In the game 2048, what is the biggest theoretical tile?

Question

In the game 2048 what is the biggest tile that can be achieved, assuming a player playing optimally and tile spawning at the most optimal place?

Naively I would say that the biggest achievable tile is 65536 * 2 = 131072 because it seems that the best possible board is the following:

  4     4     8     16
 256   128    64    32
 512   1024  2048  4096
65536 32768 16384  8192

But I'm not sure if

1. it's correct
2. how to prove that my intuition is indeed correct.

(sorry if I should have asked on gaming.stackexchange, but this is more of a CS question than a gaming one afaict)

Answer 1

you haven't finished yet with the board you propose: you can slide to the right, going all the way down and obtaining 131072. So your analysis was correct although you missed a spot:

This will be your final board:

  4     8     16    32
 512   256   128    64
 1024  2048  4096  8192
131072 65536 32768 16384