Problem - 2809
A regiment had 48 soldiers but only half of them had uniforms. During inspection, they form a 6 × 8 rectangle, and it was just enough to conceal in its interior everyone without a uniform. Later, some new soldiers joined the regiment, but again only half of them had uniforms. During the next inspection, they used a different rectangular formation, again just enough to conceal in its interior everyone without a uniform. How many new soldiers joined the regiment?
Let the dimensions of the rectangle be $x$ by $y$, with $x \ge y$. Then the number of soldiers on the outside is $2x + 2y − 4$ while the number of those in the interior is $(x − 2)(y − 2)$. From $xy − 2x − 2y + 4 = 2x + 2y − 4$, we have $(x − 4)(y − 4) = xy − 4x − 4y + 16 = 8$. If $x − 4 = 2$ and $y − 4 = 4$, we obtain the original $6 \times 8$ rectangle. If $x − 4 = 1$ and $y − 4 = 8$, we obtain the new $5\times 12$ rectangle. Thus the number of new soldiers
is $5\times 12 − 6 \times 8 = \boxed{12}$.