Henry Ernest Dudeney/Puzzles and Curious Problems/155 - The Orchard Problem/Solution

by : $155$

 * The Orchard Problem
 * A market gardener was planting a new orchard.
 * The young trees were arranged in rows so as to form a square,
 * and it was found that there were $146$ trees unplanted.
 * To enlarge the square by an extra row each way he had to buy $31$ additional trees.


 * How many trees were there in the orchard when it was finished?

Solution

 * $7921$ trees.

Proof
Let $n$ be the number of trees on each side of the square of already planted trees.

Then there are $n^2$ trees in that square.

From the Odd Number Theorem, to make a square of $\paren {n + 1}^2$ trees we need $2 n + 1$ more trees.

We are told we need $146 + 31 = 177$ more trees.

Then we have that:
 * $177 = 2 \times 88 + 1$

and so:
 * $n = 88$

and the total number of trees in the new orchard is $\paren {88 + 1}^2 = 89^2 = 7921$.