Comment by mightybyte

Finally won with a score of 20548. At first I was using a strategy of getting the biggest number into a corner. This follows a pattern of mostly up-right-up-right-... with an occasional left or down thrown in to get unstuck. This approach can pretty reliably get you to 512 in the corner without much trouble. But eventually this method saturates the available "storage" space with an inefficient pattern of tiles up to the diagonal. Then you have to start changing things up and continuing becomes tricky.

Then I tried the approach mentioned here of left-down-right-down-left... in a very mechanical fashion. I believe that this is very likely the optimal mindless strategy. It works amazingly well. As long as you never hit up, it fills things very efficiently and the biggest numbers percolate down towards the middle of the bottom row.

The way I finally won was to take this second approach and augment it with some periods of trying to get the biggest number in the corner. Start with ldrd until you get some 64s or 128s, then slow down and play catch-up with a corner. Then go back to ldrd. Occasionally stopping at strategic places to consolidate things also seems to help quite a bit.

All in all, a very fun game with some interesting properties.