Has anyone actually managed to complete this? It's the Bulletin Board level in the computer chapter, second or third level I think. I've tried plenty, but at this point I'm fairly convinced that this level boils down to pure luck. Either that or I'm getting some really weird angles from where I'm shooting. Funny thing is, I got 12 just fine, so I'm not quite sure what I'm doing wrong.

I get to the first "base" just fine, I drag out a single cube to make a triangle on "top" (to the right of) of the base, and try to shoot through the cogs from there. Every time, I wait for the goo balls to get to the tip of the triangle, and I always aim for the inside edge of the right hand cog. Most of the time they get through, but quite often they'll hit either the left or right cog because there's a random component to the aiming. What am I supposed to do, just keep playing it like a lottery, or am I totally off?

Hmm, you have a point. I just tried it, and it took considerably more effort than I expected.
But I do have 14 balls now.

You could choose not to build the triangle on first base, that gives you an error margin of 1.
Shooting from the bottom right corner of first base worked best for me.
Good luck :)

Finally got it! Took a few tries, but firing from the base and getting 14 seems to work. The way I did it was to fire from the centre of the base, and aim with the cursor directly in between the two rightmost squares on the "landing platform" (the long left-right section). Worked the first time I tried it and I don't care to try it again, so I can't confirm it's 100%.