Let $x$ be the number of girls and $y$ the number of boys. Then $y = 3x$ and there are $4x$ children altogether.

Let $a$ be the number of boy-girl pairs. Then the number of same-sex pairs is equal to $2a$ and there are $3a$ pairs altogether. The number of pairs equals the number of all children. Indeed, every child belongs to two pairs, so there are $2 \cdot 4x$ pairs, but every pair was counted twice, which gives us $\frac{2 \cdot 4x}{2} = 4x$ distinctive pairs. Hence, $3a = 4x$. The smallest $x$ for which this equation can hold is

$$\begin{array}{cccc} & & F & D \\ & & F & F \\ F & & & D \\ F & & & D \\ F & & & F \\ F & F & F & F \end{array}$$

$x = 3$. We conclude there was at least 12 children at the table. If they sat down as shown in the figure (D represents girls, F represents boys), then the number of same-sex pairs was twice the number of boy-girl pairs.