Henry Ernest Dudeney/Puzzles and Curious Problems/128 - Forming Squares/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $128$
- Forming Squares
- An officer arranged his men in a solid square, and had $39$ men left over.
- He then started increasing the number of men on a side by one, but found that $50$ new men would be needed to complete the new square.
- Can you tell me how many men the officer had?
Solution
- $1975$ men.
Proof
Let the original square have $n$ men on each side.
Then we have:
\(\ds \paren {n + 1}^2 - n^2\) | \(=\) | \(\ds 39 + 50\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 89\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 n + 1\) | \(=\) | \(\ds 89\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds n\) | \(=\) | \(\ds 44\) |
So the officer had $44^2 + 39 = 1975$ men.
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $128$. -- Forming Squares
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $201$. Forming Squares