If your first high number is in the middle of the board, don't panic, just focus on the top half of the board
Say your first 12 or 24 is in the first column, but the second or third row. Don't worry about it, just focus on keeping a healthy board and combining numbers primarily on that side. I don't generally worry unless I've made a 192 and it's not at the top of the board.
Fill up your leftmost column with 4 white tiles (preferably in descending order) as quickly as possible
Once you've gotten your biggest tile positioned in the corner, now it's time to think about combining bigger tiles. I like to combine upwards, with my smallest numbers in the bottom left corner (say, a 3), middle numbers in the 2nd and 3rd row (6 and 12), and the biggest number in the top left.
I'll also try and stack a similar scenario in my second column, though I try to ensure that my 2nd-4th columns only ever have a max of three tiles per row so I have full up and down vertical movement without moving my corner piece. (Because we've put four tiles in that first column, it shouldn't ever move except when combining tiles upward.)
WARNING: When you're combining tiles upward in your leftmost column, pay close attention to the "Next" tile. I've gotten 1s and 2s stuck in this column, surrounded by white tiles, because I wasn't paying attention to what would pop up when I combined two 12s.
Middle and Endgame strategies
Once you've set up your board, you can keep trucking for awhile with the basic corner, but here are some more advanced suggestions if you get into trouble.
Work vertically for big numbers, horizontally for small ones
Once you get to 384 tiles and above, it becomes a little cumbersome to just combine big numbers in the leftmost column. To combat that, I try to combine my biggest numbers (48+) vertically in columns 1 and 2, and my smaller numbers (3-24) horizontally in columns 3/4 and rows 1-4.
If you can't work bottom-up, work top-down
When you get to higher and higher numbers, you'll often have to work in both vertical directions to get enough combinations. That's totally fine, just make sure you're lining up your combos properly and not staggering them. For instance, 12 / 24 / 12 / 24 is a bad column; 12 / 24 / 24 / 12 is a better column; and 24 / 24 / 24 / 12 is an ideal column.
Build multiple combinations
One of the reasons I like working on my middle-to-big numbers in the first two vertical columns is that I can build multiple combos that can then be made vertically (for the first column) or horizontally (second column into the first column). The more stacks you can build, the better your board clear will be when you finally have the right numbers on the board to make a double of your biggest tile.
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