Answer
The twin that always lies is talking.
Work Step by Step
Let the first lucky number be x, the second y.
Symbolically, what the twin is saying is:
"There are two numbers, x and y, such that
$\left\{\begin{array}{l}
3x+6y=12\\
x+2y=5
\end{array}\right.\qquad$"
If we subtract $3\times($Eq.2$)$ from Eq.1,
$(3-3)x+(6-6)y=12-15$
$0=3$
which is false.
There are no such two lucky numbers.
The twin that always lies is talking.
