It's clear that if the result is true for $668$ boys, then it is true for any smaller number of boys (just switch some girls into boys till you have $668$ boys, then the result is true and by switching back the boys into girls it stays true). We'll prove the result for $2n+1$ girls and $n$ boys. This can be done by induction. Suppose the result is true for $1, 2, ..., n-1$. Now take $2n+1$ girls and $n$ boys. Take any girl $G$ which has a boy for a neighbor (it's clear there is one), and then go round the table till you find another $G$, so you'll have something like $GBBB...BBBG$. Now take away $G, B, G$ where $B$ is any of the boys in that interval. Among the $2n-1$ girls and $n-1$ boys left there's a girl in a strong position by hypothesis; it's easy to see she's still strong when we put back $G, B, G$.