Delay your combos
When you're down to one or two spaces and you have combinations to make, delay them as long as possible. (Generally, until a move will break them up.) It's likely you'll get more combinations from waiting than making the move despite it being bad for the "Next" tile.
When you're down to one space left, what move will get you the most additional moves?
Like the previous tip on delaying your combinations, use the preview slide to figure out what move will get you the most additional moves to your game. I've been down to one move before and been able to get out of it by lining up one tile that subsequently allowed me to combine numbers all the way to the first column.
For bonus points: Rules are made to be broken
When you're down to your last move, if moving the corner might get you a few more moves, then you move the corner. You may mess up potential multiple combinations, but 10 more moves is better than one.
The other strategies
No guide would be complete without a hat tip to other successful strategies, of which there are several.
The Wall:TouchArcade's forum has kind of gone a little stir-crazy trying to beat Threes, like attempting to crack its card draw algorithm. One TouchArcade member grabbed the 6144 card by using the walls and praying that the random "+" white cards would work in her favor to instantly combine high-numbered cards rather than relying on traditional stacking or cornering.
The Center: Don't like the walls or the corners? You might prefer the center float strategy, which I've seen mentioned over the past few weeks in my Twitter feed. I used to try and keep my biggest tile in the center, but found it didn't work so well for me past 384; others may have better luck, however.
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