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

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$12$ trees, $3$ in a row

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.

$\blacksquare$

Also known as

The orchard planting problem can be seen hyphenated: orchard-planting problem.

It is also known as the orchard problem or tree-planting problem.

Note, however, that there exists another, completely different, orchard problem: the Orchard Visibility Problem, with which this one may be confused.

Also see

• Orchard Visibility Problem

Historical Note

The Orchard Planting Problem appears to have first been set by John Jackson in his $1821$ collection Rational Amusement for Winter Evenings, where he worded it poetically as:

1. Your aid I want, nine trees to plant

In rows just half a score;

And let there be in each row three.

Solve this: I ask no more.

2. 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.

3. Ingenious artists, if you please

To plant a grove, pray show,

In twenty-three rows with fifteen trees

And three in every row.

4. It is required to plant $17$ trees in $24$ rows,

and to have $3$ trees in every row.

5. Ingenious artists, pray dispose

Twenty-four trees in twenty-four rows.

Three trees I'd have in every row;

A pond in midst I'd have also.

A plan thereof I fain would have,

And therefore your assistance crave.

6. Fam'd arborists, display your power

And show how I may plant a bower

With verdant fir and yew:

Twelve trees of each I would dispose,

In only eight-and-twenty rows;

Four trees in each to view.

7. Plant $27$ trees in $15$ rows, $5$ to a row.

8. Ingenious artists, if you please,

Now plant me five-and-twenty trees,

In twenty-eight rows, nor less, nor more;

In some rows five, some three, some four.

9. It is required to plant $90$ trees in $10$ rows, with $10$ trees in each row; each tree equidistant from the other, also each row equidistant from a pond in the centre.

10. A gentleman has a quadrangular irregular piece of ground, in which he is desirous of planting a quincunx, in such a manner, that all the rows of trees, whether transversal or diagonal, shall all be right lines. How must this be done?

Note: A real quincunx is a plantation of trees disposed in a square, consisting of five trees, one at each corner, and the fifth in the middle; but in the present case, the trees are to be disposed in a quadrangle, one at each corner (as in the square), and the fifth at the point of intersection of the two diagonals.

Sources

• 1821: John Jackson: Rational Amusement for Winter Evenings

• Weisstein, Eric W. "Orchard-Planting Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Orchard-PlantingProblem.html