game - What is the largest tile possible in 2048?

I can get to 4096 max. There are videos on youtube of guys making it to 8192. What is the theoretical maximum limit on this?

PS: if you haven't played this, here is the link to addiction.

Answer

In the original game, sometimes the new tile that pops up is a 4. Let's ignore that possibility for the sake of simplicity.

We've 16 squares in total and considering the new tile popped up at the position we chose, to make it to 4096, we need minimum these tiles:

2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2

These can be combined from right to left to get to 4096. That's a total of 12 tiles. Since that's less than 16 (max number of tiles that can fit), 4096 is possible. Following along:

• 8192: 4096, 2048, 1024, ..., 4, 2, 2 Total 13


• 16384: 8192, 4096, 2048, ..., 4, 2, 2 Total 14


• 32768: 16384, 8192, 4096, ..., 4, 2, 2 Total 15


• 65536: 32768, 16384, 8192, 4096, ..., 4, 2, 2 Total 16

Next is 131072 which will require at least 17 tiles which can't fit on the board.

So, the largest possible tile is 65536.

EDIT: Since the game pops up some 4's at times, 131072 is still theoretically possible if the last tile that pops up is a 4 making the board look like this:

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

that's 16 tiles in total.