#### Answer

The liar twin is talking.

#### Work Step by Step

Let us assume x and y to be the first and second lucky number:
So, according to the first condition:
$3x+6y=12$
$x+2y=4$ …… (I)
Then, from the second condition:
$x+2y=5$ …… (II)
Then, from equation (I), $x+2y=4$
But from the second equation, $x+2y=5$
So, this twin is a liar.
Thus, the speaking twin is a liar.