Police Squared

Algebra Level 2

An officer wishing to arrange his men in a solid square found by his first arrangement that he had 39 men left over. He then started increasing the number of men on a side by one, but found that 50 additional men would be needed to complete a new square.

How many men did the officer originally have?

2,079 1,975 200 10,500

Noah Clausen by Noah Clausen , 16, USA

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1 solution

Jordan Cahn
Mar 28, 2019

Let n n be the number of men along one side of the square in the first arrangement. Then n 2 + 39 = ( n + 1 ) 2 − 50 n 2 + 39 = n 2 + 2 n − 49 2 n = 88 n = 44 \begin{aligned} n^2+39 &= (n+1)^2-50 \\ n^2+39 &= n^2 + 2n - 49 \\ 2n &= 88 \\ n &= 44 \end{aligned} Therefore, the officer has 4 4 2 + 39 = 1 , 975 44^2+39 = \boxed{1{,}975} men.

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