Edit November 2017 to address questions of if this is possible without hacks:
A computer processes at a few gigahertz. That means that, even if everything else about this game was perfect and you had master ninja robot hands, you cannot get to 5 billion moves per second.
If you add up all the tiles on the board, before and after moving, the increase in that sum is equal to the tile it spawned. The game spawns a 4 a tenth of the time, a 2 the rest of the time. Thus on average, you add 2.2 points to the board each move. You need to add 9 quadrillion points to the board before you win, so you need to move about 4 quadrillion times.
Making 4 quadrillion moves at the speed of the fastest computer processor takes about 10 days. So, if the game was re-coded to be massively parallel and resolve each input in one process cycle, and you could feed it optimal moves at the same speed, that’s how long it would take.
The highest Actions Per Minute on record from a human is around 1500, so let’s be really generous and say that 4000 APM, sustained throughout the entirety of this game, is possible. That means a person would require a trillion minutes to complete this game, or roughly 2 million years. That’s far beyond a human lifespan, so, no.
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