To solve this question, the user needs to understand basic arithmetic and fractions.
Firstly, if we lose half of the 12 balls, we will be left with 6 balls (12/2=6).
Secondly, if we retrieve 2/3 of the lost balls, we will get:
(2/3) x (1/2) x 12 = 4.
So, we retrieve 4 balls.
Therefore, the total number of balls we have now is the sum of the balls we had initially (12) minus the balls we lost (6) plus the balls we retrieved (4), which equals to 12 - 6 + 4 = 10.