Orchard Planting Problem/Historical Note

Historical Note on the Orchard Planting Problem

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