31st Junior Turkish Mathematical Olympiad

Answer: $n=11k$, where $k$ is a positive integer.

Suppose that in $m$ identically numbered pairs of red and white boxes each red box contains more balls than the white box. Then the difference between the total number of balls in red boxes and the total number of balls in white boxes is $6m-16(n-m)$. On the other hand, since the total number of balls in red boxes and the total number of balls in white boxes increases by $1$ at each step, the difference should be equal to $0$. Therefore, $11m=8n$. Thus, $n$ should be a multiple of $11$.

Firstly, we give an example for $n=11$. Let each pair $(i,j)$ of red box numbered $i$ and white box numbered $j$, $i=1,2,\dots ,8$ and $j=9,10,11$ be chosen exactly two times. Then after these $48$ steps the problem conditions will be satisfied.

If $n=11k$, we partition boxes into groups numbered $1,2,\dots ,k$ where group number $s$ contains $11$ red and $11$ white boxes numbered $11s-10,11s-9,\dots ,11s$, change box numbers to their values in $(\mathrm{mod}11)$ and apply $48$ steps defined above to each of these groups.