Peggle

The easiest way to do this is by doing the challenge that gives you 15 balls to beat 3 levels (thanks to the commenters, this is confirmed to be Challenge 12-3 of the Nights DLC). You can use all 15 balls on the first level to earn this achievement. No free balls necessary, but those are still helpful in case the purple peg is inaccessible on a shot.

This is actually a lot easier than it sounds in the description - it took me only 2 or 3 goes to do it once I found the best screen to use.

I found the best screen to be Das Bucket - Screen 1.4, the one often used to boost the "Win a Peggle Party by 100,000 points" achievement.

The first thing to remember is that you will need to earn 3 extra balls at the minimum to ensure you have 13 chances to hit the pink peg. Das Bucket is great in that you can get extra balls pretty easily either through earning 25K or 75K in a shot, or by catching the ball in the bucket.

On Das Bucket, the pink peg or tile is almost always accessible on the first shot, which means you can hit it and at the same time clear away a lot of the pegs which form the "side" of the bucket, giving you an extra ball or two and points for a slide.

If the peg isn't accessible with your first shot, restart the challenge.

From there, just keep hitting the pink peg. It becomes easier as there are fewer and fewer pegs to hit - also, when you hit a green peg, Marina's skill makes the pink peg very easy to hit - and will often earn you another ball for scoring over 25K in a turn.

One other thing to remember is this - don't run out of blue pegs. A couple of times when I tried to do this, I had hit the pink peg 10 or 11 times, but had actually run out of blue pegs for the pink peg to "turn into", meaning I'd only have orange pegs left and I'd be stuck.

The keys are to try and earn those extra balls by scoring points off catching the ball in the bucket, and to use Marina's skill well to not only electrocute a lot of balls and get lots of points, but to hit the pink ball as well.

Once you've got enough balls to ensure you hit the pink peg 13 times - ie: you've hit it 7 times and have 6 balls still left (7+6=13) just pick off the pink peg only each turn.

Hope this helps.