Henry Ernest Dudeney/Puzzles and Curious Problems/128 - Forming Squares/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $128$

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?

$1975$ men.

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.

$\blacksquare$