Naturally, a big thumbs up to Daftmouse. I used his approach except that I used a unique color scheme for each column and numbered each column down from 1 to 4. I also made sure that the topmost left corner was very unique so I could identify it immediately when starting to solve.
As Daftmouse said, a lot depends on luck of the draw. I recommend making that part of your strategy. Instead of trying to solve each problem on its own merits, come up with a technique of how you want to solve the puzzle. For example, I wanted to solve the puzzle working each row top to bottom starting with the leftmost row and moving right. That makes sense because the empty square is the bottom right square. Then, you just keep resetting the board until luck aligns with your approach.
Because of my approach, it was important to identify the first piece--the top left corner. That's why I made it unique so it would stand out. It's important to realize when pieces will warp to the other side. This occurs when the border shimmers like a rainbow. At the beginning of the game the left and right sides of the puzzle will shimmer meaning you can warp the rows horizontally. After a short while, the tops and bottoms will warp meaning you can warp the columns vertically.
A good luck based setup is when you can quickly establish the first column by either relying on one or more pieces to be in place to begin with or for the pieces to be in different rows in the right order. This doesn't happen as rarely as you might expect. If you get a bad setup just touch the folder icon on the bottom left corner of the screen and choose the 'play again' option. That way you can quickly dispense with setups not to your liking.
The point is to use a strategy you're comfortable with. With me, it was to use horizontal warping to get the first column sequenced correctly and then, when warping shifts from horizontal to vertical, to align it correctly. For example, if you get 3 on the first row, 4 on the second row, 1 on the third row, and 2 on the fourth row; they are sequenced correctly but you have to warp two spaces vertically to get 1 on the first row. All the other pieces will then align correctly.
With a good head start, you can work the next two columns a little more deliberately. With me, the second column was a unique color and the pieces were numbered 1 through four top to bottom. Ideally, your empty space is in one of the two middle columns. An easy way to place pieces correctly in sequence is to put the piece you're trying to position in the third column. Then, position the empty space where you want the piece to go in the second column. You then use the vertical warp on the third column to get the piece on the correct row and slide it to the left into its slot.
This is easier than it sounds especially because we are counting on luck. As before, if we don't like the setup, we'll exit by touching the bottom left icon and selecting 'play over'.
If you're able to take care of the first two columns with a mixture of luck and skill, the last two columns are relatively simple because the multitude of choices have narrowed considerably. You'll be able to spot immediately whether the pieces are arranged in your favor or not. If the former, go after your trophy. If the latter, try again.
If you rely on a luck based approach, the trophy requires more patience than skill. Don't play the board. Wait for the board to align itself to your preferred approach. Under normal circumstances, a time of 50 seconds is unrealistic. Even with luck, such a time is unrealistic. However, if you have a predetermined approach and you wait for the stars or pieces to align, beating 50 seconds is readily attainable. Occasionally, you'll hit the jackpot and will solve it in less than 30 seconds.