According to the question, the ring is passed clockwise starting from child 1, then child 2, then child 4, then child 7, and so on.

Therefore, the series is formed by adding $+1$, $+2$, $+3$, $+4$, $+5$, $+6$, $+7$, $\ldots \ldots \ldots \ldots$and so on.

So, the series is: 1 2 4 7 11 16 22 29 37 6 16 27 39 12 26 1

So, after 15 changes the ring will again be in the hands of child 1.

Hence, option (B) is correct.