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

by : $128$

 * 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:

So the officer had $44^2 + 39 = 1975$ men.