Henry Ernest Dudeney/Puzzles and Curious Problems/155 - The Orchard Problem/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $155$
- 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 $\paren {88 + 1}^2 = 89^2 = 7921$.
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $155$. -- The Orchard Problem
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $228$. The Orchard Problem