The closed and open organ pipes have same length. When they are vibrating simultaneously in first overtone, they produce four beats. The length of open pipe is made half and that of the closed pipe is made two times the original. Now the number of beats produced if the two pipes are vibrating in their fundamental modes simultaneously is

Beats with modified organ pipes
Let $L$ be the original length of both pipes and $v$ the speed of sound. The first overtone of an open pipe is $\frac{v}{L}$, while for a closed pipe it is $\frac{3v}{4L}$.

The initial beat frequency gives $\left|\frac{v}{L} - \frac{3v}{4L}\right| = 4$, which simplifies to $\frac{v}{4L} = 4$ and thus $\frac{v}{L} = 16$.

After modification, the open pipe length becomes $\frac{L}{2}$ and the closed pipe becomes $2L$. The fundamental frequency of the open pipe is then $\frac{v}{L}$, while for the closed pipe it is $\frac{v}{8L}$.

The new beat frequency is $\left|\frac{v}{L} - \frac{v}{8L}\right| = \frac{7v}{8L}$. Substituting $\frac{v}{L} = 16$ yields $\frac{7}{8} \cdot 16 = 14$.

$\boxed{14}$