Comment by ronaldx

This is actually a good method: by using this method (avoiding up and favouring down), you tend to cluster 2s and 4s towards the top while larger numbers percolate downwards (and towards the middle). If you encourage this natural tendency, you can do even better.

When you're nearly stuck, you can often pick left or right so that a new tile can access both 2 and 4 and save yourself.

I got comfortably to 1024 with one go of this method, scoring 16744.